# Magneto Rotational Instability in Magnetized AGN Tori

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

^{2}/s]. The gravitational potential $\mathrm{\Phi}$ is $GM/\sqrt{{r}^{2}+{z}^{2}}$, where M is the mass of SMBH, ${10}^{7}{M}_{\odot}$. We ignore the self-gravity of the gas.

^{−3}] for the number density, and ${B}_{0}=\sqrt{4\pi {m}_{\mathrm{H}}{n}_{0}{v}_{0}^{2}}=3.3$ [mG] for the magnetic field, respectively.

## 3. Results

^{−3}], and the field strength is amplified. In the right panels ($t=1.53$ [Myr]), the low resolution model (${N}_{\phi}=128$) indicates that the field strength does not depend on the number density, ${B}_{\mathrm{total}}\propto {n}^{0}$. This means that the magnetic field structure leaves unchanged. This is agreement with the left panel of Figure 4. On the other hand, the high resolution model (${N}_{\phi}=512$) forms the field amplification with the relation, ${B}_{\mathrm{total}}\propto {n}^{1/2}$. The right side of maximum field strength is stronger than the middle by a factor of ∼2. The higher the resolution, the stronger the magnetic field generated in the dense region ($1<n<4$).

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Time evolution of the model of ${N}_{\phi}=512$ in the slice of rz-plane. Color contour denotes temperature. Snapshots are (

**1**) $t=0.00$ [Myr], (

**2**) $t=0.60$ [Myr], and (

**3**) $t=1.53$ [Myr].

**Figure 2.**The same of time snapshots as Figure 1 but for the distribution of ${B}_{\phi}$ (

**top panels**) and plasma $\beta $ (

**bottom panels**).

**Figure 3.**Distribution of Q-value defined as the resolution of typical wave length growing MRI. The

**left panel**denotes the low azimuth-resolution ${N}_{\phi}=128$, and the

**right panel**denotes the high azimuth-resolution ${N}_{\phi}=512$. The criteria for resolution is selected by ${Q}_{\phi}=20$, shown in [25].

**Figure 4.**Butterfly diagram: time evolution of ${B}_{\phi}$ averaged by azimuthal direction at $r=1$ [pc]. Color contour of red and blue denotes the different sign of ${B}_{\phi}$. The

**left and right panels**denote the low resolution ${N}_{\phi}=128$ and high resolution ${N}_{\phi}=512$, respectively. Time evolutions from the adiabatic phase to the cooling/heating phase are switched with $t\sim 0.60$ [Myr].

**Figure 5.**2D histogram of total field strength ${B}_{\mathrm{total}}=\sqrt{{B}_{r}^{2}+{B}_{\phi}^{2}+{B}_{z}^{2}}$ and number density n. Color contour denotes the mass occupying cells of $\Delta (logn)=\Delta (log{B}_{\mathrm{total}})$=0.01.

**Top panels**show the low resolution model and the

**bottom panels**show the high resolution model.

**Left**: $t=0.60$ [Myr] of 20 rotation periods in the pure MHD;

**Middle**: $t=1.10$ [Myr] of 37 rotation periods in the MHD including cooling and heating effects;

**Right**: $t=1.53$ [Myr] of 52 rotation periods.

**Figure 6.**Inflow rate after cooling. The

**left panel**is the model of ${N}_{\phi}=128$, and the

**right panel**is the model of ${N}_{\phi}=512$. Red, blue, and black curves denote the radius of $r=0.6,0.8,$ and $1.0$, respectively.

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**MDPI and ACS Style**

Kudoh, Y.; Wada, K. Magneto Rotational Instability in Magnetized AGN Tori. *Galaxies* **2018**, *6*, 139.
https://doi.org/10.3390/galaxies6040139

**AMA Style**

Kudoh Y, Wada K. Magneto Rotational Instability in Magnetized AGN Tori. *Galaxies*. 2018; 6(4):139.
https://doi.org/10.3390/galaxies6040139

**Chicago/Turabian Style**

Kudoh, Yuki, and Keiichi Wada. 2018. "Magneto Rotational Instability in Magnetized AGN Tori" *Galaxies* 6, no. 4: 139.
https://doi.org/10.3390/galaxies6040139