Stability and instability of hydromagnetic Taylor-Couette flows

Decades ago S. Lundquist, S. Chandrasekhar, P.H. Roberts and R. J.~Tayler first posed questions about the stability of Taylor-Couette flows of conducting material under the influence of large-scale magnetic fields. These and many new questions can now be answered numerically where the nonlinear simulations even provide the instability-induced values of several transport coefficients. The cylindrical containers are axially unbounded and penetrated by magnetic background fields with axial and/or azimuthal components. The influence of the magnetic Prandtl number $Pm$ on the onset of the instabilities is shown to be substantial. The potential flow subject to axial fields becomes unstable against axisymmetric perturbations for a certain supercritical value of the averaged Reynolds number $\overline{Rm}=\sqrt{Re\cdot Rm}$ (with $Re$ the Reynolds number of rotation, $Rm$ its magnetic Reynolds number). Rotation profiles as flat as the quasi-Keplerian rotation law scale similarly but only for $Pm\gg 1$ while for $Pm\ll 1$ the instability instead sets in for supercritical $Rm$ at an optimal value of the magnetic field. Among the considered instabilities of azimuthal fields, those of the Chandrasekhar-type, where the background field and the background flow have identical radial profiles, are particularly interesting. They are unstable against nonaxisymmetric perturbations if at least one of the diffusivities is non-zero. For $Pm\ll 1$ the onset of the instability scales with $Re$ while it scales with $\overline{Rm}$ for $Pm\gg 1$. - Even superrotation can be destabilized by azimuthal and current-free magnetic fields; this recently discovered nonaxisymmetric instability is of a double-diffusive character, thus excluding $Pm= 1$. It scales with $Re$ for $Pm\to 0$ and with $Rm$ for $Pm\to \infty$.