Impact of the four-wave quasi-resonance on freak wave shapes in the ocean

Abstract

Two freak waves were observed a day apart in October 2009 at a 5000-m deep moored station in the northwest Pacific Ocean. As the typhoon passed by, the wave system transitioned within a day from a narrow and unimodal spectrum to the broad and bi-modal spectrum. The occurrence probability of a freak wave is known to increase due to a modulational instability; however, whether the modulational instability survives under a realistic directional sea state has not been conclusively determined yet. In this study, a phase-resolving wave model was used to obtain ensembles of realizations based on observed and simulated directional spectra. Unlike previous studies that focused only on the probability of freak wave occurrence, this study focuses on wave shape. It reveals that the front-to-rear asymmetry and crescent shape deformation of the crest are more pronounced for narrower spectrum and longer-lifetime freak waves; this distortion of wave shape and extended lifetime are both characteristics of nonlinear wave groups. This study also shows that the distribution of the lifetime of a freak wave depends on the sea state and that the number of nonlinear wave groups increases for a narrower spectrum. We therefore conjecture that both the four-wave quasi-resonance and dispersive focusing are responsible for freak wave generation, but their relative significance depends on the spectral broadness. Investigating the total kurtosis or occurrence probability alone is insufficient to unravel the underlying mechanisms of individual freak-wave generation.

Appendix: relationship between HOSM and Zakharov equation

Onorato et al. (2007) illustrated that the HOSM with M = 2 and M = 3 is equivalent to the second and third-order Hamiltonian dynamical equations, respectively, for $\widehat{\eta}\left(\boldsymbol{k}\right)$ and $\widehat{\varPhi}\left(\boldsymbol{k}\right)$. Likewise, as illustrated on page 16 of Krasitskii (1994), the nonlinear order of the Zakharov equation A(k) coincides with that of the Hamiltonian dynamical equations. The second-order kernels of the Zakharov equation, ${U}_{0,1,2}^{(3)}$ and ${U}_{0,1,2}^{(1)}$ (corresponding to ${V}_{1,2,3}^{\left(+\right)}$ and ${V}_{1,2,3}^{\left(-\right)}$ in this paper), are expressed by the second-order kernel, ${E}_{0,1,2}^{(3)}$, of the Hamiltonian dynamical equation. Similarly, the third-order kernel, ${V}_{0,1,2,3}^{(2)}$ (corresponding to ${W}_{0,1,2,3}^{(2)}$), is defined from the third-order kernel, ${E}_{0,1,2,3}^{(4)}$, of the Hamiltonian dynamical equation. Therefore, the nonlinear order of the HSOM is consistent with that of the Zakharov equation Eq. (13) from Sect. 4.1.