Instability and Evolution of Nonlinearly Interacting Water Waves

We consider the modulational instability of nonlinearly interacting two-dimensional waves in deep water, which are described by a pair of two-dimensional coupled nonlinear Schrödinger equations. We derive a nonlinear dispersion relation. The latter is numerically analyzed to obtain the regions and the associated growth rates of the modulational instability. Furthermore, we follow the long term evolution of the latter by means of computer simulations of the governing nonlinear equations and demonstrate the formation of localized coherent wave envelopes. Our results should be useful for understanding the formation and nonlinear propagation characteristics of large-amplitude freak waves in deep water.

Figure 1

The normalized growth rate $\Gamma /\omega$ plotted as a function of $K/\kappa$ and $L/\kappa$. Here we have used $\theta =\pi /8$ and the wave amplitudes $A_0=B_0=0.1/\kappa$. The left upper and lower panels show the cases with a single wave $A_0$ and $B_0$, respectively, while the right panel shows the case of interacting waves.

Figure 2

The normalized growth rate $\Gamma /\omega$ plotted as a function of $K/\kappa$ and $L/\kappa$. Here we have used $\theta =\pi /4$ and the wave amplitudes $A_0=B_0=0.1/\kappa$. The left upper and lower panels show the cases with a single wave $A_0$ and $B_0$, respectively, while the right panel shows the case of interacting waves.

Figure 3

The normalized growth rate $\Gamma /\omega$ plotted as a function of $K/\kappa$ and $L/\kappa$. Here we have used $\theta =\pi /2$ and the wave amplitudes $A_0=B_0=0.1/\kappa$. The left upper and lower panels show the cases with a single wave $A_0$ and $B_0$, respectively, while the right panel shows the case of interacting waves.

Figure 4

The interaction between two waves, initially with equal amplitudes $|A|=|B|=0.1{\kappa }^{-1}$ and a propagation angle of $\theta =\pi /8$ relative to the dichotome. Added to the initially homogeneous wave envelopes is a low-amplitude noise of order ${10}^{-3}/\kappa$ to give a seed to the modulational instability.