Governing Equations

Two types 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 fully nonlinear interactions more than third-order (Dommermuth and Yue, 1987) and another one is the second-order approximated equations which excluded 3rd order nonlinear terms (Yasuda et al., 1992).

It is assumed a periodic boundary condition and is assigned to spatial coordinates $(x,z)$; the origin is located at the mean water level, $x$ 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 $\eta(x,t)$ and the velocity potential on the surface $\phi^s(x,t)$= $\phi(x,\eta,t)$(Zakharov, 1968);

$$\eta_t+\ensuremath{\nabla_{x} \phi^s}\eta_{x} - (1 + \nabla_{x}\eta\!\cdot\!\!\nabla_{x}\eta)\ensuremath{\phi_z}=0, \hskip 2.44cm (z=\eta)$$ (1)

$$\phi_t+g\eta+\ensuremath{\frac{1}{2}}\ensuremath{\nabla_{x} \phi^... ...{x}\eta\!\cdot\!\!\nabla_{x}\eta)\ensuremath{\phi_z}^2=0, \hskip 0.5cm (z=\eta)$$ (2)

where $\nabla x$= $(\partial/\partial x)$, the subscript $t$ denotes the partial differentiation with $t$, $\phi_z$ the vertical gradient of the velocity potential $\phi$, $t$ the time and $g$ the acceleration due to the gravity. Dommermuth and Yue (1987) directly solved eqs.(1) and (2) for quasi-monochromatic waves by using a pseudo-spectral method. They considered an approximation $\phi_z$ 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

$$\ensuremath{\phi_z}(x,\eta,t)=\sum^{M}_{l=1}\sum^{M-l}_{\ka... ...kappa+1}\over\ensuremath{\partial z}^{\kappa+1}}\psi_n(x,0),$$ (3)

where $M$ is the order of nonlinearity. As a result, it can be solved the eqs.(1) and (2) with the approximated $\phi_z$ in the Fourier space by using pseudo-spectral method. The spatial derivations of $\ensuremath{\nabla_{x} \phi^s}$ and $\nabla\eta$ 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 of the Fourier modes of $\eta$ and $\ensuremath{\nabla_{x} \phi^s}$ 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 $2.39\times 10^{-5}$ and $6.70\times 10^{-4}$, respectively. It is hence expect the high-order nonlinear wave propagation with the sufficient accuracy solving eqs.(1), (2) and (3).

Figure 1: Initial profile of wavenumber spectra for 2D simulations given by the Wallops type spectra as a function of spectrum bandwidth $m$.