Numerical Investigation of the Instability and Nonlinear Evolution of Narrow-Band Directional Ocean Waves

The instability and nonlinear evolution of directional ocean waves is investigated numerically by means of simulations of the governing kinetic equation for narrow-band surface waves. Our simulation results reveal the onset of the modulational instability for long-crested wave trains, which agrees well with recent large-scale experiments in wave basins, where it was found that narrower directional spectra lead to self-focusing of ocean waves and an enhanced probability of extreme events. We find that the modulational instability is nonlinearly saturated by a broadening of the wave spectrum, which leads to the stabilization of the water-wave system. Applications of our results to other fields of physics, such as nonlinear optics and plasma physics, are discussed.

Figure 1

The time evolution of the maximum intensity $k_p^2{I}_{\max }$ for (a) $N=840$ (black), (b) $N=200$ (blue), (c) $N=90$ (red), (d) $N=50$ (green), (e) $N=24$ (magenta), and the case of a narrow-band normal velocity distribution (the inset). The spatial distributions of wave intensity for (a)–(d) are shown in Fig. 3 at the times indicated here with arrows.

Figure 2

(a) The linear growth rate ${\omega }_I$ of the fastest growing wave mode and (b) maximum kurtosis for $N=840$ (black), $N=200$ (blue), $N=90$ (red), $N=50$ (green), and $N=24$ (magenta), for ${\alpha }_P=0.02$ (dashed line), ${\alpha }_P=0.025$ (solid line), and ${\alpha }_P=0.03$ (dash-dotted line). The solid line (${\alpha }_P=0.025$) corresponds to curves a–e in Fig. 1.

Figure 3

The spatial distribution of the normalized wave intensity $k_p^{2}I$ for (a) $N=840$ at $t=1.27\times {10}^3{\tau }_p$, (b) $N=200$ at $t=1.46\times {10}^3{\tau }_p$, (c) $N=90$ at $t=1.81\times {10}^3{\tau }_p$, and (d) $N=50$ at $t=2.90\times {10}^3{\tau }_p$, corresponding to curves a–d in Fig. 1.

Figure 4

The velocity distribution ${\omega }_p^{2}f$ of the wave energy, averaged over space, at $t=0$ (left column) and $t=3.2\times {10}^3{\tau }_p$ (right column), for (a) $N=840$, (b) $N=200$, (c) $N=90$, and (d) $N=50$. Panel (e) shows the narrow-band normal velocity distribution at $t=0$ (left) and $t=640{\tau }_p$ (right).