Regenerative centrifugal instability on a vortex column

The limitation and renewal of centrifugal instability of a vortex column (due to a sheath of negative axial vorticity, -\textit{$\Omega $}$_{z}$, surrounding the +\textit{$\Omega $} core, i.e. a circulation overshoot) is studied via the transport dynamics of perturbations to the initially unstable vortex using DNS of the incompressible Navier-Stokes equations for a range of vortex Reynolds numbers (Re=circulation/viscosity). Any radial perturbation vorticity, \textit{$\omega $'}$_{r}$, is tilted by the column's mean shear to form filaments with azimuthal vorticity, \textit{$\omega $'}$_{\theta }$, generating positive Reynolds stress, +$u'v'$ ($u'$,$ v'$ are the radial and azimuthal perturbation velocities), required for energy growth. This \textit{$\omega $'}$_{\theta }$ in turn tilts -\textit{$\Omega $}$_{z}$ to amplify \textit{$\omega $'}$_{r}$ (and consequently \textit{$\omega $'}$_{\theta })$ -- thus causing instability. Limitation of \textit{$\omega $'}$_{r}$ growth, thus also energy production, occurs as the perturbation transports angular momentum (\textit{rV}) radially outward from the overshoot, moving the overshoot outward, hence lessening and shifting -\textit{$\Omega $}$_{z}$, while also transporting core +\textit{$\Omega $}$_{z}$, around the location of the filament. After the overshoot shifts, tilting of -\textit{$\Omega $}$_{z}$ reverses \textit{$\omega $'}$_{r}$ (hence reducing \textit{$\omega $'}$_{\theta })$, causing the filament to generate --$u'v'$, i.e. energy decay, and hence self-limitation of growth. Associated with $--u'v'$ is the filament's radially inward transport of \textit{rV}, which can produce a new circulation overshoot and renewed instability. New overshoot formation and renewed generation of $+u'v'$ is examined using a helical ($m=1)$ mode -- a promising scenario for regenerative transient growth and possible turbulence generation on a vortex column.