Frequency downshift in a viscous fluid

TL;DR

This paper introduces a viscous generalization of the Dysthe system to model weakly viscous water waves, demonstrating its ability to replicate frequency downshifting observed in experiments.

Contribution

It derives the viscous Dysthe system from Euler equations, incorporating viscosity effects and providing a mechanism for frequency downshifting without wind or breaking.

Findings

Spectral mean decreases over time in simulations.

Waves with wave numbers near zero decay more slowly.

The model aligns well with experimental data on frequency downshifting.

Abstract

In this paper, we derive a viscous generalization of the Dysthe (1979) system from the weakly viscous generalization of the Euler equations introduced by Dias, Dyachenko, and Zakharov (2008). This "viscous Dysthe" system models the evolution of a weakly viscous, nearly monochromatic wave train on deep water. It contains a term which provides a mechanism for frequency downshifting in the absence of wind and wave breaking. The equation does not preserve the spectral mean. Numerical simulations demonstrate that the spectral mean typically decreases and that the spectral peak decreases for certain initial conditions. The linear stability analysis of the plane-wave solutions of the viscous Dysthe system demonstrates that waves with wave numbers closer to zero decay more slowly than waves with wave numbers further from zero. Comparisons between experimental data and numerical simulations of the NLS, dissipative NLS, Dysthe, and viscous Dysthe systems establish that the viscous Dysthe system accurately models data from experiments in which frequency downshifting was observed and experiments in which frequency downshift was not observed.