A Physical Interpretation of the Wind-Wave Instability as Interacting Waves

Abstract

One mechanism for the growth of ocean surface waves by wind is through a shear instability that was first described by Miles in 1957. A physical interpretation of this wind-wave instability is provided in terms of the interaction of the surface gravity wave with perturbations of vorticity within the critical layer—a near-singularity in the airflow where the background flow speed matches that of the surface gravity wave. This physical interpretation relies on the fact that the vertical velocity field is slowly varying across the critical layer, whereas both the displacement and vorticity fields vary rapidly. Realizing this allows for the construction of a physically intuitive description of the critical layer vorticity perturbations that may be approximated by a simple vortex sheet model, the essence of the wind-wave instability can then be captured through the interaction of the critical layer vorticity with the surface gravity wave. This simple model is then extended to account for vorticity perturbations in the airflow profile outside of the critical layer and is found to lead to an exact description of the linear stability problem that is also computationally efficient. The interpretation allows, in general, for the incorporation of sheared critical layers into the “wave interaction theory” that is commonly used to provide a physical description and rationalization of results in the stability of stratified shear flows.