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

Athanassoulis, Agissilaos; Athanassoulis, G. A.; Ptashnyk, Mariya; Sapsis, Themistoklis P.

doi:10.3934/krm.2020024

Cited by 7 publications

(23 citation statements)

References 30 publications

(65 reference statements)

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“…The Penrose criterion ( 14) can be numerically computed for any type of background. We recall here that the realistic spectra, when become narrow, switch to instability from stable state and vice versa happens when they become broader [27]. Hence in order to achieve our task, we consider only the narrow spectrum as background.…”

Section: Instability Arising Out Of Narrow Spectral Backgroundmentioning

confidence: 99%

Penrose instabilities and the emergence of rogue waves in Sasa–Satsuma equation

2021

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In this paper, we calculate the region of emergence of rogue waves in the Sasa-Satsuma equation by performing Penrose stability analysis. We consider Wigner-transformed Sasa-Satsuma equation and separate out unstable solutions, namely Penrose instability modes. We superpose these modes in a small region. With the help of marginal property of the Wigner transform, we identify the region in which rogue wave solution can emerge in the Sasa-Satsuma equation and calculate the amount of spatial localization. We also formulate a condition for the emergence of rogue wave solution in the Sasa-Satsuma equation.

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Section: Instability Arising Out Of Narrow Spectral Backgroundmentioning

confidence: 99%

Penrose instabilities and the emergence of rogue waves in Sasa–Satsuma equation

2021

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“…The Alber equation, its derivation and interpretation have been widely studied and explained. We refer readers who are interested to [23,[28][29][30] and the references therein for more details. Here, we will briefly present its derivation and main features in order to make the paper self-contained.…”

Section: Ocean Wave Modelling With the Alber Equationmentioning

confidence: 99%

“…This was called an "eigenvalue relation" in [23]; we will call it an "instability condition". In [30], it was further shown that if the instability condition does not hold, then linear stability follows. The instability condition (7) itself can be refined in two ways: One concerns a technical issue related to X " 0.…”

Section: The Stability-of-homogeneity Questionmentioning

confidence: 99%

“…This is not straightforward in general, and historically it has been a challenge to use the Alber equation more widely [29]. In [30], this condition is reformulated so that a more constructive method to check it can be found: by dividing both sides of the fraction by X and setting…”

Section: The Stability-of-homogeneity Questionmentioning

confidence: 99%

“…The benefit is that now the argument principle (for a holomorphic function f defined on a closed domain A Ď C, f : A Ñ C, it follows that z P f pAq if and only if the curve f pBAq is circumscribed around z P C) can be used to reformulate this to a constructive condition that we can directly check [30,37,38]. To this end, we will also need to introduce the signal transform Sruspxq :" Hruspxq´iupxq; (10) we are now ready to state the equivalent instability condition: Definition 1 (Penrose-Alber condition).…”

Section: The Stability-of-homogeneity Questionmentioning

confidence: 99%

See 2 more Smart Citations

Modelling of Ocean Waves with the Alber Equation: Application to Non-Parametric Spectra and Generalisation to Crossing Seas

Athanassoulis

Gramstad

2021

Fluids

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The Alber equation is a phase-averaged second-moment model used to study the statistics of a sea state, which has recently been attracting renewed attention. We extend it in two ways: firstly, we derive a generalized Alber system starting from a system of nonlinear Schrödinger equations, which contains the classical Alber equation as a special case but can also describe crossing seas, i.e., two wavesystems with different wavenumbers crossing. (These can be two completely independent wavenumbers, i.e., in general different directions and different moduli.) We also derive the associated two-dimensional scalar instability condition. This is the first time that a modulation instability condition applicable to crossing seas has been systematically derived for general spectra. Secondly, we use the classical Alber equation and its associated instability condition to quantify how close a given nonparametric spectrum is to being modulationally unstable. We apply this to a dataset of 100 nonparametric spectra provided by the Norwegian Meteorological Institute and find that the vast majority of realistic spectra turn out to be stable, but three extreme sea states are found to be unstable (out of 20 sea states chosen for their severity). Moreover, we introduce a novel “proximity to instability” (PTI) metric, inspired by the stability analysis. This is seen to correlate strongly with the steepness and Benjamin–Feir Index (BFI) for the sea states in our dataset (>85% Spearman rank correlation). Furthermore, upon comparing with phase-resolved broadband Monte Carlo simulations, the kurtosis and probability of rogue waves for each sea state are also seen to correlate well with the PTI (>85% Spearman rank correlation).

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Modulation Instability and Convergence of the Random-Phase Approximation for Stochastic Sea States

Athanassoulis,

Kyza

2024

Water Waves

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The nonlinear Schrödinger equation is widely used as an approximate model for the evolution in time of the water wave envelope. In the context of simulating ocean waves, initial conditions are typically generated from a measured power spectrum using the random-phase approximation, and periodized on an interval of length L. It is known that most realistic ocean waves power spectra do not exhibit modulation instability, but the most severe ones do; it is thus a natural question to ask whether the periodized random-phase approximation has the correct stability properties. In this work, we specify a random-phase approximation scaling, so that, in the limit of $$L\rightarrow \infty ,$$ L → ∞ , the stability properties of the periodized problem are identical to those of the continuous power spectrum on the infinite line. Moreover, it is seen through concrete examples that using a too short computational domain can completely suppress the modulation instability.

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

Cited by 7 publications

References 30 publications

Penrose instabilities and the emergence of rogue waves in Sasa–Satsuma equation

Penrose instabilities and the emergence of rogue waves in Sasa–Satsuma equation

Modelling of Ocean Waves with the Alber Equation: Application to Non-Parametric Spectra and Generalisation to Crossing Seas

Modulation Instability and Convergence of the Random-Phase Approximation for Stochastic Sea States

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