Effect of viscosity on wind-driven gravitation waves

We address the question of how viscosity impacts the growth of gravitation waves, such as those on the ocean, when they are driven by wind. There is so far no general rigorous theory for this energy transfer. We extend Miles' approach [J. W. Miles, "On the generation of surface waves by shear flows," J. Fluid Mech. 3, 185-204 (1957)], using the same logarithmic wind profile, to incorporate bulk viscosity and derive modified growth rates. Exploiting the fact that water waves fall into the "weak viscosity" regime, we produce analytical expressions for the growth rate, which we solve using the numerical method proposed by Beji and Nadaoka ["Solution of Rayleigh's instability equation for arbitrary wind profiles," J. Fluid Mech. 500, 65-73 (2004)]. Our results confirm that corrections to the growth rates are significant for wavelengths below a meter, and for weak to modest wind strengths. We show that all wave growth is suppressed, due to viscous effects, below a critical wind strength. We also show that the wave age corresponding to a developed sea is reduced by viscosity. We quantitatively characterize the zones, in terms of wind strength and wavelength, for which the wave growth is suppressed by viscosity.