  Employing a poloidal-toroidal projection technique, a multi-scale analysis of
resonant wave triads in columnar vortices is performed to obtain the governing
equations of the triadically coupled wave amplitudes. For inviscid flows, we
establish that resonance between neutral, smooth waves is conservative, and the
temporal evolution of wave amplitudes is either bounded or explosively unstable
based on the signs of the triad's interaction coefficients. Assessing the onset
of weakly nonlinear instabilities through the pseudoenergy criterion introduced
by Cairns (1979, J. Fluid Mech., vol. 92), we use the large-axial-wavenumber
asymptotic approach by Le Dizes and Lacaze (2005, J. Fluid Mech., vol. 542) to
evaluate each triad member's pseudoenergy and argue against the possibility of
explosive conservative three-wave resonance involving only regular Kelvin
waves. Additionally, extending our investigation to specific vortices, such as
the Lamb-Oseen vortex and the Batchelor vortex, we find that triadic resonance
among their neutral modes consistently results in bounded behaviour.
