High-wavenumber instability of a non-monotone Prandtl layer ($\gamma =\alpha \mathrm{Im}c=O(\sqrt{\alpha })$)

A stationary point in a non-monotone velocity profile can support a quarter-power critical layer whose phase-speed correction has positive imaginary part. The resulting normal mode grows at a rate proportional to the square root of the wavenumber. Such unbounded high-frequency amplification demonstrates linear instability of the frozen layer; nonlinear evolution and precise well-posedness consequences require separate analysis.