Sausage Instabilities on Top of Kinking Lengthening Current-Carrying Magnetic Flux Tubes

We theoretically explore the possibility of sausage instabilities developing on top of a kink instability in lengthening current-carrying magnetic flux tubes. Observations indicate that the dynamics of magnetic flux tubes in our cosmos and terrestrial experiments can involve topological changes faster than time scales predicted by resistive magnetohydrodynamics. Recent laboratory experiments suggest that hierarchies of instabilities, such as kink and Rayleigh-Taylor, could be responsible for initiating fast topological changes by locally accessing two fluid and kinetic regimes. Sausage instabilities can also provide this coupling mechanism between disparate scales. Flux tube experiments can be classified by the flux tube's evolution in a configuration space described by a normalized inverse aspect-ratio $\bar{k}$ and current-to-magnetic flux ratio $\bar{\lambda}$. A lengthening current-carrying magnetic flux tube traverses this $\bar{k}$ - $\bar{\lambda}$ space and crosses stability boundaries. We derive a single general criterion for the onset of the sausage and kink instabilities in idealized magnetic flux tubes with core and skin currents. The criterion indicates a dependence of the stability boundaries on current profiles and shows overlapping kink and sausage unstable regions in the $\bar{k}$ - $\bar{\lambda}$ space with two free parameters. Numerical investigation of the stability criterion reduces the number of free parameters to a single one that describes the current profile, and confirms the overlapping sausage and kink unstable regions in $\bar{k}$ - $\bar{\lambda}$ space. A lengthening, ideal current-carrying magnetic flux tube can therefore become sausage unstable after it becomes kink unstable.