Nonlinear stage of Benjamin-Feir instability in forced/damped deep water waves

TL;DR

This paper investigates the nonlinear behavior of deep-water waves under wind and viscosity effects using a three-wave model, revealing regimes of spectral shifts and attractor dynamics consistent with experimental observations.

Contribution

It introduces a three-wave truncation model for a damped/forced high-order nonlinear Schrödinger equation, providing new insights into wave spectral shifts and attractor regimes.

Findings

Identification of three wind-viscosity regimes based on attractor dynamics

Coexistence of spectral downshift with net forcing and damping

Validation of the model's applicability to long wave-tank experiments

Abstract

We study a three-wave truncation of a recently proposed damped/forced high-order nonlinear Schr\"odinger 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 is associated with stronger winds, i.e., to a net forcing where the attractor is always a period-1 solution. The applicability of our classification to experiments in long wave-tanks is verified.