Stability analysis of stratified shear flows with a monotonic velocity profile without inflection points

Abstract

The two-layer model of a stably stratified medium has been used to investigate the stability of flows without inflection points on the profile of the velocity $V_x\,{=}\,u(y)$, which is monotonically increasing from zero at the bottom ($y\,{=}\,0$) to its maximum value $U_0$ (when $y\,{\to}\,\infty$). It is shown that in the case of flows of a general form (in which $u''(y)\,{<}\,0$ everywhere) an instability sets in for an arbitrarily small density difference; furthermore, perturbations of all scales build up simultaneously. With an enhancement of stratification, the real part $c_r$ of the phase velocity of unstable perturbations increases. The upper boundary of the instability domain is determined by the fact that at a certain stratification level (a particular one for perturbations of each scale), $c_r$ reaches $U_0$, the perturbation is no longer in phase resonance with the flow and turns into a neutral oscillation of the medium.

For flows of a special kind, having points of zero curvature (where $u''=0$) on the velocity profile (but having no inflection points as before), the influence of neutral modes, associated with these points, on the formation of the instability domain configuration is analysed, and an interpretation of this influence is given in terms of the resonance and non-resonance contributions to shear flow instability.