Spatial stability and the onset of absolute instability of Batchelor vortex for high swirl numbers

Batchelor's vortex has been commonly used in the past as a model for aircraft trailing vortices. Using a temporal stability analysis, Fabre and Jacquin [\em J. Fluid Mech. \bf 500, \rm 239 (2004)] have recently found new viscous unstable modes for the high swirl numbers of interest in actual large-aircraft vortices. We look here for these unstable viscous modes occurring at large swirl numbers ($q>1.5$), and large Reynolds numbers ($Re>10^3$), using a spatial stability analysis, thus characterizing the frequencies at which these modes become convectively unstable for different values of $q$, of $Re$, and for different intensities of the uniform axial flow. We consider both jet-like and wake-like Bartchelor's vortices, and are able to reach values of $Re$ as high as $10^8$. We also characterize the onset of absolute instability of these unstable viscous modes for large $q$.