Summary: The absolute/convective nature of the instability of the Batchelor vortex is determined in the plane swirl and co-flow parameters for different azimuthal wavenumbers. Swirl is shown to induce a transition to absolute instability for jets and wakes without requiring any counter-flow. The transitional helical mode is \(n=-1\) for wakes and \(n=-2\) or \(n=-3\) for co-flowing jets.