  We investigate the effect of a light turbulent wind on a liquid surface,
below the onset of wave generation. In that regime, the liquid surface is
populated by small disorganised deformations elongated in the streamwise
direction. Formally identified recently by Paquier et al. (2015), the
deformations that occur below the wave onset were named wrinkles. We provide
here a theoretical framework for this wrinkle regime, using the viscous
response of a free surface liquid submitted to arbitrary normal and tangential
interfacial stresses at its upper boundary. We relate the spatio-temporal
spectrum of the surface deformations to that of the applied interfacial
pressure and shear stress fluctuations. For that, we evaluate the
spatio-temporal statistics of the turbulent forcing using Direct Numerical
Simulation of a turbulent air channel flow, assuming no coupling between the
air and the liquid flows. Combining theory and numerical simulation, we thus
obtain synthetic wrinkles that reproduce previous experimental investigations.
We show that the wrinkles are a multi-scale superposition of random wakes
generated by the turbulent fluctuations. They result mainly from the nearly
isotropic pressure fluctuations generated in the boundary layer, rather than
from the elongated shear stress fluctuations. The wrinkle regime described in
this paper naturally arises as the viscous-saturated asymptotic of the inviscid
growth theory of Phillips (1957). Experiments indicate that the onset of
exponential wave growth depends on the liquid viscosity. Our theory suggests
that the empirical criterion for the onset is satisfied when the wrinkle
amplitude reaches a given fraction of the viscous sublayer thickness. It
indicates that the turbulent fluctuations near the onset may play a role in the
triggering of exponential wave growth.
