Nonlinear stability of a stratified shear flow in the regime with an unsteady critical layer

Abstract

In a previous paper (Churilov & Shukhman 1987a) we investigated the nonlinear development of disturbances to a weakly supercritical, stratified shear flow; we now report a continuation of that study. The degree of supercriticality of the flow is assumed not too small so that — unlike Paper 1 — the critical layer that appears in the region of resonance of the wave with the flow is an unsteady rather than viscous one. The evolution equation with cubic and quintic nonlinearity has been derived. The nonlinear term is non-local in time, i.e. depends on the entire preceding development of the disturbance. This equation has been used in the analysis of the evolution of an initially small disturbance. It is shown that where wave amplitude A is small enough (A [Lt ] ν^½, ν is the inverse of the Reynolds number), cubic nonlinearity dominates. In this case, as in Paper 1, the character of the evolution essentially depends on the sign of the quantity (η − 1), where η is the Prandtl number. However, independently of this sign the disturbance reaches — as it increases — the level A ∼ O(ν^½) and then quintic nonlinearity becomes dominant. At this stage an ‘explosive’ regime occurs and amplitude grows as $A \sim (t_0 - t)^{-\frac{7}{4}}$. The results obtained, together with the findings of Paper 1, provide a full description of the development of small disturbances at a large (but finite) Reynolds number in different regimes which are determined by the degree of flow's supercriticality.