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The linear stability of the laminar boundary layer flow of a Stokes wave indeep waters is investigated by means of a 'momentary' criterion of instabilityfor unsteady flows (Blondeaux and Seminara, 1979). In the parameter rangeinvestigated, it is found that the flow is stable to 2-D perturbations. Theleast stable eigenmode of the resulting Orr-Sommerfield spectrum attains itsmaximum beneath the boundary layer of the Stokes wave. Moreover, an analysis ofthe associated pseudospectrum indicates that the laminar flow when modified byimperfections is unstable due to the non-normality of the Orr-Sommerfeldoperator, and the unstable pseudo-eigenmodes tend to peak within the boundarylayer. The laminar flow of the Stokes wave is also stable to 3-Dstreamwise-independent perturbations. Instability is observed for the laminarflow with imperfections. The associated unstable pseudo-eigenmodes arestreamwise vortical rolls similar to Langmuir cells. The laminar boundary layerflow of a Stokes wave appears to be stable to infinitesimal perturbations, butit may likely be unstable to finite perturbations, as in Poiseuille pipe flows.The present results are supportive of the recent experimental evidence ofspontaneous occurrence of turbulence beneath unforced non-breaking surfacewaves.

On the stability of the laminar boundary layer beneath a Stokes wave