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[Lt ]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
(Pr−1)Ri/γ2L,
where Pr is the Prandtl number and γL is the wave's
linear growth rate (which is a measure of supercriticality), and this effect
is important only when γL<Ri1/2,
Pr≠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
γL. 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.