Bound-waves due to sea and swell trigger the generation of freak-waves
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RESEARCH ARTICLE
Bound-waves due to sea and swell trigger the generation of freak-waves David Andrade1
· Michael Stiassnie1
Received: 4 August 2020 / Accepted: 5 October 2020 © Springer Nature Switzerland AG 2020
Abstract Averaged bound-waves, arising from the interaction of a stormy-sea with a marginal swell, are used as an initial inhomogeneous disturbance which, as a result of an instability inherent in narrow homogeneous JONSWAP spectra, is amplified exponentially. This drives the system away from the equilibrium. Finally, by looking into the statistics of the underlying sea state we find that, throughout the non linear long time evolution, there is an increase in the probability of freak wave occurrence. Keywords Freak-waves · C.S.Y. equation · Inhomogeneous random waves · Bound-waves
1 Introduction In this article, we propose a mechanism for the generation of freak waves in a sea state consisting of a local wind sea and a swell. This mechanism is probabilistic in nature, i.e. it does not show how a freak wave would emerge out of a single realization of the surface elevation in combination with swell. Rather, it shows that throughout the non linear, long time evolution of the sea state, treated as a Gaussian process, the probability of extreme waves is increased considerably. Dealing with non-linear random inhomogeneous waves is the crux of the matter. In the context of deep water waves, there are currently two equations at hand. We have the Alber equation, see Alber (1978), and the equation derived by Crawford, Saffman and Yuen, see Crawford et al. (1980), that we call the C.S.Y. equation. The main difference is that Alber’s equation is a narrow banded model whereas the C.S.Y. is not. Actually, Alber’s equation can be derived directly from the C.S.Y. equation, see Crawford et al. (1980). We choose the C.S.Y. equation as the model for non linear evolution of random inhomogeneous waves.
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David Andrade [email protected] Michael Stiassnie [email protected]
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There are two steps needed for the mechanism to generate freak waves. First, one must find the instability generator of the underlying wave spectrum. This means considering a special inhomogeneous perturbation of a homogeneous spectrum, defined in terms of a wave number and some frequencies. Whenever any of such frequencies have a non-zero imaginary part, the wave vector of the disturbance destabilizes the spectrum. The set of all such unstable wave vectors act as the instability generator of the spectrum. Here is where the swell enters the picture. It turns out that, by taking two-wave vector correlations, between the bound-waves (arising from the sea and the swell) and the free waves of the sea, one obtains a similar type of inhomogeneous perturbation required as a generator of instability. The wave vector of the disturbance is the wave vector of the swell. In case of a destabilizing swell, the next step is to compute the non linear, long time evolution of the spectrum. This is achieved by using a discretized version of the C.S.Y. equation as
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