Extreme wave events at the initial stage of the modulational instability in a forced and damped wave medium

TL;DR

This paper investigates the initial formation of extreme wave events during the early phase of modulational instability in a forced and damped wave medium, using a specialized nonlinear model based on $S$-theory.

Contribution

It introduces a model that captures the nonlinear stage of modulational instability considering external forcing and dissipation, highlighting the emergence of abnormal waves initially.

Findings

Abnormal waves appear predominantly at the initial stage of instability.

The amplitude of abnormal waves decreases as the instability develops.

The model accounts for interactions between modes with slowly varying phases.

Abstract

The modulational instability of waves in a medium under the action of an external monochromatic force and dissipation is considered. The model which describes the nonlinear stage of the modulation instability was constructed with using methods of so-called $S$-theory. Following this theory, we take into account only interactions between modes with slowly varying relative phases. Within framework of this model, it is shown that the waves satisfying the criterion of abnormality appears as a rule at the initial stage of the instability. With further development of the process, the abnormal waves can be also observed but their amplitudes decrease gradually with decreasing of the average wave height.