1. Global Magnetohydrodynamic Simulations of Disk Dynamos
5th East-Asia School and Workshop on Laboratory, Space, Astrophysical Plasmas, Aug 17-22,
Global Magnetohydrodynamic Simulations of Disk Dynamos
Ryoji Matsumoto (Chiba University)

2. Outline of this Talk Magnetic Fields in Astrophysical Disks
Global 3D MHD Simulations of Accretion Disks
and Galactic Gas Disks
New simulation code based on HLLD+MP5
Applications to State Transitions in Black Hole Accretion Disks
Applications to the interaction of an infalling cloud with Galactic center accretion disk

3. Magnetic Activities in Astrophysical Disks
In astrophysical disks, magnetic fields amplified in the disk will be transported to the corona by magnetic buoyancy and will trigger flares, jets, and outflows.
Magnetic fields amplified in the disk buoyantly rise into the disk corona. They drive various magnetic activities

4. Galactic Magnetic Fields
M51, Fletcher et al. (2011)
Top left panel shows the optical image of M51 galaxy overlayed with the direction of magnetic fields obtained by radio oplarization observations. Magnetic fields are strong in spiral arms and their direction is parallel to the spiral arms. The left bottom panel shows the radio intensity and direction of magnetic fields in M31. Magnetic fields in this galaxy is axisymmetric and ring-like.
Magnetic fiedls in our galaxy is studied extensively by Planck satellite. The misson of this satellite is to measure the polarization of the cosmic microwave background radiation. In the top right figure, orange shows the background radiation., blue is the radiation from dust, and pink shows the synchrotron radiation. It is expected that the existence of primordial gravitational wave formed during the inflation stage of our universe can be detected by quadrapole pattern of polarization of CMB. For that purpose, we have to separate the contribution of our galactic magnetic field from CMB. The right bottom panel shows the polarized emisson at 30 GHz. Curves show magnetic field lines. We can identify loop structure in the galactic corona, which may be formed by the buoyant escape of magnetic fields from the galactic gas disks.
M31 Berkhuijsen, Beck, and Hoernes (2003). MPIfR Bonn.
Polarized emission at 30GHz
ESA and Planck Collaboration

5. Magneto-rotational Instability (MRI) in Differentially Rotating Disks
Angular momentum
Balbus and Hawley (1991), Velikhov (1959)
When a differeintally rotating disk is threaded by weak magnetic fields, the magnetic field subjects to the magneto-rotational instability. This instability grows even when the initial field is purely azimuthal.
Radial field can be created from toroidal field

6. Global 3D MHD Simulation of Differentially Rotating Torus
After 10 Rotation Period
Initial Condition
200*64*240 grid points
b = Pgas/Pmag=100
Matsumoto 1999

7. Dependence on the Initial Magnetic Field
b ~ 10
ORBIT

8. Formation of Magnetic Loops in the Disk Corona
The formation of magnetic loops in differentially rotating disks was simulated by Machida et al. The left panel shows the initial condition. We assume azimuthal field. The plasma beta at the density maximum is unity. Red curves in the right panel show magnetic field lines after the disk becomes turbulent. Yellow surfaces show the isosurface of magnetic field strength. Buoyantly rising magnetic fields create magnetic loops in the disk halo.
Buoyancy
Tension
Shock Wave
Machida et al. 2000

9. MRI-Parker Dynamo MRI Parker Instability Ω effect MRI
This viewgraph schematically shows how the magnetic field is amplified and maintained in astrophysical disks. Radial field is produced by MRI from vertical field and azimuthal field. Azimuthal field is amplifield by streching the radial field by differential rotation. When the plasma beta becomes less than 5-10, magnetic fields buoyantly escape from the disk by Parker instability. Parker instability also produces the vertical field. These processes are essentially the alpha-Omega dynamo but we cannot ignore the back reaction from the magnetic field to gas motion because the angular momentum transport through magnetic fields is essential for MRI.
MRI

10. Global 3D MHD Simulation of Galactic Gas Disks in Axisymmetric Gravitational Potential
Potential by Miyamoto (1980)
No spiral arm, No self-gravity
No supernovae, No cosmic ray
No multiphase interstellar gas
Constant angular momentum torus at the initial state
Absorbing boundary condition at r=0.8kpc
Symmetric boundary condition at equatorial plane
Nishikori et al. carried out global 3D MHD simulations of galactic gas disks by assuming an axisymmetric gravitational potential of our galaxy. Initial condition is a torus threaded by weak azimuthal magnetic fields. We impose symmetric boundary condition at the equatorial plane and absorbing boundary condition at 0.8kpc from the galactic center. The volution of the disk was studied by using an 3D MHD code based on the modified LW method in cylindrical coordinates.
The number of mesh points is 250 in radial direction, 64 in azimuthal direction, and 319 in vertical direction.
250*64*320mesh
Nishikori, Machida, and Matsumoto 2006
ApJ 641, 862

11. Result of Simulation 2Gyr 3.5Gyr ρ＋B t = 3.8Gyr
Left panels show density and magnetic field lines at 2Gry and 3.5 Gyr . The right panel shows the magnetic field lines projected onto the equatorial plane. Mean azimuthal magnetic field and turbulent magnetic field is produced. The strength of the mean magnetic field is maicro gauss. . .
3.5Gyr
　ρ＋B
t = 3.8Gyr

12. Reversal of Mean Azimuthal Magnetic Fields
after
1Gyr…
1Gyr
We found that the direction of mean azimuthal magnetic field inside the disk reverses with time scale about 10 rotation period of the disk. Color in left panel shows the mean azimuthal magnetic field. Blue and Orange show different polarity. As the azimuthal magnetic flux rises from the disk to the corona, the direction of mean azimuthal magnetic field reverses. The right panel shows the time variation of mean azimuthal magnetic field near the equatorial plane and in the disk corona. They reverse their sign quasi periodically with time scale of 1Gyr.
5Gyr
Time variation of mean azimuthal magnetic field at 5kpc < r < 6kpc

13. Result of Simulation without Assuming Equatorial Symmetry
Emg
2μG
<Bφ>
Machida et al. carried out global 3D MHD simulation of galactic gas disks without assuming euqatorial symmetry. We doubled the azimuthal resolution and used 128 grids in azimuthal direction and 640 grids in vertical direction. Left panel shows the density distribution and magnetic field lines. Color in the magnetic field line shows the azimuthal direction. Near the galactic center, vertical magnetic field is produced by ejection of outflows. Long wavelength magnetic loops are formed in the disk corona.The right panel shows the dependence on the azimuthal resolution. Dashed curve shows the result with 256 grid points. Although the growth of the magenetic energy begins earlier in the simulation with higher azimuthal resolution, the saturation level of the magnetic energy, amplitude of the azimuthal field, and the interval of field reversals do not depend much on the azimuthal resolution.
Density and Magnetic Field Lines
(Nr,Nφ,Nz)=(250,128,640) grids
Machida et al. (2013) ApJ 764, 81
5Gyr

14. Butterfly Diagram Height R=7kpc Height R=2kpc time time
These viewgraphs show the butterfly diagram obtained from numerical results without assuming the symmetry with respect to the galactic midplane. Colar shows the mean azimuthal magnetic fields. They are amplified, and buoyantly rise with time scale of 10 rotation period. Since the rotation angular speed is faster near the galactic center, the time scale of buoyant rise and field reversal is shorter. The bottom left panel shows the symmetry with respect to the midplane. White shows the symmetric, qudrupole field, and black shows anti-symmetric dipolar field. Although we start from the quadrupole field, dipole field is formed from time to time.
time
time
Azimuthal Magnetic fields
Machida et al. 2013

15. Solar Dynamo Cycle corona sunspot
Magnetic loop
sunspot
Sunspot (HINODE) Formation of Sunspots
Soft X-ray Image of the Sun
We can learn from magnetic activities of the sun. The sun shows 11 year cycle of activities. The bottom left panel shows the change of the solar corona observed in X-rays. The X-ray intensity changes year by year. The right bottom panel shows the time variation of the number of sunspots and the butterfly diagram which shows the latitude where the sunspot appears. It clearly shows the 11year cycle. The sunspots are cross section of the emerging magnetic loops. The number of sunspots correlates with the magnetic flux buoyantly rising to the solar corona. Understanding the solar dynamo is one of the most important challenge in astrophysics.
Solar cycle observed in X-ray
Butterfly diagram （NASA)

16. Dynamo Cycle obtained by Local 3D MHD Simulations
Miller and Stone 2000
white：β＝１　
The cyclic disk dynamo appears in local 3D MHD simulations of accretion disks. The top right panel shows the result of Miller and Stone The color shows the magnetic energy. The bottom right panel shows the result reported by Shi et al. in Color shows the azimuthal magnetic field. Symmetric, quadrupole field seems to be dominant in their model.
Shi et al. 2010
Time Variabilities of Azimuthal Field

17. How are Azimuthal Magnetic Fields Reversed ?
Amplification by MRI
This viewgraph schematically shows how the azimuthal magnetic field reverses. As the MRI grows, radial field is produced. The radial field is stretched by differential rotation, and the azimuthal field is amplified. When the strength of the azimuthal field exceeds the threshold for the nonlinear growth of the Parker instability, the azimuthal field escapes from the disk and form coronal magnteic loops. Since the total azimuthal magnetic flux is conserved, azimuthal field with opposite polarity remains in the disk, and amplified by MRI. The growth of MRI and the Parker instability drives cyclic dynamo. z
Buoyant rise by Parker Instability

18. Comparison with Observation
The right panel shows the rotation measure at 8kpc from the galactic center obtained by post processing the result of global MHD simulations of the galactic disk. The reversal of sign of RM with respect to the equatorial plane, and the point symmetry with respect to the galactic center is reproduced. The reversal of mean azimuthal fields in our galaxy will be confirmed by detecting the reversal of RM between magnetic loops in the disk corona.
Rotation Measure at 8kpc from the galactic center obtained from our simulation
(Machida et al. 2013)
All Sky map of Faraday Rotation Measure in Galactic Coordinates
(Oppermann et al. 2012)

19. New MHD Simulation Code CANS+
HLLD Scheme (Miyoshi and Kusano 2005)
MP5 Scheme (Suresh and Huynh 1997) UL UR * ** t UL UR U* t
HLLD Scheme
HLL Scheme
Preserve peaks and gradients by relaxing the monotonicity condition
j-2
j-1 j
j+1
j+2
So far, we presented the results of global 3D MHD simulations using an MHD code based on the modified Lax-Wendroff scheme with artificial viscosity. We have implemented a new simulation engine based on the HLLD scheme. Proposed by Miyoshi and Kusano. In HLLD scheme, Riemann problem at cell interface is solved approximately by using 4 states instead of 1 state in HLL scheme. Thus, HLLD scheme can resolve contact discontinuity and rotational discontinuiteis. In order to avoid numerical dissipation of smooth part of the waves, we applied MP5 scheme, in which cell interface values are determined by usng 4th order polynomials using vales at 5 points, and enlarge the range of the interface values allowed to preserve monotonicity by using the information for 2nd order derivatives.
By determining 4th order polynomial using values at 5 points, compute the value at cell interface at j+1/2

20. Disk Dynamo Simulation using CANS+
Initial State
Distribution of β=Pgas/Pmag　
Left: Second Order, Right: MP5 (5th Order)
HLLD+MP5
このコードを用いた円盤ダイナモシミュレーション結果を示します。初期条件は弱い方位角磁場に貫かれたトーラスです。左上図は空間二次精度の従来の解法とＭＰ５法で計算したときのプラズマβ値の分布を示します。ＭＰ５法の方が細かい構造が捕らえられています。この図は磁気エネルギーの増幅と飽和を示し、町田らの結果を再現できています。右下図は方位角磁場方向を黄色と青で示した図で１０回転周期で方位角磁場方向が反転する準周期ダイナモが励起されていることがわかります。
64mesh
128mesh
32mesh
Dynamo Cycle obtained by HLLD+MP5 code. Color shows azimuthal field.
Amplification of Magnetic Energy

21. State Transitions in Black Hole Candidates
State transitions of black hole candidates has been studied extensively by RXTE satellites and MAXI aboard the ISS
MAXI X-ray sky
MAXI aboard ISS (2009-)

22. State Transitions Observed in XTE J1752-223
1/21
This viewgraph shows the X-ray light curve of the black hole candidate XTE J The source changed from X-ray hard, blue state to the X-ray soft red state on January 21, this year. Radio emission indicating the Jet ejection is observed during this transition.
Jet
ejection
Nakahira et al. 2010
MAXI Science News #17

23. Thermal Equilibrium Curves of Accretion Disks
Classical Accretion Disk models give too low transition luminosity. We need to explain, why the source stays in the X-ray hard state even when its luminosity exceeds this threshold.
Solid curves are thermal equilibrium curves by Abramowicz et al. (1995)

24. Global MHD Simulations of the Hard-to-Soft Transition
Include cooling term after quasi-steady disk is formed for simulations using （256,64,256） mesh points
Formation of cool, dense region after cooling instability grows

25. Schematic Picture of the Growth of the Cooling Instability
Radiative Cooling
Cool Down
　b < 1
This veiwgraph schematically shows the growth of the cooling instability. Before the transition, an optically thin, hot disk is supported by gas pressure. As the instability grows, the magnetic pressure increases and the disk is supported by the magnetic pressure. Since the disk cannot shrink due to the magnetic pressure, the disk still stays in optically thin state.
　b ~ 10
Optically Thin Hot Disk Supported by Gas Pressure
Optically Thin Cool Disk Supported by Magnetic Pressure

26. Application to the Galactic Center
NRAO

27. Infall of G2 Cloud toward Sgr A*
L’band Image （Gillessen et al. 2013)
P-V diagram by VLT (Gillessen et al. 2013)

28. Tidal Disruption by a Black Hole
Tidal Force : GMR/r3
Self Gravity : Gm/R2 m 2R
Condition for disruption
GMR/r3 > Gm/R2
r/R < (M/m)1/3
M=106Msun
m=Msun
R=1011cm
r<100Rs r
The earth mass cloud can be tidally disrupted when its distance is more than 10 thousand times of its radius. The debris of the cloud will collide with the accretion disk surrounding the black hole M
Rees （1988）

29. 3D MHD Simulation of G2 Infall
Kawashima et al. 2015
Magnetic energy

30. Time Evolution of the Magnetic Energy
delay
pericenter passage
pericenter passage
20yr
20yr
time
time
The increase of the magnetic energy may be observed via the synchrotron emission in the radio band after 5-10 yrs.

31. Summary
In astrophysical gas disks, magnetic fields are amplified by MRI. The buoyant escape of the magnetic flux drives quasi-periodic reversal of azimuthal magnetic fields
A new simulation code CANS+ based on HLLD+MP5 scheme has been implemented.
Simulations of the hard-to-soft transition in black hole candidates indicate that magnetic energy release around the interface of hot and cool disks generate X-ray flares and mass ejections.
G2 cloud passage around the galactic center black hole Sgr A* may drive delayed radio activity
I would like to summarize.

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