  The recent analytical of multi-layer analyses proposed by Sajjadi, Hunt and
Drullion (2014) (SHD14) is solved numerically for atmospheric turbulent shear
fows blowing over unsteady Stokes water waves, of low to moderate steepness. 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_* is
very small, namely slow moving waves the energy-transfer rate to the waves,
\beta, computed here using an eddy-viscosity model, agrees with the asymptotic
steady state analysis of Belcher and Hunt (1993) and the earlier model of
Townsend (1980). Computations for the cases when the waves are traveling faster
and growing significantly show a critical shear layer forms outside the inner
surface shear layer for steeper waves. We show 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. Over peaked waves, the inner layer flow on the lee-side
tends to slow and separate, which over a growing fast wave defects the
streamlines and the critical layer upwards on the lee side. Hence, 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 compared with growth rates for
monochromatic waves. This is because in non- linear waves individual harmonics
have stronger positive and weaker negative growth rates.
