Abstract: | The linear stability properties of an incompressible axisymmetrical vortex of axial velocity W 0( r ) and angular velocity Ω0( r ) are considered in the limit of large Reynolds number. Inviscid approximations and viscous WKBJ approximations for three-dimensional linear normal modes are first constructed. They are then shown to be singular at the critical points rc satisfying ω= m Ω0( rc ) + kW 0( rc ) , where ω is the frequency, k and m the axial and azimuthal wavenumbers. The goal of this paper is to resolve these singularities. We show that a viscous critical-layer analysis is analytically tractable. It leads to a single sixth-order equation for the perturbation pressure. This equation is identical to the one obtained in stratified shear flows for a Prandtl number equal to 1. Integral expressions for typical solutions of this equation are provided and matched to the outer inviscid and viscous approximations in the complex plane around rc . As for planar flows, it is proved that the critical layer solution with a balanced behavior matches a non-viscous approximation in a 4π/3 sector of the complex-plane. As a consequence, the Frobenius expansions of a non-viscous mode on each side of a critical point rc differ by a π phase jump. |