# Spectral Analysis of Non-Ideal MRI Modes: The effect of Hall diffusion

## TL;DR

This paper investigates how Hall diffusion influences the stability and eigenmodes of accretion disks, revealing unique stress and energy behaviors that differ from ideal MRI, supported by analytical and simulation results.

## Contribution

It provides a systematic linear analysis of eigenmodes affected by Hall diffusion and develops a geometrical representation of their polarization properties.

## Key findings

- Kinetic stresses dominate when magnetic and angular momentum vectors are anti-parallel.
- Analytical expressions for stresses and energies in non-ideal MRI are derived.
- Simulation results agree well with linear analysis, indicating potential nonlinear implications.

## Abstract

The effect of magnetic field diffusion on the stability of accretion disks is a problem that has attracted considerable interest of late. In particular, the Hall effect has the potential to bring about remarkable changes in the dynamical behavior of disks that are without parallel. In this paper, we conduct a systematic examination of the linear eigenmodes in a weakly magnetized differentially rotating gas with special focus on Hall diffusion. We first develop a geometrical representation of the eigenmodes and provide a detailed quantitative description of the polarization properties of the oscillatory modes under the combined influence of the Coriolis and Hall effects. We also analyze the effects of magnetic diffusion on the structure of the unstable modes and derive analytical expressions for the kinetic and magnetic stresses and energy densities associated with the non-ideal MRI. Our analysis explicitly demonstrates that, if the dissipative effects are relatively weak, the kinetic stresses and energies make up the dominant contribution to the total stress and energy density when the equilibrium angular momentum and magnetic field vectors are anti-parallel. This is in sharp contrast to what is observed in the case of the ideal or dissipative MRI. We conduct shearing box simulations and find very good agreement with the results derived from linear analysis. As the modes in consideration are also exact solutions of the non-linear equations, the unconventional nature of the kinetic and magnetic stresses may have significant implications for the non-linear evolution in some regions of protoplanetary disks.