Instability of a stratified boundary layer and its coupling with internal gravity waves. Part 1. Linear and nonlinear instabilities

Abstract

In this paper, we consider a viscous instability of a stratified boundary layer that is a form of the familiar Tollmien–Schlichting (T-S) waves modified by a stable density stratification. As with the usual T-S waves, the triple-deck formalism was employed to provide a self-consistent description of linear and nonlinear instability properties at asymptotically large Reynolds numbers. The effect of stratification on the temporal and spatial linear growth rates is studied. It is found that stratification reduces the maximum spatial growth rate, but enhances the maximum temporal growth rate. This viscous instability may offer a possible alternative explanation for the origin of certain long atmospheric waves, whose characteristics are not well predicted by inviscid instabilities. In the high-frequency limit, the nonlinear evolution of the disturbances is shown to be governed by a nonlinear amplitude equation, which is an extension of the well-known Benjamin–Davis–Ono equation. Numerical solutions indicate that as a spatially isolated disturbance evolves, it radiates a beam of long gravity waves, and meanwhile small-scale ripples develop on its front to form a well-defined wavepacket. It is also shown that for jet-like velocity profiles, the standard triple-deck theory must be adjusted to account for both the displacement and transverse pressure variation induced by the inviscid flow in the main layer. The nonlinear evolution of high-frequency disturbances is governed by a mixed KdV–Benjamin–Davis–Ono equation.