Nonlinear stability of a stratified shear flow: a viscous critical layer

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Abstract
The nonlinear stability of a weakly supercritical shear flow with vertical temperature (density) stratification is investigated. It is shown that the usual Lin's rule of ‘indenting’ a singularity at the point of wave-flow resonance (the so-called critical layer, CL) is inapplicable for evaluating the nonlinear effects. To this end, a consistent procedure for deriving a nonlinear evolution equation is suggested and realized for the viscous critical-layer regime. The procedure takes into account the interaction of the fundamental harmonic with the second harmonic as well as with the zeroth one (i.e. with the mean-flow distortion). It is shown that the nonlinear factors both act in the same manner - at Prandtl number η ≤ 1 they limit the instability but at η > 1 they enhance it and convey a ‘burst-like’ character to it.It is found that CL is the region of strongest interactions between the harmonics. Hence the nonlinear contribution does not actually depend on the type of original flow model chosen. A simple physical interpretation is given to illustrate the mechanism governing the nonlinearity effects on the stability in the viscous critical-layer regime.

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