The current-modified nonlinear Schrödinger equation

Stocker, Jenny; Peregrine, D. H.

doi:10.1017/s0022112099006618

Cited by 45 publications

(51 citation statements)

References 22 publications

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“…Trulsen & Dysthe 1996) that only allows for a limited time horizon, τ , for the calculations of the order of (ω p s 3 ) −1 , where ω p = 2π/T p . Quantitative comparisons with wave tank experiments and other numerical wave models (Lo & Mei 1985;Trulsen & Stansberg 2001;Shemer et al 2001Shemer et al , 2002Stocker & Peregrine 1999;Clamond & Grue 2002) have indicated that a reasonable time horizon is indeed τ ≈ (ω p s 3 ) −1 , which for the present simulations (with…”

Section: The Simulation Modelsupporting

confidence: 73%

Probability distributions of surface gravity waves during spectral changes

et al. 2005

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Simulations have been performed with a fairly narrow band numerical gravity wave model (higher-order NLS type) and a computational domain of dimensions 128 × 128 typical wavelengths. The simulations are initiated with s 6 × 10 4 Fourier modes corresponding to truncated JONSWAP spectra and different angular distributions giving both short-and long-crested waves. A development of the spectra on the so-called Benjamin-Feir timescale is seen, similar to the one reported by Dysthe et al. (J. Fluid Mech. vol. 478, 2003, P. 1). The probability distributions of surface elevation and crest height are found to fit theoretical distributions found by Tayfun (J. Geophys. Res. vol. 85, 1980Res. vol. 85, , p. 1548) very well for elevations up to four standard deviations (for realistic angular spectral distributions). Moreover, in this range of the distributions, the influence of the spectral evolution seems insignificant. For the extreme parts of the distributions a significant correlation with the spectral change can be seen for very long-crested waves. For this case we find that the density of large waves increases during spectral change, in agreement with a recent experimental study by Onorato et al. (J. Fluid Mech. 2004 submitted).

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Section: The Simulation Modelsupporting

confidence: 73%

Probability distributions of surface gravity waves during spectral changes

et al. 2005

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“…Note that a term of that kind arises in the Eulerian description of potential wave oscillations of the free surface as well. In the paper by Stocker and Peregrine (1999), U = U * sin (kx − ω t) was chosen and interpreted as a harmonically changing surface current induced by an internal wave. We shall further take U = 0.…”

Section: Cubic Approximationmentioning

confidence: 99%

Lagrange form of the nonlinear Schrödinger equation for low-vorticity waves in deep water

Abrashkin

Pelinovsky

2017

Nonlin. Processes Geophys.

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Abstract. The nonlinear Schrödinger (NLS) equation describing the propagation of weakly rotational wave packets in an infinitely deep fluid in Lagrangian coordinates has been derived. The vorticity is assumed to be an arbitrary function of Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. The vorticity effects manifest themselves in a shift of the wave number in the carrier wave and in variation in the coefficient multiplying the nonlinear term. In the case of vorticity dependence on the vertical Lagrangian coordinate only (Gouyon waves), the shift of the wave number and the respective coefficient are constant. When the vorticity is dependent on both Lagrangian coordinates, the shift of the wave number is horizontally inhomogeneous. There are special cases (e.g., Gerstner waves) in which the vorticity is proportional to the squared wave amplitude and nonlinearity disappears, thus making the equations for wave packet dynamics linear. It is shown that the NLS solution for weakly rotational waves in the Eulerian variables may be obtained from the Lagrangian solution by simply changing the horizontal coordinates.

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“…Higher-order expansions in terms of nonlinearity and wave dispersion have been used to derive more general Schrödinger-like models for water waves including currents [22]. In the formalism of Ref.…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…In the formalism of Ref. [22], the current velocity is of order 2 , while the time and space derivatives and Ak 0 are of order , where is a nonlinearity parameter. Since the dispersion and nonlinearity coefficients D x and −ξ have equal signs, Eq.…”

Section: Mathematical Formulationmentioning

confidence: 99%

Trapping and instability of directional gravity waves in localized water currents

Eliasson

Haas

2014

Phys. Rev. E

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The influence of localized water currents on the nonlinear dynamics and stability of large amplitude, statistically distributed gravity waves is investigated theoretically and numerically by means of an evolution equation for a Wigner function governing the spectrum of waves. It is shown that water waves propagating in the opposite direction of a localized current channel can be trapped in the channel, which can lead to the amplification of the wave intensity. Under certain conditions the wave intensity can be further localized due to a self-focusing (Benjamin-Feir) instability. The localized amplification of the wave intensity may increase the probability of extreme events in the form of freak waves, which have been observed in connection with ocean currents.

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The current-modified nonlinear Schrödinger equation

Cited by 45 publications

References 22 publications

Probability distributions of surface gravity waves during spectral changes

Probability distributions of surface gravity waves during spectral changes

Lagrange form of the nonlinear Schrödinger equation for low-vorticity waves in deep water

Trapping and instability of directional gravity waves in localized water currents

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