  We study the weakly non-linear development of shear-driven gravity waves, and
investigate the mixing properties of the finite amplitude solutions.
Calculations to date have been restricted to the linear theory, which predicts
that gravity waves are amplified by an influx of energy through the critical
layer, where the velocity of the wind equals the wave phase velocity. Because
of the presence of a critical layer, ordinary weakly non-linear methods fail;
in this paper, we use a rescaling at the critical layer and matched asymptotics
to derive an amplitude equation for the most unstable wave, under the
simplifying assumption that the physical domain is periodic. These amplitude
equations are solved numerically, in their quasi-steady limit, for the cases of
small density ratio (applicable to oceanography), and for arbitrary density
ratio but strong stratification (for more general physical/astrophysical
situations). In addition to the familiar asymptotic growth found in other
inviscid flow, we find that, for the air over water case (provided the maximum
wind velocity is in the range of $0.2\mpers \sim 1\mpers$), the single mode
transitions from exponential to algebraic growth when the amplitude of the wave
is as small as $h\sim 10^{-5}\lambda$; hence, it may be difficult to observe
the linear regime for this case in numerical simulations. We also find that the
weakly non-linear flow allows for super-diffusive particle transport with an
exponent $\sim 3/2$, consistent with Venkataramani's results.
