Non-equilibrium Ionization
Degree and MRI
in Protoplanetary Disks
Yuri I. Fujii (ELSI)S. Okuzumi (Tokyo Tech), T. K. Suzuki, & S. Inutsuka (Nagoya University)

Magnetorotational Instability (MRI)
Balbus & Hawley (1991)

magnetic field+

weakly ionized disk
turbulentrotating
mass angularmomentum

angular momentum transportmass accretion

Disk Dynamics due to MRI
protoplanetary disk

turbulence

dead zonestarturbulence

affects dead-zone boundary

Turbulent mixing may change dead-zone size (dust-free case).
Inutsuka & Sano (2005), Turner et al. (2007), Ilgner & Nelson (2008)	 

Disk Dynamics due to MRI
disk winddead zonestarturbulence

affects dead-zone boundary

Turbulent mixing may change dead-zone size (dust-free case).
Inutsuka & Sano (2005), Turner et al. (2007), Ilgner & Nelson (2008)	 

disk wind: mass loss from disk surface

Suzuki & Inutsuka (2009), Suzuki et al. (2010), Bai & Stone (2013) 	 

Is equilibrium ionization applicable everywhere?	 

Timescales to Reach Equilibrium(mid-plane)

neutral gas

T~300Kn~1014 /cc

CRs, X-rays［10 s］［0.1 s］

ionizationions

electrons

recombination
absorption［1014 s］
［10 s］［0.1 s］

dust grains

dynamical timescaletKep=1/Ω~107 s

Timescales to Reach Equilibrium(high altitude)

neutral gas

T~300Kn~105 /cc

CRs, X-rays［1010 s］

ionization［1010 s］
electrons

absorption

ionsrecombination［1010 s］
［1010 s］

［1010 s］

dust grains

dynamical timescaletKep=1/Ω~107 s

MHD Simulation Setupl Local 3D simulation• Athena MHD code with time-dependent ionization(Stone et al., 2008) 	 

h=√2c /Ωs• box size
(x, y, z) = (±0.5h, ±2h, ±4h)

• 16x32x192 grids

r0

• 0.01MMSN @2AU• parameters: βmid=106, fdg=10-4, a=1µm

yxMHD Equations
mass conservation 	 
equation of state 	 
momentum conservation 	 magnetic diffusivity	 
induction equation	 

chemical reactions 	 

(Blaes & Balbus 1994)

Ionization degree
density1	 

10-3	 

ionization
degree10-6	 

10-8	 

ionization degree

Propagation of Ionization Degreeheight [h]horizontally averaged

Averaged Ionization Degree
ionizationxe degree

10-5

averageanalytic

equilibrium

10-6

simulation
10-7

10-8

10-9-4-3-2-101z [h] [h]height234

Lower than equilibrium ionization at high altitude

ionization degree

Comparison with Equilibrium xeheight [h]horizontally averaged

Cause of ionization-degree propagation	 

Disk Wind by Turbulencehorizontally averaged

Wind Velocity
2
1.5
1vz/cs0.5
0-0.5-1-1.5-2-4-3-2-10z [h]

1

2

34

Timescales
10•
Advection
bywind
4

104

noise

103

102
10

tdiff1

tadv1

103• Turbulent mixing

chemicaltimescale1102

chemicaltimescale

101

100

100-4-3-2-101
z [h] [h]height234-4-3-2-1

01z [h] [h]height23

4

timescale [1/Ω]Variation of Chemical Timescale

time to travel 1htimescale tobecome equilibrium

height [h]x-y averaged

Time-variation at 4H
10-3xe, d

10

10-4

density	 

-5

10-6

10xe,4Hd,4Hxe	 

-7700

720

740

760
time

780

800

Rate Equation

M+ : metal ion
e : electrond : dust grains
Z : dust charge

electron’s number density+	 ionization+	 

recombination

absorption by grains⇒	 (when ionization degree is smaller than equilibrium)	 

Summary• We implemented time-dependent ionization calculation
into Athena MHD code
• We performed 3D simulations and investigated effects
of non-equilibrium ionization• Poorly-ionized gas inside disk is brought to surface• Ionization degree in surface layer have time variation• This may affect chemical evolution and MRI activity• Other non-ideal MHD effect must be considered