Dissipative Taylor-Couette flows under the influence of helical magnetic fields

Abstract

The linear stability of magnetohydrodynamic Taylor-Couette flows in axially unbounded cylinders is considered for magnetic Prandtl number unity. Magnetic background fields varying from purely axial to purely azimuthal are imposed, with a general helical field parametrized by β=B(ϕ)/B(z). We map out the transition from the standard magnetorotational instability (MRI) for β=0 to the nonaxisymmetric azimuthal magnetorotational instability for β→∞. For finite β, positive and negative wave numbers m , corresponding to right and left spirals, are no longer degenerate. For the nonaxisymmetric modes, the most unstable mode spirals in the opposite direction to the background field. The standard (β=0) MRI is axisymmetric for weak fields (including the instability with the lowest Reynolds number) but is nonaxisymmetric for stronger fields. If the azimuthal field is due in part to an axial current flowing through the fluid itself (and not just along the central axis), then it is also unstable to the nonaxisymmetric Tayler instability which is most effective without rotation. For purely toroidal fields the solutions for m=±1 are identical so that in this case no preferred helicity results. For large β the wave number m=-1 is preferred, whereas for β≲1 the mode with m=-2 is most unstable. The most unstable modes always spiral in the same direction as the background field. For background fields with positive and not too large β the kinetic helicity of the fluctuations proves to be negative for all the magnetic instabilities considered.