  We study a mathematical model of a perturbed stratified shear mean flow in
the presence of eddy coefficients of turbulent viscosity. We adopt the standard
Boussinesq approximation in the natural convection of the buoyancy-driven flow
and neglect the influence of the eddy coefficients of turbulent diffusivity.
Comprising both the vertical and horizontal viscosity effects, a model for the
vertical velocity perturbation corresponds to a fourth-order Taylor-Goldstein
(TG) differential equation. Considering only the latter, we obtained a modified
TG equation with the same order as the classical, inviscid one. Under an
assumption of the slowly varying Brunt-V\"ais\"al\"a frequency and background
horizontal velocity, we discuss the corresponding geometrical and physical
optics approximations of the modified TG equation using the WKB method. We
further investigate the behavior of these asymptotic solutions near singular
values of a turning point and critical level.
