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

On physical optics approximation of viscous stratified shear flows