Local instabilities in magnetized rotational flows: a short-wavelength approach

Abstract

We perform a local stability analysis of rotational flows in the presence of a constant vertical magnetic field and an azimuthal magnetic field with a general radial dependence. Employing the short-wavelength approximation we develop a unified framework for the investigation of the standard, the helical, and the azimuthal version of the magnetorotational instability, as well as of current-driven kink-type instabilities. Considering the viscous and resistive setup, our main focus is on the case of small magnetic Prandtl numbers which applies, e.g., to liquid metal experiments but also to the colder parts of accretion disks. We show that the inductionless versions of MRI that were previously thought to be restricted to comparably steep rotation profiles extend well to the Keplerian case if only the azimuthal field slightly deviates from its current-free (in the fluid) profile. We find an explicit criterion separating the pure azimuthal inductionless magnetorotational instability from the regime where this instability is mixed with the Tayler instability. We further demonstrate that for particular parameter configurations the azimuthal MRI originates as a result of a dissipation-induced instability of the Chandrasekhar's equipartition solution of ideal magnetohydrodynamics.

On properties of Velikhov–Chandrasekhar MRI in ideal and non-ideal plasma

“…The situation is different when the azimuthal magnetic field is present. This case should be considered separately and has been investigated for a range of astrophysical applications in other papers (see, e.g., Acheson (1978); Sano & Miyama (1999); Ruediger et al (2014); Kirillov, Stefani & Fukumoto (2014)). We conclude that in thin Keplerian accretion discs the adding of viscosity can strongly restrict the MRI conditions once the mean free path of ions becomes comparable with the disc thickness.…”

Quasi-two-dimensional nonlinear evolution of helical magnetorotational instability in a magnetized Taylor–Couette flow

“…In some respect, more progress has been made recently on the nonlinear dynamics of a 'sibling' of HMRIthe azimuthal magnetorotational instability (AMRI). This non-axisymmetric instability, which emerges in the presence of an imposed purely azimuthal magnetic field in TC liquid metal flows, had been first identified after HMRI [22,57] (see also [46] and references therein). Using numerical simulations, Guseva et al [58][59][60] probed much broader ranges of Reynolds and Hartmann numbers than those done for HMRI so far and identified different regimes of nonlinear saturation, from supercritical Hopf bifurcation near the linear instability threshold up to a catastrophic transition to spatio-temporal defects, which are mediated by a subcritical subharmonic Hopf bifurcation, and ultimately to turbulence.…”

“…Previous linear analysis showed that HMRI is effective for relatively strong azimuthal fields, 1  b (e.g. [28,40,41,46]). Consider perturbations of the velocity, pressure and magnetic field about the equilibrium, u…”

“…Like the SMRI, the HMRI also draws free energy from the background shear flow and transports angular momentum outward. Subsequently, HMRI has been widely studied theoretically by means of linear modal stability analysis [28,[41][42][43][44][45][46][47]. It was shown that HMRI represents a destabilized inertial wave [41] and its growth rate is governed by the Reynolds number and the Hartmann number, Ha, characterizing the magnetic field strength.…”

Subcritical excitation of the current-driven Tayler instability by super-rotation

“…of a local approximation for Pm → 0 where Re * , Ha * and m * represent the slightly modified Reynolds number, Hartmann number and azimuthal wave number 28 . The Rossby number Ro = (1/2)dlogΩ /dlogR represents the differential rotation, it is positive for super-rotation and negative for sub-rotation.…”