Wave Motion Induced By Turbulent Shear Flows Over Growing Stokes Waves

The recent analytical of multi-layer analyses proposed by Sajjadi et al. (J Eng Math 84:73, 2014) (SHD14 therein) is solved numerically for atmospheric turbulent shear flows blowing over growing (or unsteady) Stokes (bimodal) water waves, of low-to-moderate steepness. For unsteady surface waves, the amplitude a(t)∝ekcita(t)∝ekcit, where kcikci is the wave growth factor, k is the wavenumber, and cici is the complex part of the wave phase speed, and thus, the waves begin to grow as more energy is transferred to them by the wind. This will then display the critical height to a point, where the thickness of the inner layer kℓikℓi becomes comparable to the critical height kzckzc, where the mean wind shear velocity U(z) equals the real part of the wave speed crcr. It is demonstrated that as the wave steepens further the inner layer exceeds the critical layer, and beneath the cat’s-eye, there is a strong reverse flow which will then affect the surface drag, but at the surface, the flow adjusts itself to the orbital velocity of the wave. We show that in the limit as cr/U∗cr/U∗ is very small, namely, slow-moving waves (i.e., for waves traveling with a speed crcr which is much less than the friction velocity U∗U∗), the energy-transfer rate to the waves, ββ (being proportional to momentum flux from wind to waves), computed here using an eddy-viscosity model, agrees with the asymptotic steady-state analysis in Belcher and Hunt (J Fluid Mech 251:109, 1993), and the earlier model in Townsend (J Fluid Mech 98:171, 1980). The non-separated sheltering flow determines the drag and the energy transfer and not the weak critical shear layer within the inner shear layer