E®ects 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 non- linearities in two and three dimensional domain are performed by us- ing 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 SchrÄodinger 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 scienti¯c 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 e®ects 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 con¯ned 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 o®- shore 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 e®ects are primary pos- sible 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 di®raction[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 prop- erties, occurrence probabilities and e®ects 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 equa- tions of a potential flow by pseudo spectrum method. On the basis of the nu- merical 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 consid- eration 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 bound- ary conditions on the free surface are rewritten into the evolution equations as a function of the free surface pro¯le ´(x; t) and the velocity potential on the surface Ás(x; t)=Á(x; ´; t)[14]; s ´t + rxÁ ´x ¡ (1 + rx´¢rx´)Áz = 0; (z = ´) (1) 1 1 Á + g´ + r Ás ¢r Ás ¡ (1 + r ´¢r ´)Á2 = 0; (z = ´) (2) t 2 x x 2 x x z where rx=(@=@x; @=@y), the subscript t denotes the partial di®erentiation with t, Áz the vertical gradient of the velocity potential Á, 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 Áz up to the order M in relative wave steepness. To skip the detail of the formulation, ¯nally, it is formulated the vertical gradient of velocity potential on the surface as ¡ M M l ´· N @·+1 Á (x; ´; t) = Á(l)(t) à (x; 0); (3) z ·! n @z·+1 n ·=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 Áz in the Fourier space by using pseudo-spectral s method. The spatial derivations of rxÁ and r´ 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 ´ and rxÁ is evaluated in the Fourier space with the fourth-order Runge-Kutta-Gill method. The order of nonlinearity M was ¯xed 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 veri¯ed by prop- agating the exact solution of the Stoke wave. The maximum error of the total energy leak and the surface pro¯le change were 2:39 £ 10¡5 and 6:70 £ 10¡4, respectively. It is hence expect the high-order nonlinear wave propagation with the su±cient accuracy solving eqs.(1), (2) and (3). ¤ ¤ ¢¡¤£ § ¤ ¢¡¤£ ¢¡¤£ # " ! ¢¡¤£ © ¢¡¤£¦¥¦¨ ¢¡¤£¦¥¦§ ¡ $&%'$&( Fig. 1 Initial pro¯le of wavenumber spectra for 2D simulations given by the Wallops type spectra as a function of spectrum band- width 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 ® ¡ m m ¡ 4¼ 2kh 2 2 2 S(k) dk = H1=3· exp ¡ · 1 + dk (4) 2 4 skkpthkhthkph sh2kh ³ ´ · ¸ · = kthkh=kpthkph (5) where, m the spectral bandwidth parameter, kp the peak wavenumber of spectra, h the water depth and ® is a constant satisfying a following relation with H1=3: 1 H1=3 = 4:004 S(k)dk: (6) sZ0 Eq.(4) with m=5 and kph=1 is equivalent to JONSWAP spectra and the shape of spectrum is getting narrower as increasing the value of m. The Wallops spec- trum 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 pro¯le 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 ¯xed characteristic wave steep- ness: 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=1(deep-water), 3.0, 2.0, 1.36, 1.0. The total time integration was calcu- lated up to t=100Tp. 2.3 Initial Conditions: Stokes Wave in 3D An initial wave pro¯le 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 ¯xed 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 E®ects 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.