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 ] ν½, 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(ν½) 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.

Nonlinear evolution of spiral density waves generated by the instability of the shear layer in a rotating compressible fluid

“…The rapid increase of disturbances leads to the fact that a nonlinear CL cannot be produced here during the course of the evolution, so that the evolution is proceeding only through regimes with either an unsteady or viscous CL (which, however, with increasing amplitude is also replaced by an unsteady one). The same picture emerges in the case of the evolution of a stratified shear flow (Churilov & Shukhman 1988). Thus, the objective of this paper is twofold.…”

From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators

“…Yes, all these things were familiar and standard in the context of collisionless plasmas [38][39][40][41][42][43][44][45][46][47][48]. Not only that, he explained, but similar phenomena occur in many other parts of science, in connection with instabilities of ideal shear flows [49][50][51], solitary waves [52,53], bubbly fluids [54], and resonance poles in atomic systems [55]. Wow -I was talking to the right guy.…”

Nonlinear evolution of a weakly unstable wave in a free shear flow with a weak parallel magnetic field

“…The structure of such an equation has been investigated on numerous occasions (e.g. Hickernell 1984;Churilov & Shukhman 1988;Goldstein & Choi 1989). It is easy to describe the character of its solution.…”

“…If, however, a disturbance 'starts' from the viscous CL region (γ L < ν 1/3 ), it is either stabilized in the viscous CL regime (with a stabilizing sign of the nonlinear term in the Landau-Stuart-Watson NEE that holds in this case) or (with a negative sign) passes into an explosive stage with the law of growth |A| ∝ (s 1 − s) −1/2 , while remaining initially still in the viscous CL regime and subsequently switching over to the unsteady CL regime with the law of growth (1.2). Such a 'fast' scenario is followed by the evolution of disturbances with the singular neutral mode in a stratified flow (Churilov & Shukhman 1988) and in compressible flows (Goldstein & Leib 1989;Leib 1991;Shukhman 1991), and three-dimensional disturbances in a homogeneous incompressible flow (Goldstein & Choi 1989;Wu, Lee & Cowley 1993;Wu 1993a, b;Churilov & Shukhman 1994;Wu & Cowley 1995;Wu, Lieb & Goldstein 1997).…”