Viscous shear instability at air-water interface as a function of wind velocity profile

Recently, we demonstrated that the temporal fetch-dependent wind-wave growth under abruptly applied wind forcing can be accurately described by considering a stochastic ensemble of multiple unstable harmonics (submitted to PRL). In that study, the two-phase viscous shear flow instability at the air-water interface was examined using the energy growth rates β and the group velocities c_g of the unstable harmonics obtained by solving the coupled Orr-Sommerfeld (OS) equations in air and water with appropriate boundary and initial conditions. The predictions of this unidirectional model compare well with measurements of random time-and space-dependent wave field performed in our laboratory (JFM 828, 459, 2017). The eigenvalues of the model equations determine β and c_g of each harmonic defined by its wavenumber; the suggested model then allows computation of variation with time and with fetch of the statistical wave field parameters such as the characteristic wave amplitude and the instantaneous dominant frequency. The eigenvalues of the OS system however depend strongly on the adopted mean vertical velocity profile in air and in water. The water velocity is assumed to decay exponentially with depth from the maximum value corresponding to the drift velocity at the interface. In air, we assumed the lin-log suggested by Miles that consists of a linear segment in the viscous sublayer connected smoothly to a logarithmic turbulent velocity profile over smooth water surface. The assumption of smooth water surface is reasonable at the onset of wind. However, emerging wind-waves render the surface rough; the surface roughness becomes more pronounced at higher wind forcing and larger fetches. In the present study, we extend our previous study and apply the developed OS solver to investigate the dependence of the viscous shear-flow stability on the shape of air velocity profile. We take advantage of the detailed wind-velocity profiles measured in our facility at various wind velocities and a number of fetches (JGR 117, C00J19, 2012)that demonstrated the significant deviations of the actual air velocity profiles over waves from the shape corresponding to smooth-surface. The surface drift velocities under different operational conditions were also measured. The effect of the evolving wind-wave field on eigenvalues of the OS system of equation and thus on the domains of instability, the energy growth rates β and the group velocities c_g is studied. These results extend our understanding of the interrelation between the varying in time and space wind-wave field and the turbulent airflow above the water surface and shed light on momentum and energy exchange between air and water.