In studies of the turbulent air flow over water waves it is usually assumed that
the effect of viscosity near the water surface is negligible, i.e. the Reynolds number,
Re = u∗λ/v, is considered to be high. However, for short waves or low wind speeds this
assumption is not valid. Therefore, a second-order turbulence closure that takes into
account viscous effects is used to simulate the air flow. The model shows reasonable
agreement with laboratory measurements of wave-induced velocity profiles. Next, the
dependence of the dimensionless energy flux from wind to waves, or growth rate,
on Re is investigated. The growth rate of waves that are slow compared to the
wind is found to increase strongly when Re decreases below 104, with a maximum
around Re = 800. The numerical model predictions are in good agreement with
analytical theories and laboratory observations. Results of the study are useful in
field conditions for the short waves in the spectrum, which are particularly important
for remote sensing applications.