Properties of Rogue Waves and the Shape of the Ocean Wave Power Spectrum

We discuss a method for the determination of the shape of the ocean wave power spectrum that is based on the physics of the modulational instability for the nonlinear Schrödinger and the Zakharov equations. We find that the form of the spectrum includes an enhanced spectral peak and modulational channels that extend to both high and low frequency. Furthermore, this fundamental shape of the spectrum is found to also be contained in the kinetic equation commonly used for wind-wave models provided that the full Boltzmann four-wave interactions are included. We discuss a number of numerical simulations that demonstrate the modulational form of the power spectrum. We furthermore discuss how the enhanced spectral peak governs the formation of rogue wave packets. We provide ways to compute the properties of the rogue waves directly from the nonlinear spectrum of analyzed time series data or from wave forecasts and hindcasts.