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A spectral method using associated Legendre functions with algebraic mappingis developed for a linear stability analysis of wake vortices. These functionsserve as Galerkin basis functions, capturing correct analyticity and boundaryconditions for vortices in an unbounded domain. The incompressible Euler orNavier-Stokes equations linearised on a swirling flow are transformed into astandard matrix eigenvalue problem of toroidal and poloidal streamfunctions,solving perturbation velocity eigenmodes with their complex growth rate aseigenvalues. This reduces the problem size for computation and distributescollocation points adjustably clustered around the vortex core. Based on thismethod, strong swirling $q$-vortices with linear perturbation wavenumbers oforder unity are examined. Without viscosity, neutrally stable eigenmodesassociated with the continuous eigenvalue spectrum having critical-layersingularities are successfully resolved. The inviscid critical-layer eigenmodesnumerically tend to appear in pairs, implying their singular degeneracy. Withviscosity, the spectra pertaining to physical regularisation of critical layersstretch out toward an area, referring to potential eigenmodes with wavepacketsfound by Mao & Sherwin (2011). However, the potential eigenmodes exhibit nospatial similarity to the inviscid critical-layer eigenmodes, doubting thatthey truly represent the viscous remnants of the inviscid critical-layereigenmodes. Instead, two distinct continuous curves in the numerical spectraare identified for the first time, named the viscous critical-layer spectrum,where the similarity is noticeable. Moreover, the viscous critical-layereigenmodes are resolved in conformity with the $Re^{-1/3}$ scaling law. Theonset of the two curves is believed to be caused by viscosity breaking thesingular degeneracy.

Linear stability analysis of wake vortices by a spectral method using mapped Legendre functions