Effect of slight stratification on the nonlinear spatial evolution of a weakly unstable wave in a free shear layer

We examine the spatial evolution of an instability wave excited by an external source in a free, nearly non-dissipative, stably stratified shear flow with a small Richardson number Ri much less than 1. It turns out that at the nonlinear stage of evolution even so small a stratification modifies greatly the evolution behaviour compared with the case of a homogeneous flow which was studied in detail by Goldstein & Hultgren (1988). We have investigated (analytically and numerically) different stages of evolution corresponding to different critical layer regimes, and determined the formation conditions and structure of a quasi-steady nonlinear critical layer. It is shown that the stratification influence upon the nonlinear evolution is governed by the parameter (PI - 1)Ri/gamma(L)(2), where Pr is the Prandtl number and gamma(L) is the wave's linear growth rate (which is a measure of supercriticality), and this effect is important only when gamma(L) < Ri(1/2), Pr not equal 1. The character of this influence radically depends on the sign of (Pr - 1). Thus, when Pr < 1 the amplitude in the course of the evolution varies in a limited range and either reaches saturation, when the supercriticality is small enough or, at higher supercriticality, performs quasi-periodic oscillations, whose structure becomes increasingly complicated with increasing gamma(L). When Pr > 1 stratification leads to the appearance of new evolutionary stages, namely the stage of explosive growth in the unsteady critical layer regime, and the stage of essentially unsteady evolution in the nonlinear critical layer regime, and to a modification of the power-law growth in the regime of a quasi-steady nonlinear critical layer.