  Asymptotic multi-layer analyses and computation of solutions for turbulent
flows over steady and unsteady monochromatic surface wave are reviewed, in the
limits of low turbulent stresses and small wave amplitude. The structure of the
flow is defined in terms of asymptotically-matched thin-layers, namely the
surface layer and a critical layer, whether it is elevated or immersed,
corresponding to its location above or within the surface layer. The results
particularly demonstrate the physical importance of the singular flow features
and physical implications of the elevated critical layer in the limit of the
unsteadiness tending to zero. These agree with the variational mathematical
solution of Miles (1957) for small but finite growth rate, but they are not
consistent physically or mathematically with his analysis in the limit of
growth rate tending to zero. As this and other studies conclude, in the limit
of zero growth rate the effect of the elevated critical layer is eliminated by
finite turbulent diffusivity, so that the perturbed flow and the drag force are
determined by the asymmetric or sheltering flow in the surface shear layer and
its matched interaction with the upper region. But for groups of waves, in
which the individual waves grow and decay, there is a net contribution of the
elevated critical layer to the wave growth. Critical layers, whether elevated
or immersed, affect this asymmetric sheltering mechanism, but in quite a
different way to their effect on growing waves. These asymptotic multi-layer
methods lead to physical insight and suggest approximate methods for analyzing
higher amplitude and more complex flows, such as flow over wave groups.
