Nonlinear stage of the Benjamin-Feir instability: Three-dimensional coherent structures and rogue waves

Abstract

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.

11 pages, 7 figures, submitted to PRL

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• Two different kinds of rogue waves in weakly-crossing sea states
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• Anomalous wave as a result of the collision of two wave groups on sea surface
• Canonical variables for steep planar water waves over nonuniform bed
• Extreme Events in Nonlinear Lattices
• Cross-phase modulation instability in mode-locked laser based on reduced graphene oxide