Wind wave growth in the viscous regime

We investigate the growth of short gravity-capillary waves due to wind forcing, solving the two-phase Navier-Stokes equations. The numerical method features a momentum conserving scheme, interface reconstruction using volume of fluid, and adaptive mesh refinement. A 2D laminar wind profile is used to force short gravity-capillary waves in the viscous regime, and the growth of the wave amplitude and subsurface drift layer are analyzed. The threshold for wave growth is found to depend on a balance between the growth rate and viscous dissipation rate, while the wave growth for all data can be described as a scaling depending on wind stress and a viscous correction accounting for the growth threshold. Together with the wave growth, the subsurface drift layer develops and can be described in terms of a similarity solution. The nonlinear stage of wave growth is discussed for increasing wavelength, and we recover steep capillary waves, parasitic capillary waves, and spilling breakers depending on the ratio of gravity to surface tension forces.