Energy integral description of the development of Kelvin-Helmholtz billows

In this paper we study the development of instability waves in a stratified shear flow which occurs, for instance, in frontal zones and on large scale internal waves where the Richardson number is sufficiently low. The model is based on splitting the flow into the mean flow and instability wave components. The basis for the interaction between the mean flow and the wave is their respective vertically integrated energy flux equations. The wave description is obtained through a shape assumption: the time dependent wave amplitude is determined by its energy equation solved jointly with the mean flow and the vertical shape function is given by the local linear theory. The instability wave kinetic energy development is determined by the balance between energy production from the mean flow and the conversion of fluctuation kinetic to potential energy. From the energy integral considerations, the numerical results show moderately good agreement with observations in our estimate of the lifetime of the wave and the doubling of the thickness of the mean shear layer during this time. The modification of the mean flow velocity and temperature profiles is explained via the effects of the wave generated vertical momentum and heat (or buoyancy) fluxes, respectively.