  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.
