Frequency downshift in a viscous fluid

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 et al. (2008). This “viscous Dysthe” system models the evolution of a weakly viscous, nearly monochromatic wave train on deep water. It includes only one free parameter, which can be determined empirically. It contains a term that provides a mechanism for frequency downshifting in the absence of wind and wave breaking. The system 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 frequencies closer to zero decay more slowly than waves with frequencies further from zero. Comparisons between experimental data and numerical simulations of the nonlinear Schrödinger, dissipative nonlinear Schrödinger, 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.