Extreme wave phenomena in down-stream running modulated waves

TL;DR

This paper investigates the formation of extreme waves in downstream modulated surface gravity waves using the NLS equation, revealing phase singularities and wave dislocations that distinguish rogue waves from smaller waves, and proposes a laboratory generation strategy.

Contribution

It introduces a detailed analysis of extreme wave phenomena using explicit NLS solutions, highlighting phase singularities and wave dislocations as key mechanisms.

Findings

Phase singularities occur at large waves for long modulation lengths.

Wave dislocations cause large amplitude differences and energy flow reversal.

A strategy for generating rogue waves in laboratories is proposed.

Abstract

Modulational, Benjamin-Feir, instability is studied for the down-stream evolution of surface gravity waves. An explicit solution, the soliton on finite background, of the NLS equation in physical space is used to study various phenomena in detail. It is shown that for sufficiently long modulation lengths, at a unique position where the largest waves appear, phase singularities are present in the time signal. These singularities are related to wave dislocations and lead to a discrimination between successive `extreme' waves and much smaller intermittent waves. Energy flow in opposite directions through successive dislocations at which waves merge and split, causes the large amplitude difference. The envelope of the time signal at that point is shown to have a simple phase plane representation, and will be described by a symmetry breaking unfolding of the steady state solutions of NLS. The results are used together with the maximal temporal amplitude MTA, to design a strategy for the generation of extreme (freak, rogue) waves in hydrodynamic laboratories.