On the appearance of non-local MRI in Keplerian accretion discs

TL;DR

This paper analyzes the magneto-rotational instability in Keplerian accretion disks using a non-local approach, revealing how radial variations in magnetic fields affect the stability and growth rates of perturbations.

Contribution

It introduces a non-local modal analysis considering radial magnetic field dependence, showing qualitative changes in the effective potential and stability conditions compared to local models.

Findings

No unstable modes in shallow potentials with certain background conditions.

The critical Alfvén speed above which MRI ceases is identified.

MRI growth rates are suppressed in thin disks with low magnetic field strength.

Abstract

We revisit the modal analysis of small perturbations in Keplerian ideal gas flows leading to magneto-rotational instability (MRI) using the non-local approach. We consider the case of constant vertical background magnetic field, as well as the case of radially dependent background Alfv\'en velocity. In the case of constant Alfv\'en velocity, MRI modes are described by a Schr\"odinger-like differential equation with some effective potential including 'repulsive' ($1/r^2$) and 'attractive' ($-1/r^3$) terms. Taking into account the radial dependence of the background Alfv\'en speed leads to a qualitative change in the shape of the effective potential. It is shown that there are no stationary energy levels corresponding to unstable modes $\omega^2 < 0$ in ``shallow'' potentials. In thin accretion disks, the wavelength of the disturbance $\lambda=2\pi/k_z$ is smaller than the half-thickness $h$ of the disk only in ``deep'' potentials. The limiting value of the background Alfv\'en speed $(c_A)_\mathrm{cr}$, above which the magnetorotational instability does not occur, is found. In thin accretion disks with low background Alfv\'en speed $c_A\ll (c_A)_\mathrm{cr}$, the increment of the magnetorotational instability $\omega\approx -\sqrt{3}\mathrm{i}c_Ak_z$ is suppressed compared to the value obtained in the local perturbation analysis.