An asymptotic model for the Kelvin-Helmholtz and Miles mechanisms of water wave generation by wind

The generalized Kelvin-Helmholtz (KH) and Miles mechanisms of the water wave generation by wind are investigated for two-layer piecewise linear model of the wind profile. It is shown by asymptotic expansions in small air-to-water density ratio that two mechanisms of the instability operate in quite different scales. Miles' short waves are generated by weak winds, in particular, Miles' regime is responsible for initiation of the instability at the minimum wind speed, while the generalized KH regime dominates at strong winds and raises moderately short waves.