Numerical and analytical studies of wind-wave growth

In 1957, Miles proposed a linear theory for the growth of wind-induced small ripples on water. In the framework of hydrodynamic stability, these ripples are regarded as perturbations of the air flow. There is an energy transfer from the wind to the waves in the critical layer, which is the height where the phase speed is equal to the wind speed. We unify the many approaches of Miles' theory and provide a physical mechanism for the wave growth and we develop a simple method to compute the growth rate for an arbitrary wind profile. With this numerical scheme, we revisit the work of Morland and Saffman (1993) on Squire's theorem for the Miles instability, and study various turbulent boundary layer profiles as wind models. Moreover, for short and long waves we obtain uniform approximations of the perturbation stream function by the method of Matched Asymptotic Expansions, and infer approximate formulae for the growth rate. Finally, we simulate a quasi-linear model proposed by Janssen (1982), which takes into account the feedback of growing waves on the wind.