# Analysis of azimuthal magnetorotational instability of rotating MHD flows and Tayler instability via an extended Hain-Lust equation

## TL;DR

This paper extends the Hain-Lust equation to include effects of magnetic fields, viscosity, and resistivity, providing a new dispersion relation for analyzing the stability of rotating magnetohydrodynamic flows and identifying a novel long-wavelength instability.

## Contribution

The authors derive an extended Hain-Lust equation incorporating magnetic fields, viscosity, and resistivity, and develop a comprehensive dispersion relation for stability analysis of rotating MHD flows.

## Key findings

- Flows with angular velocity beyond Liu limit become unstable under azimuthal magnetic fields.
- Keplerian flows are also susceptible to instability.
- Identification of a new long-wavelength instability in magnetized Taylor-Couette flow.

## Abstract

We consider a differentially rotating flow of an incompressible electrically conducting and viscous fluid subject to an external axial magnetic field and to an azimuthal magnetic field that is allowed to be generated by a combination of an axial electric current external to the fluid and electrical currents in the fluid itself. In this setting we derive an extended version of the celebrated Hain-Lust differential equation for the radial Lagrangian displacement that incorporates the effects of the axial and azimuthal magnetic fields, differential rotation, viscosity, and electrical resistivity. We apply the Wentzel-Kramers-Brillouin method to the extended Hain-Lust equation and derive a new comprehensive dispersion relation for the local stability analysis of the flow to three-dimensional disturbances. We confirm that in the limit of low magnetic Prandtl numbers, in which the ratio of the viscosity to the magnetic diffusivity is vanishing, the rotating flows with radial distributions of the angular velocity beyond the Liu limit, become unstable subject to a wide variety of the azimuthal magnetic fields, and so is the Keplerian flow. In the analysis of the dispersion relation we find an evidence of a new long-wavelength instability which is caught also by the numerical solution of the boundary value problem for a magnetized Taylor-Couette flow.