Separatrix crossing and symmetry breaking in NLSE-like systems due to forcing and damping

AbstractWe theoretically and experimentally examine the effect of forcing and damping on systems that can be described by the nonlinear Schrödinger equation (NLSE), by making use of the phase-space predictions of the three-wave truncation. In the latter, the spectrum is truncated to only the fundamental frequency and the upper and lower sidebands. Our experiments are performed on deep water waves, which are better described by the higher-order NLSE, the Dysthe equation. We therefore extend our analysis to this system. However, our conclusions are general for NLSE systems. By means of experimentally obtained phase-space trajectories, we demonstrate that forcing and damping cause a separatrix crossing during the evolution. When the system is damped, it is pulled outside the separatrix, which in the real space corresponds to a phase-shift of the envelope and therefore doubles the period of the Fermi–Pasta–Ulam–Tsingou recurrence cycle. When the system is forced by the wind, it is pulled inside the separatrix, lifting the phase-shift. Furthermore, we observe a growth and decay cycle for modulated plane waves that are conventionally considered stable. Finally, we give a theoretical demonstration that forcing the NLSE system can induce symmetry breaking during the evolution.

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On the subharmonic instabilities of steady three-dimensional deep water waves

Journal of Fluid Mechanics ◽

10.1017/s0022112094000509 ◽

1994 ◽

Vol 262 ◽

pp. 265-291 ◽

Cited By ~ 21

Author(s):

Mansour Ioualalen ◽

Christian Kharif

Keyword(s):

Gravity Waves ◽

Deep Water ◽

Water Waves ◽

Transverse Direction ◽

Plane Waves ◽

Three Dimensional ◽

Numerical Procedure ◽

Two Dimensional ◽

Wave Steepness ◽

Oblique Angle

A numerical procedure has been developed to study the linear stability of nonlinear three-dimensional progressive gravity waves on deep water. The three-dimensional patterns considered herein are short-crested waves which may be produced by two progressive plane waves propagating at an oblique angle, γ, to each other. It is shown that for moderate wave steepness the dominant resonances are sideband-type instabilities in the direction of propagation and, depending on the value of γ, also in the transverse direction. It is also shown that three-dimensional progressive gravity waves are less unstable than two-dimensional progressive gravity waves.

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Spatial Form of a Hamiltonian Dysthe Equation for Deep-Water Gravity Waves

Fluids ◽

10.3390/fluids6030103 ◽

2021 ◽

Vol 6 (3) ◽

pp. 103

Author(s):

Philippe Guyenne ◽

Adilbek Kairzhan ◽

Catherine Sulem ◽

Boyang Xu

Keyword(s):

Gravity Waves ◽

Deep Water ◽

Water Waves ◽

Short Wave ◽

Birkhoff Normal Form ◽

Spatial Form ◽

Periodic Groups ◽

Dimensional Gravity ◽

Nonlinear Modulation ◽

Dysthe Equation

An overview of a Hamiltonian framework for the description of nonlinear modulation of surface water waves is presented. The main result is the derivation of a Hamiltonian version of Dysthe’s equation for two-dimensional gravity waves on deep water. The reduced problem is obtained via a Birkhoff normal form transformation which not only helps eliminate all non-resonant cubic terms but also yields a non-perturbative procedure for surface reconstruction. The free surface is reconstructed from the wave envelope by solving an inviscid Burgers’ equation with an initial condition given by the modulational Ansatz. Particular attention is paid to the spatial form of this model, which is simulated numerically and tested against laboratory experiments on periodic groups and short-wave packets. Satisfactory agreement is found in all these cases.

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Bound Coherent Structures Propagating on the Free Surface of Deep Water

Fluids ◽

10.3390/fluids6030115 ◽

2021 ◽

Vol 6 (3) ◽

pp. 115

Author(s):

Dmitry Kachulin ◽

Sergey Dremov ◽

Alexander Dyachenko

Keyword(s):

Free Surface ◽

Deep Water ◽

Coherent Structures ◽

Water Waves ◽

Nonlinear Models ◽

Ideal Incompressible Fluid ◽

Equation Model ◽

System Of Nonlinear Equations ◽

Potential Flows ◽

Deep Water Waves

This article presents a study of bound periodically oscillating coherent structures arising on the free surface of deep water. Such structures resemble the well known bi-soliton solution of the nonlinear Schrödinger equation. The research was carried out in the super-compact Dyachenko-Zakharov equation model for unidirectional deep water waves and the full system of nonlinear equations for potential flows of an ideal incompressible fluid written in conformal variables. The special numerical algorithm that includes a damping procedure of radiation and velocity adjusting was used for obtaining such bound structures. The results showed that in both nonlinear models for deep water waves after the damping is turned off, a periodically oscillating bound structure remains on the fluid surface and propagates stably over hundreds of thousands of characteristic wave periods without losing energy.

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Five-Wave Resonances in Deep Water Gravity Waves: Integrability, Numerical Simulations and Experiments

Fluids ◽

10.3390/fluids6060205 ◽

2021 ◽

Vol 6 (6) ◽

pp. 205

Author(s):

Dan Lucas ◽

Marc Perlin ◽

Dian-Yong Liu ◽

Shane Walsh ◽

Rossen Ivanov ◽

...

Keyword(s):

Normal Form ◽

Numerical Simulations ◽

Gravity Waves ◽

Deep Water ◽

Plane Waves ◽

Surface Elevation ◽

Wave Interactions ◽

Frequency Space ◽

Target Mode ◽

The Hilbert Transform

In this work we consider the problem of finding the simplest arrangement of resonant deep-water gravity waves in one-dimensional propagation, from three perspectives: Theoretical, numerical and experimental. Theoretically this requires using a normal-form Hamiltonian that focuses on 5-wave resonances. The simplest arrangement is based on a triad of wavevectors K1+K2=K3 (satisfying specific ratios) along with their negatives, corresponding to a scenario of encountering wavepackets, amenable to experiments and numerical simulations. The normal-form equations for these encountering waves in resonance are shown to be non-integrable, but they admit an integrable reduction in a symmetric configuration. Numerical simulations of the governing equations in natural variables using pseudospectral methods require the inclusion of up to 6-wave interactions, which imposes a strong dealiasing cut-off in order to properly resolve the evolving waves. We study the resonance numerically by looking at a target mode in the base triad and showing that the energy transfer to this mode is more efficient when the system is close to satisfying the resonant conditions. We first look at encountering plane waves with base frequencies in the range 1.32–2.35 Hz and steepnesses below 0.1, and show that the time evolution of the target mode’s energy is dramatically changed at the resonance. We then look at a scenario that is closer to experiments: Encountering wavepackets in a 400-m long numerical tank, where the interaction time is reduced with respect to the plane-wave case but the resonance is still observed; by mimicking a probe measurement of surface elevation we obtain efficiencies of up to 10% in frequency space after including near-resonant contributions. Finally, we perform preliminary experiments of encountering wavepackets in a 35-m long tank, which seem to show that the resonance exists physically. The measured efficiencies via probe measurements of surface elevation are relatively small, indicating that a finer search is needed along with longer wave flumes with much larger amplitudes and lower frequency waves. A further analysis of phases generated from probe data via the analytic signal approach (using the Hilbert transform) shows a strong triad phase synchronisation at the resonance, thus providing independent experimental evidence of the resonance.

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Analysis of PFG Anomalous Diffusion via Real-Space and Phase-Space Approaches

Mathematics ◽

10.3390/math6020017 ◽

2018 ◽

Vol 6 (2) ◽

pp. 17 ◽

Cited By ~ 3

Author(s):

Guoxing Lin

Keyword(s):

Phase Space ◽

Anomalous Diffusion ◽

Real Space

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Wave action on currents with vorticity

Journal of Fluid Mechanics ◽

10.1017/s0022112099004565 ◽

1999 ◽

Vol 386 ◽

pp. 329-344 ◽

Cited By ~ 13

Author(s):

BENJAMIN S. WHITE

Keyword(s):

Phase Shift ◽

Deep Water ◽

Current Velocity ◽

Wave Action ◽

Ray Theory ◽

Wkb Method ◽

Interaction Of Waves ◽

New Equation ◽

Spatially Varying

The interaction of waves on deep water with spatially varying currents may be described by a ray theory, with the wave amplitudes determined by the principle of conservation of wave action (CWA). However, all previous deep water derivations of CWA are restricted to the case of an irrotational current. In this paper, both the ray theory and CWA are derived by a WKB method without the assumption of irrotationality. Also derived is a new equation for a spatially varying phase shift which is not predicted by the usual ray theory, and which, in general, displaces the positions of the wave crests by a distance on the order of a wavelength. This phase shift, which is caused by variations of the current velocity with depth, vanishes in the irrotational case, and so is in accord with the irrotational theory.

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Steep gravity waves in water of arbitrary uniform depth

Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences ◽

10.1098/rsta.1977.0113 ◽

1977 ◽

Vol 286 (1335) ◽

pp. 183-230 ◽

Cited By ~ 215

Keyword(s):

Gravity Waves ◽

Deep Water ◽

Water Waves ◽

Perturbation Parameter ◽

Intermediate Energy ◽

Finite Amplitude ◽

Wave Profile ◽

Accurate Calculation ◽

Theoretical Developments ◽

Deep Water Waves

Modern applications of water-wave studies, as well as some recent theoretical developments, have shown the need for a systematic and accurate calculation of the characteristics of steady, progressive gravity waves of finite amplitude in water of arbitrary uniform depth. In this paper the speed, momentum, energy and other integral properties are calculated accurately by means of series expansions in terms of a perturbation parameter whose range is known precisely and encompasses waves from the lowest to the highest possible. The series are extended to high order and summed with Padé approximants. For any given wavelength and depth it is found that the highest wave is not the fastest. Moreover the energy, momentum and their fluxes are found to be greatest for waves lower than the highest. This confirms and extends the results found previously for solitary and deep-water waves. By calculating the profile of deep-water waves we show that the profile of the almost-steepest wave, which has a sharp curvature at the crest, intersects that of a slightly less-steep wave near the crest and hence is lower over most of the wavelength. An integration along the wave profile cross-checks the Padé-approximant results and confirms the intermediate energy maximum. Values of the speed, energy and other integral properties are tabulated in the appendix for the complete range of wave steepnesses and for various ratios of depth to wavelength, from deep to very shallow water.

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Effective Power and Speed Loss of Underwater Vehicles in Close Proximity to Regular Waves

Volume 7A: Ocean Engineering ◽

10.1115/omae2017-62056 ◽

2017 ◽

Author(s):

Stefan Daum ◽

Martin Greve ◽

Renato Skejic

Keyword(s):

Free Surface ◽

Deep Water ◽

Water Waves ◽

Underwater Vehicles ◽

Total Resistance ◽

Wave Load ◽

Regular Waves ◽

Effective Power ◽

Close Proximity ◽

Deep Water Waves

The present study is focused on performance issues of underwater vehicles near the free surface and gives insight into the analysis of a speed loss in regular deep water waves. Predictions of the speed loss are based on the evaluation of the total resistance and effective power in calm water and preselected regular wave fields w.r.t. the non-dimensional wave to body length ratio. It has been assumed that the water is sufficiently deep and that the vehicle is operating in a range of small to moderate Froude numbers by moving forward on a straight-line course with a defined encounter angle of incident regular waves. A modified version of the Doctors & Days [1] method as presented in Skejic and Jullumstrø [2] is used for the determination of the total resistance and consequently the effective power. In particular, the wave-making resistance is estimated by using different approaches covering simplified methods, i.e. Michell’s thin ship theory with the inclusion of viscosity effects Tuck [3] and Lazauskas [4] as well as boundary element methods, i.e. 3D Rankine source calculations according to Hess and Smith [5]. These methods are based on the linear potential fluid flow and are compared to fully viscous finite volume methods for selected geometries. The wave resistance models are verified and validated by published data of a prolate spheroid and one appropriate axisymmetric submarine model. Added resistance in regular deep water waves is obtained through evaluation of the surge mean second-order wave load. For this purpose, two different theoretical models based on potential flow theory are used: Loukakis and Sclavounos [6] and Salvesen et. al. [7]. The considered theories cover the whole range of important wavelengths for an underwater vehicle advancing in close proximity to the free surface. Comparisons between the outlined wave load theories and available theoretical and experimental data were carried out for a submerged submarine and a horizontal cylinder. Finally, the effective power and speed loss are discussed from a submarine operational point of view where the mentioned parameters directly influence mission requirements in a seaway. All presented results are carried out from the perspective of accuracy and efficiency within common engineering practice. By concluding current investigations in regular waves an outlook will be drawn to the application of advancing underwater vehicles in more realistic sea conditions.

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