Modulational Instability in Crossing Sea States: A Possible Mechanism for the Formation of Freak Waves

Here we consider a simple weakly nonlinear model that describes the interaction of two-wave systems in deep water with two different directions of propagation. Under the hypothesis that both sea systems are narrow banded, we derive from the Zakharov equation two coupled nonlinear Schrödinger equations. Given a single unstable plane wave, here we show that the introduction of a second plane wave, propagating in a different direction, can result in an increase of the instability growth rates and enlargement of the instability region. We discuss these results in the context of the formation of rogue waves.

Figure 1

Instability diagram for $B_0=0$ as a function of angle $\theta$ and perturbation wave number. See text for details.

Figure 2

Instability diagram for $B_0=A_0$ as a function of angle $\theta$ and perturbation wave number. See text for details.

Figure 3

Nondimensional growth rates for longitudinal perturbation of a single carrier wave (solid line) and of two carrier waves (dotted line).

Figure 4

Growth rates as a function of perturbation wave number for different angles $\theta$; in the legend angles are expressed in degrees.

Figure 5

Evolution in time of mode $K^{\prime }=0.226$ for $\theta =15.3°$ for the case of $B_0=0$ and $A_0\kappa =B_0\kappa =0.1$. Time has been normalized with $\omega (\kappa )$. The exponential curves are of the form of $e^{0.01028t}$ and $e^{0.005t}$.