Helical magnetorotational instability in magnetized Taylor-Couette flow

Hollerbach and Rüdiger have reported a new type of magnetorotational instability (MRI) in magnetized Taylor-Couette flow in the presence of combined axial and azimuthal magnetic fields. The salient advantage of this “helical” MRI (HMRI) is that marginal instability occurs at arbitrarily low magnetic Reynolds and Lundquist numbers, suggesting that HMRI might be easier to realize than standard MRI (axial field only), and that it might be relevant to cooler astrophysical disks, especially those around protostars, which may be quite resistive. We confirm previous results for marginal stability and calculate HMRI growth rates. We show that in the resistive limit, HMRI is a weakly destabilized inertial oscillation propagating in a unique direction along the axis. But we report other features of HMRI that make it less attractive for experiments and for resistive astrophysical disks. Large axial currents are required. More fundamentally, instability of highly resistive flow is peculiar to infinitely long or periodic cylinders: finite cylinders with insulating endcaps are shown to be stable in this limit, at least if viscosity is neglected. Also, Keplerian rotation profiles are stable in the resistive limit regardless of axial boundary conditions. Nevertheless, the addition of a toroidal field lowers thresholds for instability even in finite cylinders.

Figure 1

Selected roots of full dispersion relation (9) for $\eta =2000{\mathrm{cm}}^2{\mathrm{s}}^{-1}$ [gallium], $r_1=9\mathrm{cm}$, $r_2=11\mathrm{cm}$, vertical periodicity $2h=16\mathrm{cm}$, ${\Omega }_1=100\mathrm{rpm}$, ${\Omega }_2=68.1\mathrm{rpm}$, $B_z=500\mathrm{G}$, $B_{\theta }=10\mathrm{kG}$ at $r=(r_1+r_2)∕2$. The two rapidly damped modes are omitted. (a) Growth rate $\gamma =\mathrm{Im}\omega$ vs. wave number $k_z$ (b) Real frequency ${\omega }_r=\mathrm{Re}\omega$ vs. wave Number $k_z$