Effect of an Insoluble Surface Film on the Drift Velocity of Capillary–Gravity Waves

Abstract

The drift velocity due to capillary-gravity waves in a deep ocean is investigated theoretically. The surface is covered by an insoluble, inextensible film, and the analysis is based an a Lagrangian description of motion. Attenuated as well as nondecaying, or permanent warm, are discussed. The strong temporal attenuation due to the inextensible film is shown to have a profound influence on the drift problem. It causes the induced mean virtual wave stress at the surface to decay quite rapidly, thereby limiting strongly the Growth of the Eulerian part of the drift current. The drift problem for permanent waves is demonstrated to fall into two different categories, depending on the boundary conditions to second order (i) If the mean tangential wind stress vanishes, our results confirm Craik's criticism of the analyses by Phillips and (ii) if tie mean horizontal wind stress vanishes, there is an increased shear in the viscous boundary layer at the surface, as suggested by Phillips. Below the boundary layer the mean flow is essentially dot of Stokes. But the surface velocity, and hence the motion of the film, is in the opposite direction of the wave. Finally, for temporally attenuated waves, it is demonstrated that the difference between the mean drift velocity at a clean surface and the mean drift velocity at a film-covered surface depends very much on the wavelength.