Three-dimensional instability of isolated vortices

We study the three-dimensional stability of the family of vortices introduced by Carton and McWilliams [Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence, edited by Nikhoul and Jamart (Elsevier, New York, 1989)] describing isolated vortices. For these vortices, the circulation vanishes outside their core over a distance depending on a single parameter, the steepness α. We proceed to the direct numerical simulation of the linear impulse response to obtain both temporal and spatio-temporal instability results. In the temporal instability framework, growth rates are calculated as a function of the axial wavenumber k and the azimuthal wavenumber m. The stability analysis is performed at a Reynolds number of Re=667. It is shown that the most unstable mode is the axisymmetric mode m=0, regardless of the steepness parameter in the investigated range. When the steepness α is increased the band of unstable azimuthal modes widens, i.e., larger m are destabilized. The study of the spatio-temporal spreading of the wave packet shows that the m=2 mode is always the fastest traveling mode, for all studied values of the steepness parameter.