  The linear stability of the laminar boundary layer flow of a Stokes wave in
deep waters is investigated by means of a 'momentary' criterion of instability
for unsteady flows (Blondeaux and Seminara, 1979). In the parameter range
investigated, it is found that the flow is stable to 2-D perturbations. The
least stable eigenmode of the resulting Orr-Sommerfield spectrum attains its
maximum beneath the boundary layer of the Stokes wave. Moreover, an analysis of
the associated pseudospectrum indicates that the laminar flow when modified by
imperfections is unstable due to the non-normality of the Orr-Sommerfeld
operator, and the unstable pseudo-eigenmodes tend to peak within the boundary
layer. The laminar flow of the Stokes wave is also stable to 3-D
streamwise-independent perturbations. Instability is observed for the laminar
flow with imperfections. The associated unstable pseudo-eigenmodes are
streamwise vortical rolls similar to Langmuir cells. The laminar boundary layer
flow of a Stokes wave appears to be stable to infinitesimal perturbations, but
it may likely be unstable to finite perturbations, as in Poiseuille pipe flows.
The present results are supportive of the recent experimental evidence of
spontaneous occurrence of turbulence beneath unforced non-breaking surface
waves.
