David Andrade: New solutions of the C.S.Y. equation reveal increases in freak-wave occurrence
In this talk we present recent results on the time evolution of broad banded, random inhomogeneous fields of deep water waves. Our study is based on solutions of the equation derived by Crawford, Saffman and Yuen in 1980. (“Evolution of a random inhomogeneous fields of nonlinear deep-water gravity waves”, Wave Motion, 2(1), 1 - 16).
This talk covers three major aspects. The instability of a homogeneous spectrum to inhomogeneous disturbances, the long time evolution of such instabilities, and their impact on the probability of encountering freak waves.
The main results are the following. First, it is shown, from a broad banded model, that a JONSWAP spectrum is unstable when it is narrow and it stabilizes as it broadens. Then, from the spectral time evolution, we obtained the evolution of the variance of the free surface. In case of instability, it is observed that the variance and thus, the energy in the wave field, localizes in regions of space and time. Initially, the evolution of the variance exhibits a recurrent pattern, akin to the one found in solutions of Alber’s equation. Interestingly, in the most unstable cases, such recurrent pattern fades away giving rise to a localized pattern that dominates the later stages of the evolution. Last, we compute the probabilities of encountering freak waves. Our results suggest significantly higher probabilities than those predicted from the Rayleigh distribution surpassing those obtained from Alber’s equation.