The Landau-Alber bifurcation: implications for the analysis of metocean data, and open questions in Landau damping

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• Agis Athanassoulis (University of Dundee)
• Friday 09 September 2022, 10:30-11:30
• Seminar Room 1, Newton Institute.

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HY2W02 - Analysis of dispersive systems

It is famously known that plane wave solutions of the Nonlinear Schrödinger equation (NLS) are linearly unstable; this fact is called the modulation instability for the NLS . While recent breakthorughs have provided insights on the qualitative aspects of nonlinear evolution of the modulation instability, many fundamental questions (such as global existence of solutions for general initial perturbations) are still open. The NLS is used as an approximate model in ocean waves, where the modulation instability is also known as the Benjamin-Feir instability, and is linked with concrete physical phenomena including rogue waves. In this context of ocean engineering, linear sea states are often used as approximate stochastic solutions of the NLS , as only stochastic wavefileds—and not plane waves—are relevant for marine safety applications.  The main points of this talk are the following:

There exists a 2nd moment theory of stochastic wavefileds under the NLS equation that is compatible with the bulk of the metocean data available, namely the Alber equation. This allows concepts from the NLS to be generalised to a stochastic setting yet remain tractable. The existing well-posedness theory for the Alber equation will be briefly reviewed [1]. In the stochastic setting, the instability of the homogeneous solution (i.e. ‘generalised modulation instability’) is not the only possibility. In fact, many homogeneous sea states will give rise to Landau damping, unexpectedly stabilising themselves despite the presence of infinite energy and a focusing nonlinearity. Still, unstable sea states do exist. So a bifurcation between Landau damping and generalised modulation instability exists within measured sea states [1-4]. Since I. E. Alber was the one who initiated this line of work, we call this the Landau-Alber bifurcation. The rigorous result on the Landau damping will be briefly outlined, along with its proof which builds crucially on an esoteric property of the Hilbert transform [1]. The linear stability analysis can surprisingly predict both profiles of rogue waves in the unstable case, and the likelihood of extreme events in the stable case. These findings are in excellent agreement with empirical facts in the ocean engineering community [2,5].

A number of open questions will be discussed in the end. Includes joint work with T. Sapsis (MIT), G. Athanassoulis (NTUA), M. Ptashnyk (Heriot-Watt) and O. Gramstad (DNV). References:[1]. Athanassoulis, Agissilaos G., et al. “Strong solutions for the Alber equation and stability of unidirectional wave spectra.” Kinetic & Related Models 13.4 (2020).[2]. Athanassoulis, Agissilaos G., and Odin Gramstad. “Modelling of Ocean Waves with the Alber Equation: Application to Non-Parametric Spectra and Generalisation to Crossing Seas.” Fluids 6.8 (2021): 291.[3]. Ribal, A., et al. “Recurrent solutions of the Alber equation initialized by Joint North Sea Wave Project spectra.” Journal of Fluid Mechanics 719 (2013): 314-344.[4]. Alber, I. E. “The effects of randomness on the stability of two-dimensional surface wavetrains.” Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 363.1715 (1978): 525-546.[5]. Dematteis, Giovanni, Tobias Grafke, and Eric Vanden-Eijnden. “Rogue waves and large deviations in deep sea.” Proceedings of the National Academy of Sciences 115.5 (2018): 855-860.

This talk is part of the Isaac Newton Institute Seminar Series series.

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