Critical layer in a shear flow ($U(y_c)=c_0$)

A critical layer is a region where the leading wave phase speed coincides with the local shear-flow velocity and an outer approximation degenerates. Small viscosity or other corrections must then be retained on a distinguished transverse scale. The width depends on the order of vanishing of $U-c_0$ and on the governing mode equation; simple-shear and stationary-profile layers have different balances.

For the Prandtl normal-mode equation with viscosity scaled to one and a profile having a nondegenerate stationary point, $U-c_0\sim (y-y_c{)}^2/2$. A piecewise outer mode proportional to $U-c_0$ has a second-derivative jump. Balancing the advective and third-derivative terms smooths that jump over width ${\alpha }^{-1/4}$, with velocity amplitude ${\alpha }^{-1/2}$ and a phase-speed correction of the same order. Matching selects the eigenvalue through a third-order inner problem.

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.