  We investigate numerically the influence of a weak current on wind-generated
surface deformations for wind velocity below the onset of regular waves. In
that regime, the liquid surface is populated by small disorganised deformations
elongated in the wind direction, referred to as wrinkles. These wrinkles are
the superposition of incoherent wakes generated by the pressure fluctuations
traveling in the turbulent boundary layer in the air. In this work, we account
for the effect of a weak sheared current in the liquid, either longitudinal or
transverse, by introducing a modified Doppler-shifted dispersion relation to
lowest order in viscosity and current in the spectral theory previously derived
by Perrard et al (2019). This theory describes the simplified one-way problem
of surface deformations excited by a prescribed turbulent forcing, thereby
neglecting the retroaction of waves on turbulence in the air. The forcing is
taken from a set of direct numerical simulations of a turbulent channel flow.
We determine the wrinkle properties (size and amplitude) as a function of the
liquid viscosity and current properties (surface velocity, thickness and
orientation). We find significant modifications of the wrinkle geometry by the
currents: the wrinkles are tilted for a transverse current, and show finer
scales for a longitudinal current. However, their characteristic size is weakly
affected, and their amplitude remains independent of the current. We discuss
the implications of these results on the onset of regular waves at larger wind
velocity. In this work, we introduce a spectral interpolation method to
evaluate the surface deformation fields, based on a refined meshing close to
the dispersion relation of the waves. This method, which can be extended to any
dispersive system excited by a random forcing, strongly reduces the
discretization effects at a low computational cost.

Fluid Dynamics