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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 $$a(t)\propto e^{kc_it}$$a(t)∝ekcit, where $$kc_i$$kci is the wave growth factor, k is the wavenumber, and $$c_i$$ci 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 $$k\ell _i$$kℓi becomes comparable to the critical height $$kz_c$$kzc, where the mean wind shear velocity U(z) equals the real part of the wave speed $$c_r$$cr. 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 $$c_r/U_*$$cr/U∗ is very small, namely, slow-moving waves (i.e., for waves traveling with a speed $$c_r$$cr which is much less than the friction velocity $$U_*$$U∗), the energy-transfer rate to the waves, $$\beta $$β (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. Computations for the cases when the waves are traveling faster (i.e., when $$c_r>U_*$$cr>U∗) and growing significantly (i.e., when $$0<c_i/U_*$$0<ci/U∗) show critical shear layer forms outside the inner surface shear layer for steeper waves. Analysis, following Miles (J Fluid Mech 3:185; 1957, J Fluid Mech 256:427, 1993) and SHD14, shows that the critical layer produces a significant but not the dominant effect; a weak lee-side jet is formed by the inertial dynamics in the critical layer, which adds to the drag produced by the sheltering effect. The latter begins to decrease when $$c_r$$cr is significantly exceeds $$U_*$$U∗, as has been verified experimentally. Over peaked waves, the inner layer flow on the lee side tends to slow and separate, which over a growing fast wave deflects the streamlines and the critical layer upwards on the lee side. This also tends to increase the drag and the magnitude of energy-transfer rate $$\beta $$β (from wind to waves). These complex results, computed with relatively simple turbulence closure model, agree broadly with DNS simulations in Sullivan et al. (J Fluid Mech 404:47, 2000). It is proposed, using an earlier study SHD14, that the mechanisms identified here for wave-induced motion contributes to a larger net growth of wind-driven water waves when the waves are non-linear (e.g., bimodal waves) compared with growth rates for monochromatic waves. This is because in non-linear waves, individual harmonics have stronger positive and weaker negative growth rates.

Wave motion induced by turbulent shear flows over growing Stokes waves