Nonlinear Wave Interaction in Coastal and Open Seas: Deterministic and Stochastic Theory

Stuhlmeier, Raphael; Vrecica, Teodor; Toledo, Yaron

doi:10.1007/978-3-030-33536-6_10

Cited by 5 publications

(2 citation statements)

References 78 publications

(162 reference statements)

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“…The most well-known stochastic models for ocean waves include the CSY equation [18,3,4], Hasselmann's equation [26] and the Alber equation [2] that we will focus on here. For a recent review of various stochastic models one can see [48]. Broadly speaking, they are moment equations, starting from phase-resolved equations for the sea surface (such as Zakharov's equation or the NLS) as an approximation for deterministic wave dynamics.…”

Section: Introductionmentioning

confidence: 99%

Modelling of ocean waves with the Alber equation: application to non-parametric spectra and generalization to crossing seas

Athanassoulis,

Gramstad

2021

Preprint

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We study the Alber equation, a phase-averaged second-moment model for the statistics of a wavefield, derived under a narrowbandedness assumption. More specifically, we use the Alber equation and its associated instability condition to quantify how close a given non-parametric spectrum is to being modulationally unstable, and apply this to a dataset of 100 non-parametric spectra provided by the Norwegian Meteorological Institute. The vast majority of realistic spectra turn out to be stable, but the "proximity to instability" metric is seen to correlate strongly with their steepness and Benjamin-Feir Index (BFI). Moreover, upon comparing with phase-resolved broadband Monte Carlo simulations, the kurtosis and probability of extreme waves for each sea state wavefield are also seen to correlate well with its "proximity to instability". Finally, in order to extend this approach to crossing seas, we derive a generalized Alber equation starting from a system of nonlinear Schrödinger equations modelling crossing seas. We also derive the associated 2-dimensional instability condition.

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Section: Introductionmentioning

confidence: 99%

Modelling of ocean waves with the Alber equation: application to non-parametric spectra and generalization to crossing seas

Athanassoulis,

Gramstad

2021

Preprint

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“…The most well known stochastic models for ocean waves include the CSY equation [19][20][21], Hasselmann's equation [22] and the Alber equation [23], which we will focus on here. For a recent review of various stochastic models, one can see [24]. Broadly speaking, they are moment equations, starting from phase-resolved equations for the sea surface (such as Zakharov's equation or the NLS) as an approximation for deterministic wave dynamics.…”

Section: Introductionmentioning

confidence: 99%

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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An Introduction to the Zakharov Equation for Modelling Deep-Water Waves

Stuhlmeier

2024

Advances in Mathematical Fluid Mechanics

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Nonlinear Wave Interaction in Coastal and Open Seas: Deterministic and Stochastic Theory

Cited by 5 publications

References 78 publications

Modelling of ocean waves with the Alber equation: application to non-parametric spectra and generalization to crossing seas

Modelling of ocean waves with the Alber equation: application to non-parametric spectra and generalization to crossing seas

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

An Introduction to the Zakharov Equation for Modelling Deep-Water Waves

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