This study considers the stability of an axisymmetric vortex with respect to small wavelike perturbations. We restrict attention to a marginal stability so that the perturbation does not blow up or decay in a short time scale. An asymptotic method is used to analytically describe the flow in the neighbourhood of the radius where the wave frequency is equal to the vortex angular rotation. At this location, a singularity appears in the linearized and inviscid equation of the motion which deeply modifies the flow in the so-called critical layer. Here, nonlinear effects are invoked to resolve this singularity, since the relevant vortical flows in engineering (aeronautics) or geophysics (the atmospheric or oceanic dynamics) are of high Reynolds numbers. The formation of this layer characterizes a strong interaction between the perturbation and the rotating mean flow for which we will show the mean features.