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 $Pm\to \infty$. The presented results allow the construction of several new experiments with liquid metals as the conducting fluid. Some of them are described here and their results will be discussed together with relevant diversifications of the magnetic instability theory including nonlinear numerical studies of the kinetic and magnetic energies, the azimuthal spectra and the influence of the Hall effect.