A specific, genuinely three-dimensional mechanism of rogue wave formation, in a late stage of the modulational instability of a perturbed Stokes deep-water wave, is recognized through numerical experiments. The simulations are based on fully nonlinear equations describing weakly three-dimensional potential flows of an ideal fluid with a free surface in terms of conformal variables. Spontaneous formation of zigzag patterns for wave amplitude is observed in a nonlinear stage of the instability. If initial wave steepness is sufficiently high ($ka>0.06$), these coherent structures produce rogue waves. The most tall waves appear in ``turns'' of the zigzags. For $ka<0.06$, the structures decay typically without formation of steep waves.
We investigate the Benjamin-Feir (or modulational) instability of Stokes waves, i.e., small-amplitude, one-dimensional periodic gravity waves of permanent form and constant velocity, in water of finite and infinite depth. We develop a perturbation method to describe to high-order accuracy the unstable spectral elements associated with this instability, obtained by linearizing Euler's equations about the small-amplitude Stokes waves. These unstable elements form a figure-eight curve centered at the origin of the complex spectral plane, which is parameterized by a Floquet exponent. Our asymptoti…
We prove that a Stokes' periodic wave of sufficiently small amplitude, traveling under gravity at the free surface of a two dimensional, infinitely deep, and irrotational flow, is spectrally unstable to slow modulation, rigorously justifying Benjamin and Feir's formal argument.
We develop a general framework to describe the cubically nonlinear interaction of a unidirectional degenerate quartet of deep-water gravity waves. Starting from the discretised Zakharov equation, and thus without restriction on spectral bandwidth, we derive a planar Hamiltonian system in terms of the dynamic phase and a modal amplitude. This is characterised by two free parameters: the wave action and the mode separation between the carrier and the side-bands. The mode separation serves as a bifurcation parameter, which allows us to fully classify the dynamics. Centres of our system correspond…
We study a three-wave truncation of a recently proposed damped/forced high-order nonlinear Schrödinger equation for deep-water gravity waves under the effect of wind and viscosity. The evolution of the norm (wave-action) and spectral mean of the full model are well captured by the reduced dynamics. Three regimes are found for the wind-viscosity balance: we classify them according to the attractor in the phase-plane of the truncated system and to the shift of the spectral mean. A downshift can coexist with both net forcing and damping, i.e., attraction to period-1 or period-2 solutions. Upshift…
The nonlinear stage of the modulational (Benjamin - Feir) instability of unidirectional deep water surface gravity waves is simulated numerically by the firth-order nonlinear envelope equations. The conditions of steep and breaking waves are concerned. The results are compared with the solution of the full potential Euler equations and with the lower order envelope models (the 3-order nonlinear Schrodinger equation and the standard 4-order Dysthe equations). The generalized Dysthe model is shown to exhibit the tendency to re-stabilization of steep waves with respect to long perturbations.
It is proven that small-amplitude steady periodic water waves with infinite depth are unstable with respect to long-wave perturbations. This modulational instability was first observed more than half a century ago by Benjamin and Feir. It has been proven rigorously only in the case of finite depth. We provide a completely different and self-contained approach to prove the spectral modulational instability for water waves in both the finite and infinite depth cases.
This thesis deals with mathematical and physical aspects of deterministic freak wave generation in a hydrodynamic laboratory. We adopt the nonlinear Schrödinger (NLS) equation as a mathematical model for the evolution of the surface gravity wave packet envelopes. As predicted theoretically and observed experimentally by Benjamin and Feir (1967), Stokes waves on deep water are unstable under a sideband modulation. The plane-wave solution of the NLS equation, which represents the fundamental component of the Stokes wave, experiences a similar modulational instability under the linear perturbatio…
