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Register a new userNonlinear dynamics of trapped waves on jet currents and rogue waves
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Nonlinear dynamics of surface gravity waves trapped by an opposing jet current is studied analytically and numerically. For wave fields narrowband in frequency but not necessarily with narrow angular distributions the developed asymptotic weakly nonlinear theory based on the modal approach of (V. Shrira, A. Slunyaev, J. Fluid. Mech, 738, 65, 2014) leads to the one-dimensional modified nonlinear Schr{o}dinger equation of self-focusing type for a single mode. Its solutions such as envelope solitons and breathers are considered to be prototypes of rogue waves; these solutions, in contrast to waves in the absence of currents, are robust with respect to transverse perturbations, which suggests potentially higher probability of rogue waves. Robustness of the long-lived analytical solutions in form of the modulated trapped waves and solitary wave groups is verified by direct numerical simulations of potential Euler equations.
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We show experimentally that a stable wave propagating into a region characterized by an opposite current may become modulationaly unstable. Experiments have been performed in two independent wave tank facilities; both of them are equipped with a wavemaker and a pump for generating a current propagating in the opposite direction with respect to the waves. The experimental results support a recent conjecture based on a current-modified Nonlinear Schrodinger equation which establishes that rogue waves can be triggered by non-homogeneous current characterized by a negative horizontal velocity gradient.
A two-dimensional water wave system is examined consisting of two discrete incompressible fluid domains separated by a free common interface. In a geophysical context this is a model of an internal wave, formed at a pycnocline or thermocline in the ocean. The system is considered as being bounded at the bottom and top by a flatbed and wave-free surface respectively. A current profile with depth-dependent currents in each domain is considered. The Hamiltonian of the system is determined and expressed in terms of canonical wave-related variables. Limiting behaviour is examined and compared to that of other known models. The linearised equations as well as long-wave approximations are presented.
The formation of extreme localizations in nonlinear dispersive media can be explained and described within the framework of nonlinear evolution equations, such as the nonlinear Schrodinger equation (NLS). Within the class of exact NLS breather solutions on finite background, which describe the modulational instability of monochromatic wave trains, the hierarchy of both in time and space localized rational solutions are considered to be appropriate prototypes to model rogue wave dynamics. Here, we use the time-reversal invariance of the NLS to propose and experimentally demonstrate a new approach to construct strongly nonlinear localized waves focused both in time and space. The potential areas of applications of this time-reversal approach range from remote sensing to motivated analogous experimental analysis in other nonlinear dispersive media, such as optics, Bose-Einstein condensates and plasma, where the wave motion dynamics is governed by the NLS.
We review the study of rogue waves and related instabilities in optical and oceanic environments, with particular focus on recent experimental developments. In optics, we emphasize results arising from the use of real-time measurement techniques, whilst in oceanography we consider insights obtained from analysis of real-world ocean wave data and controlled experiments in wave tanks. Although significant progress in understanding rogue waves has been made based on an analogy between wave dynamics in optics and hydrodynamics, these comparisons have predominantly focused on one-dimensional nonlinear propagation scenarios. As a result, there remains significant debate about the dominant physical mechanisms driving the generation of ocean rogue waves in the complex environment of the open sea. Here, we review state-of-the-art of rogue wave studies in optics and hydrodynamics, aiming to clearly identify similarities and differences between the results obtained in the two fields. In hydrodynamics, we take care to review results that support both nonlinear and linear interpretations of ocean rogue wave formation, and in optics, we also summarise results from an emerging area of research applying the measurement techniques developed for the study of rogue waves to dissipative soliton systems. We conclude with a discussion of important future research directions.
We study dispersion properties of linear surface gravity waves propagating in an arbitrary direction atop a current profile of depth-varying magnitude using a piecewise linear approximation, and develop a robust numerical framework for practical calculation. The method has been much used in the past for the case of waves propagating along the same axis as the background current, and we herein extend and apply it to problems with an arbitrary angle between the wave propagation and current directions. Being valid for all wavelengths without loss of accuracy, the scheme is particularly well suited to solve problems involving a broad range of wave vectors, such as ship waves and Cauchy-Poisson initial value problems for example. We examine the group and phase velocities over different wavelength regimes and current profiles, highlighting characteristics due to the depth-variable vorticity. We show an example application to ship waves on an arbitrary current profile, and demonstrate qualitative differences in the wake patterns between concave down and concave up profiles when compared to a constant shear profile with equal depth-averaged vorticity. We also discuss the nature of additional solutions to the dispersion relation when using the piecewise-linear model. These are vorticity waves, drifting vortical structures which are artifacts of the piecewise model. They are absent for a smooth profile and are spurious in the present context.
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