Subcritical excitation of the current-driven Tayler instability by super-rotation

It is known that in a hydrodynamic Taylor-Couette system uniform rotation or a rotation law with positive shear ('super-rotation') are linearly stable. It is also known that a conducting fluid under the presence of a sufficiently strong axial electric-current becomes unstable against nonaxisymmetric disturbances. It is thus suggestive that a cylindric pinch formed by a homogeneous axial electric-current is stabilized by rotation laws with $d\Omega/dR \geq 0$. However, for magnetic Prandtl numbers Pm$\neq 1$ and for slow rotation also rigid rotation and super-rotation support the instability by lowering their critical Hartmann numbers. For super-rotation in narrow gaps and for modest rotation rates this double-diffusive instability even exists for toroidal magnetic fields with rather arbitrary radial profiles, the current-free profile $B_\phi\propto 1/R$ included. For rigid rotation and for super-rotation the sign of the azimuthal drift of the nonaxisymmetric hydromagnetic instability pattern strongly depends on the magnetic Prandtl number. The pattern counterrotates with the flow for Pm$\ll 1$ and it corotates for Pm$\gg 1$ while for rotation laws with negative shear the instability pattern migrates in the direction of the basic rotation for all Pm. An axial electric-current of minimal 3.6 kAmp flowing inside or outside the inner cylinder suffices to realize the double-diffusive instability for super-rotation in experiments using liquid sodium as the conducting fluid between the rotating cylinders. The limit is 11 kAmp if a gallium alloy is used.