On the inertial effects of density variation in stratified shear flows

In this paper, we first revisit the celebrated Boussinesq approximation in stratified flows. Using scaling arguments we show that when the background shear is weak, the Boussinesq approximation yields either (i) $A_t\ll \mathcal{O}(1)$ or (ii) $Fr_c^2 \ll \mathcal{O}(1)$, where $A_t$ is the ratio of density variation to the mean density and $Fr_c$ is the ratio of the phase speed to the long wave speed. The second clause implies, contrary to the commonly accepted notion, that a flow with large density variations can also be Boussinesq. Indeed, we show that deepwater surface gravity waves are Boussinesq while shallow water surface gravity waves are not. However, in the presence of moderate/strong shear, Boussinesq approximation implies the conventionally accepted $A_t\ll \mathcal{O}(1)$. To understand the inertial effects of density variation, our second objective is to explore various non-Boussinesq shear flows and study different kinds of stably propagating waves that can be present at an interface between two fluids of different background densities and vorticities. Furthermore, three kinds of density interfaces - neutral, stable and unstable - embedded in a background shear layer, are investigated. Instabilities ensuing from these configurations, which includes Kelvin-Helmholtz, Holmboe, Rayleigh-Taylor and triangular-jet, are studied in terms of resonant wave interactions. The effects of density stratification and the shear on the stability of each of these flow configurations are explored. Some of the results, e.g. the destabilizing role of density stratification, stabilizing role of shear, etc. are apparently counter-intuitive, but physical explanations are possible if the instabilities are interpreted from wave interactions perspective.