  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.
