 Prandtl critical layer at a stationary shear profile

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.