Effects of High-Order Nonlinear Wave-Wave Interactions on Gravity

Eﬀects of High-Order Nonlinear Wave-Wave Interactions on Gravity Waves

Nobuhito Mori1 and Takashi Yasuda2

1 Department of Hydraulics, Abiko Research Laboratory, Central Research Institute of Electric Power Industry(CRIEPI), 1646 Abiko, Abiko, Chiba 2701194, JAPAN, [email protected]. 2 Graduate Course of Environ. and Renew. Energy System, Gifu University, Yanagido 1-1, Gifu, JAPAN, [email protected].

Abstract. Numerical simulations of gravity waves with high-order nonlinearities in two and three dimensional domain are performed by using pseudo spectral method. High-order nonlinearities more than third- order excite apparently chaotic evolutions of the Fourier energy in deep- water random waves. The high-order nonlinearities increase kurtosis, wave height distribution and Hmax/H1/3 in deep-water and decrease these wave statistics in shallow water. They, moreover, can generate a single extreme high wave with an outstanding crest height in deep-water. The high-order nonlinearities more than third-order can be regarded as one of a reason of a cause of a freak wave in deep-water.

1 Introduction

Since the discovery of the Benjamin-Feir instability in the Stokes wave train, much attention has been paid attention to the behavior of nonlinear deep-water waves. Last two decades, the instability of the gravity waves have been studied many researchers by the nonlinear Schrödinger type equations[1][2][3], mode- coupling equations[4], pseudo-spectral methods[?] and experiments[5]. However, the most of them were concentrated on amplitude modulations of the Stokes wave for the purpose of scientific interests. The rest of studies were related energy transfer of random waves for the purpose of prediction of ocean wave spectra[6][7]. Little is known about the high-order resonant interaction effects on random wave statistics in deep and shallow-water. Alber mathematically demonstrated that randomness of waves makes wave trains to stabilize[8] and others stated that the instability is confined within an initially unstable range and become weak if the spectral bandwidth becomes broad[9]. However, it is not clear that how stability and instability of the gravity wave are connected between the Stokes and random waves having broad band spectra. It is found that relatively broad banded spectrum waves can transfer the Fourier mode energy in deep-water[?][?]. On the other hand, a freak wave becomes an important topic recently and is sometimes featured by a single and steep crest giving severe damage to offshore structures and ships. There is no doubt on the occurrence of a freak wave from many reports[?] and the mechanisms and detailed statistical properties of the freak wave are getting clear[?][10]. The state of the art on the freak wave was summarized at NATO Advanced Research Workshop the last decade. It was concluded that both of nonlinearity and directionality effects are primary possible causes of the freak wave[11]. Experimental studies demonstrates that the freak wave like wave can be generated in two-dimensional wave flume without current, refraction and diffraction[12]. Numerical studies also indicates that the freak wave having a single and steep crest can be generated by the third-order nonlinear interaction in deep-water[?]. It is, however, not clear statistical properties, occurrence probabilities and effects of spectrum shape and water depth for the instability generated freak wave. The purpose of this study is to investigate influence of spectrum band width and water depth on the stability of random waves solving highly nonlinear equations of a potential flow by pseudo spectrum method. On the basis of the numerical results, it is evaluated importance of the high-order nonlinearities in comparison with the second-order solution. Moreover, it is demonstrate stability of the Stokes wave in three dimensional domain.

2 Numerical Method

2.1 Governing Equations

Two type of nonlinear equations for gravity waves are numerically solved in this study. One is high-order nonlinear equations which can take into the consideration of the nonlinear interactions more than third-order are completely[13] and another one is the second-order approximated equations which excluded nonlinear terms higher than the 3rd order[?]. It is assumed a periodic boundary condition and is assigned to spatial co- ordinates (x, z); the origin is located at the mean water level, x = (x, y) the horizontal axis and z the upward vertical axis. Kinematic and Dynamic boundary conditions on the free surface are rewritten into the evolution equations as a function of the free surface proﬁle η(x, t) and the velocity potential on the surface φs(x, t)=φ(x, η, t)[14];

s ηt + ∇xφ ηx − (1 + ∇xη·∇xη)φz = 0, (z = η) (1) 1 1 φ + gη + ∇ φs ·∇ φs − (1 + ∇ η·∇ η)φ2 = 0, (z = η) (2) t 2 x x 2 x x z where ∇x=(∂/∂x, ∂/∂y), the subscript t denotes the partial diﬀerentiation with t, φz the vertical gradient of the velocity potential φ, t the time and g the acceleration due to the gravity. Dommermuth & Yue[13] directly solved eqs.(1) and (2) for quasi-monochromatic waves by using a pseudo-spectral method. They considered an approximation φz up to the order M in relative wave steepness. To skip the detail of the formulation, ﬁnally, it is formulated the vertical gradient of velocity potential on the surface as

− M M l ηκ N ∂κ+1 φ (x, η, t) = φ(l)(t) ψ (x, 0), (3) z κ! n ∂zκ+1 n κ=0 n=0 Xl=1 X X where M is the order of nonlinearity. As a result, it can be solved the eqs.(1) and (2) with the approximated φz in the Fourier space by using pseudo-spectral s method. The spatial derivations of ∇xφ and ∇η are evaluated in the Fourier space, the nonlinear products are calculated on the physical space. Therefore, this approach is useful to simulate the long time evolution of random waves having broad band spectra because it requires the CPU time as order of N log N, although the mode-coupling equation consumes the CPU time as order of N 3. All aliasing errors generated in the nonlinear terms are deleted. The time integration s of the Fourier modes of η and ∇xφ is evaluated in the Fourier space with the fourth-order Runge-Kutta-Gill method. The order of nonlinearity M was fixed four for all cases that is the fourth-order nonlinear interactions were took into consideration for the high-order nonlinear simulation. The accuracy and convergence of the numerical model are verified by propagating the exact solution of the Stoke wave. The maximum error of the total energy leak and the surface profile change were 2.39 × 10−5 and 6.70 × 10−4, respectively. It is hence expect the high-order nonlinear wave propagation with

the suﬃcient accuracy solving eqs.(1), (2) and (3).

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Fig. 1 Initial proﬁle of wavenumber spectra for 2D simulations given by the Wallops type spectra as a function of spectrum bandwidth m. 2.2 Initial Conditions: Random waves in 2D The amplitudes of initial waves were given by the Wallops type spectrum in wavenumber space transformed through the linear dispersion relation.

2 α − m m − 4π 2kh 2 2 2 S(k) dk = H1/3κ exp − κ 1 + dk (4) 2 4 skkpthkhthkph sh2kh ³ ´ · ¸ κ = kthkh/kpthkph (5) where, m the spectral bandwidth parameter, kp the peak wavenumber of spectra, h the water depth and α is a constant satisfying a following relation with H1/3:

∞ H1/3 = 4.004 S(k)dk. (6) sZ0 Eq.(4) with m=5 and kph=∞ is equivalent to JONSWAP spectra and the shape of spectrum is getting narrower as increasing the value of m. The Wallops spectrum is evaluated as a function of spectrum bandwidth m only. The phase con- stants of the initial waves were assumed a random phase approximation. This assumption is very important simulating random wave propagation based with reality. A further important point is that if the phase is given factitiously(e.g. frequency wave focusing), it is possible to generate a freak wave like surface proﬁle at an arbitrary time and location. However, such approach is out of our intention. The computations were made in the periodic space having the length of 256Lp. Initial wave statistics were comprised with ﬁxed characteristic wave steepness: kpa=0.14 and spectrum band width:m=10, 20, 30, 40, 60, 80 and 100 as shown in Figure 1. Here, a is a half of H1/3, and Lp and Tp are the wave length and wave period of spectral peak mode, respectively. The water depth was chose as kph=∞(deep-water), 3.0, 2.0, 1.36, 1.0. The total time integration was calculated up to t=100Tp.

2.3 Initial Conditions: Stokes Wave in 3D An initial wave proﬁle and a potential energy on the surface for three dimensional simulation were given by the Stokes exact solution[15] for 3D simulation. The relative amplitude ka was ﬁxed with 0.15. The amplitudes of the perturbations for the Stokes wave were given by 1/100 of the carrier wave amplitude and angle was as θ=5, 15, 30 degree, respectively. The number of the Fourier modes were given as 64X64 in wavenumber space. The total time integration was evaluated up to t=250Tp.

3 Numerical Results and Discussions

3.1 High-order Nonlinear Eﬀects on Random Wave Trains in 2D Spectral Evolutions and Dispersion Relations @ Figure 2 shows the time evolutions of the wavenumber spectra for m=10 and (a) High-order solution

(b) Second-order solution

Fig. 2 Temporal evolutions of the Fourier spectra of the simulated wave train which has steepness kpa=0.14 and spectrum bandwidth m=10, initially.

kph=∞ both of the high-order and the second-order solution. Although both simulations were started the same initial condition, there is significant differences of the spectrum evolution between them. It is found that the Fourier modes actively exchange their energy and the Fourier mode amplitudes are strongly modulated during the propagation process for the high-order solution, while the second-order solution seems to stable. The fact that the differences between the high-order solution and the second-order one in Figure 2 suggest that the Fourier modes can transfer the energy due to the high-order nonlinear interactions even if the spectrum band width is relatively broad. A similar relation was observed in various spectrum band width in deep-water condition, however the activity of energy transfer became weak if initial spectrum band width became broader. The amplitude modulation like behavior became weak as deceasing the water depth

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and it finally disappeared if the characteristic water depth kph was shallower than 1.36. Figure 3 shows the time averaged wavenumber spectra both the high-order and second-order solution in deep-water condition. The second-order solution shows a secondary peak in the higher harmonics, if initial spectrum band width is narrow. On the contrary, there is no significant differences among the time averaged wavenumber spectra of the high-order solution. This result demonstrate that spectra of random wave trains in deep-water transform their profiles through the high-order nonlinear resonant interaction and it is equivalent to phase averaged high-order equation(e.g. Hasselmann’s eq.) It is appeared that the shapes of time averaged spectra of the high-order solution are similar and independent from the initial spectrum band width. However, (a) High-order solution

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Fig. 5 Examples of simulated water surface proﬁles both of the high-order and the second-order solution at the time t/Tp=25. eﬀects of the initial spectrum band width are hid from the shape of the time averaged spectra but are found out in dispersion relations. The phase of the nonlinear wave is given by

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Surface wave profile and wave statistics @ It is important for engineering practice to make clear the high-order nonlinear effects on water surface elevations and their statistics, because understanding of the wave characteristics/statistics is valuable for engineering. Figure 5 are plotted the spatial wave profiles of the high-order and 2nd order solutions at the evolution time of t/Tp=25 for m=10 and kph=∞. The whole of the surface profiles of the high-order and second-order solution is quite similar, however, a giant and steep wave, a freak wave like, can be observed at kpx=140 of the high- order solution. It is found that the high-order nonlinear interactions is strongly

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related to the occurrence of a single extreme wave having the outstanding crest height, because such wave can be never observed in the second-order nonlinear solution. The occurrence of the steep wave is related to higher wavenumber components than 2k/kp, particularly 3k/kp[?] and has high speed velocity near the surface[?]. The fact that the high-order nonlinear interactions generate the steep wave suggests that such high-order nonlinearities also affect on wave statistics. Thus GF (Groupiness Factor) is picked up to describe the characteristics of wave train. The time histories of GF during the propagating process are shown in Figure 6 for m=10, 30 and 100. GF of the non-linear solution are always larger than the second non-linear solution, and the high-order effects is strong for initially narrow banded spectrum wave in deep-water condition. Moreover, if the water depth becomes shallower, differences between the high-order and the second-

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broader in deep-water. These differences are decreased in kph=2.0 and are van- ished in kph=1.36. Moreover, they have opposite relationship in kph=1.0. This imply that the high-order nonlinear effects play an important role to stabilize the waves in shallow-water. The effects of the high-order nonlinearities are the most remarkable in µ4. The reason why µ4 is stand out is that µ4 depends on the third-order nonlinearities in statistically[16]. Therefore, the value of µ4 is one of a milestone to check the influence of the high-order nonlinearities of the observed wave train.

Wave height distribution @ The high-order nonlinearities increase Hmax/H1/3, kurtosis and GF . Another signiﬁcant aspect of the high-order nonlinear eﬀects is an exceedance probabil-

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3.2 High-order Nonlinear Eﬀects on the Stokes Wave in 3D

The numerical simulations in three dimensional domain were performed for the Stokes wave. Figure 11 shows temporal evolutions of spatial surface proﬁle of the

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waves at t/Tp=0, 100 and 200 with the perturbations which are inclined 5 and 30 degree to the carrier wave. The horizontal and vertical axis denote x and y axis normalized by carrier wave length Lp, respectively. The Stokes wave trains become unstable due to the five wave resonance at t/Tp=200. Figure 12 shows temporal evolutions of two dimensional wavenumber spectra for the same case of Figure 11. The horizontal and vertical axis denote two dimensional wavenumber space normalized by carrier wavenumber kp, respectively. The energy of the carrier wave spread out on wavenumber space both of θ=5 and 30. There is no significant difference between θ=5 and 30 at t/Tp=200, although the influence of the initial condition still remain at t/Tp=100. Finally the temporal evolution of the wavenumber spectra of x direction is shown in Figure 12. There is no regular motion such as FPU recurrence in two dimensional case.

4 Conclusion

It is found that the high-order nonlinear interactions play very important roles in the long time evolutions of gravity waves both deep and shallow-water solving the hydrodynamic equations for gravity waves having narrow to broad banded spectra. It is concluded as follows: – The high-order nonlinear interactions can transfer energy among the Fourier modes and excite apparently chaotic mode evolutions even if wave have broad band spectrum in deep-water. – The high-order nonlinear interactions can generate a single extreme high wave having outstanding crest height such as a freak wave. – The high-order nonlinear interactions aﬀect Hmax, kurtosis and GF remark- ably. – The high-order nonlinear interactions increase the occurrence probability of large wave height in deep-water and decrease it in shallow-water in comparison with the Rayleigh theory. Consequently, the high-order nonlinear eﬀects should be taken into account in- dependently of the spectral bandwidth to predict the maximum wave and the freak wave generation. ACKNOWLEDGEMENT We would like to thank Shinichi Ohmiya and Atsushi Kawai for their supports.

References

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