On the linearity of wave energy temporal growth in the initial stage of Phillips theory

In 1957, Phillips proposed a prominent theory to elucidate the role of the resonance mechanism in the initial growth of surface waves from a flat air-water interface. The resonance between turbulent airflow and water deformation occurs when the convection velocity of air pressure fluctuations at the water surface matches the phase velocity of water waves. Near the resonance curve in the wavenumber space, the wave energy spectrum component grows quadratically with time. The overall temporal evolution of surface elevation variance exhibits a linear feature. Here, we present numerical evidence on the resonance mechanism in the initial stage of the Phillips theory based on direct numerical simulation data. We also provide a rigorous proof of the linear growth of surface elevation variance in the initial stage. We show that for gravity waves the leading order term of surface elevation variance is bounded by a linear function of time from an asymptotic analysis.