Strong solutions for the Alber equation and stability of unidirectional wave spectra

Agissilaos G. Athanassoulis; Gerassimos A. Athanassoulis; Mariya Ptashnyk; Themistoklis Sapsis

doi:10.3934/krm.2020024

Article Contents

2020, Volume 13, Issue 4: 703-737. Doi: 10.3934/krm.2020024

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Strong solutions for the Alber equation and stability of unidirectional wave spectra

1.
Department of Mathematics, University of Dundee, Dundee DD1 4HN, UK
2.
School of Naval Architecture and Marine Engineering, National Technical University of Athens, 9 Heroon Polytechniou str., Zographos 157 73, Athens, Greece
3.
School of Mathematical & Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK
4.
Department of Mechanical Engineering, Massachusetts Institute of Technology, Room 5-318, 77 Massachusetts Av., Cambridge, MA 02139-4307, USA

^* Corresponding author: Agissilaos G. Athanassoulis
^* Corresponding author: Agissilaos G. Athanassoulis

Received: June 2019

Revised: January 2020

Published: May 2020

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Abstract

The Alber equation is a moment equation for the nonlinear Schrödinger equation, formally used in ocean engineering to investigate the stability of stationary and homogeneous sea states in terms of their power spectra. In this work we present the first well-posedness theory for the Alber equation with the help of an appropriate equivalent reformulation. Moreover, we show linear Landau damping in the sense that, under a stability condition on the homogeneous background, any inhomogeneities disperse and decay in time. The proof exploits novel $ L^2 $ space-time estimates to control the inhomogeneity and our result applies to any regular initial data (without a mean-zero restriction). Finally, the sufficient condition for stability is resolved, and the physical implications for ocean waves are discussed. Using a standard reference dataset (the "North Atlantic Scatter Diagram") it is found that the vast majority of sea states are stable, but modulationally unstable sea states do appear, with likelihood $ O(1/1000); $ these would be the prime breeding ground for rogue waves.

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Mathematics Subject Classification: Primary: 35Q35, 35B35; Secondary: 81S30.

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Figure 3. Numerical investigation of the stability condition for a stable JONSWAP spectrum, cf. Section 8 for more details. We are using a target of $ 1/4\pi $ as in equation 54. Left: Plots of the curve $ \Gamma_X $ on the complex plane for different values of $ X. $ Since $ 1/4\pi $ is always outside the $ \Gamma_X, $ this spectrum is stable. Right: The span of the real parts of $ \Gamma_X $ for different values of $ X $

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Figure 4. Numerical investigation of the stability condition for an unstable JONSWAP spectrum. Left: Plots of the curve $ \Gamma_X $ on the complex plane for different values of $ X. $ Since $ 1/4\pi $ is contained in some curves $ \Gamma_X, $ the spectrum is unstable. Right: The span of the real parts of $ \Gamma_X $ for different values of $ X. $ In this case it highlights clearly the bandwidth of unstable wavenumbers $ X $

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Figure 1. The domains of integration for the integrals $ I_j, $ $ j = 1,\dots,6 $

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Figure 2. Some common profiles of JONSWAP spectra

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Figure 5. A number of points on the $ (\gamma,\alpha) $ plane are tested for stability of the corresponding JONSWAP spectrum, cf. equation 52. $ \alpha $ controls the power of the sea state (larger $ \alpha $ means larger significant wave height) and $ \gamma $ controls the effective bandwidth (larger $ \gamma $ means more narrowly peaked spectrum). The carrier wavenumber $ k_0 $ can easily be seen not to affect the (in)stability of the spectrum. $ (\gamma,\alpha) $ points found to be stable are marked with a full square, while points found to be unstable are marked with an empty square. For reference the proposed separatrices of [29] and [13] are shown (they are of the form $ \alpha\cdot\gamma/\beta = C, $ where $ \beta $ is the mean wave steepness and $ C = 0.77 $ [13] or $ C = 0.974 $ [29]). More details can be found in Section 8. Top: Linear scaling in both axes. Bottom: Log scaling in the $ \alpha $ (vertical) axis

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