MRI-Driven Turbulence
with Resistivity
Takayoshi Sano
(Osaka Univ.)
Collaborators:
S.M. Miyama, S. Inutsuka
J.M. Stone, N.J. Turner
T.K. Suzuki, Y. Masada
Outline
• MRI in Resistive Disks
– Motivation
– Lundquist Number
• Small Scale Structures in MRI Driven
Turbulence
– Characteristic Scales & Energy Spectrum
– Effect of Magnetic Field Geometry
• Comparison with MRI in Viscous Disks
– Linear Dispersion Relation & 2D Simulations
Importance of Resistivity
• Protoplanetary Disks
– Resistivity >> Viscosity
– Net Vertical Fields
Originated from
Molecular Clouds
– If ionization fraction is
high enough, MRI is
important.
• Talks by Mark Wardle
& Neal Turner
• Saturation
Mechanism of MRI
– Magnetic Reconnections
– Thermalization by Joule
Heating
Machida et al. (2007)
MRI in Resistive Disks
• Resistivity  Lundquist Number
Sano & Miyama (1999)
Linear
Dispersion
Relation
Sano & Stone (2002)
Nonlinear Saturation
Amplitude
of the Stress
3D Shearing Box
Simulations with
Ohmic Dissipation
Critical Lundquist Number
• Critical value is always unity.
– Linear Analysis, Local Box Simulations, Stratified
Disk Simulations
• But, it depends on the saturated field
strength.
Turner et al. (2007)
Saturation Amplitude of MRI
• Importance of Net Magnetic Flux
– Veritcal or Toroidal Flux
• Resolution Dependence  Higher
Resolution
Net Bz
Net By
Pessah et al. (2007)
Preliminary Result
Resolution Dependence
of Saturated Stress
in Uniform By Runs
Stronger Initial By
Weaker By
1. Small-Scale Structures
in MRI-Driven Turbulence
Collaborators:
Shuichiro Inutsuka (Kyoto)
Takeru K. Suzuki (Tokyo)
High Resolution Resistive MHD Model
• Resistive MHD
– Local Shearing Box:
0.4H x 0.4H x 0.4H
– Resolution: 5123
– Field Geometry: No Net
Flux
– Lundquist Number: 30
– Time Integration: 75-90
orbits
Stress
2563
5123
Azimuthal Component of Magnetic Field
in Radial-Vertical Plane at 90 orbits
Orbit
Origin of Small Structures?
• Channel Flow (Axisymmetric MRI mode)
– Nonlinear Growth  Exact Solution of Nonlinear
MHD Eqs. (Magnetic field is amplified efficiently.)
– Characteristic Structures of a Channel Mode
• Strong Horizontal Field & Thin Current Sheets
Color:
Toroidal Field
Arrow:
Poloidal Field
Color:
Current Density
Arrow:
Poloidal Velocity
Unit Structure of MRI Driven Turbulence
• Lots of channel-flow structures can be
seen in MRI turbulence.
Color: Current Density
Color: Toroidal Field
B
B
A
A
Micro-Channel Flow at Point A
Color: Toroidal Field
Color: Current Density
Net-vertical field is non-zero. 
Magnetic energy is larger than the average value.
Micro-Channel Flow at Point B
Color: Toroidal Field
Color: Current Density
Net-vertical field is negative in this region.
Unit Structure of MRI Turbulence  Growth and Decay of Channel Flows
Resolution Dependence
• MRI wavelength &
current thickness
decreases with
increasing resolution.
Model 1
Box Size:
L x 4L x L
Grid:
N x 4N x N
N=32,64,128
Model 2
Box Size:
LxLxL
Grid:
NxNxN
N=128,256,512
Sano et al. in prep
Field Geometry
• Channel flow
structures are much
larger in models with
a net-vertical flux.
with Net Vertical Field
Channel Flow Size
(Unit Structure)
2H x 2H x H (256 x 256 x 128)
Saturation Amplitude
of MRI Turbulence
• Quantitative Analysis
of the Size
• Channel Flow
Evolution
without Net Vertical Field
Energy Spectrum of MRI Turbulence
• Anisotropic
Turbulence
– Elongated
by Shear Flow
• Weak Field
• Toroidal Field
Dominant
– Vertical
– Azimuthal
Sano et al. in prep
Power Spectrum at Inertia Range (1)
MRI Active Range
Best Fit
of the Power
Dissipation
Dominant Range
Inertia Range
Sano et al. in prep
Power Spectrum at Inertia Range (2)
• Vertical Direction
– Kolmogorov Spectrum
• Azimuthal Direction
– Weaker Power
– Steeper Decline
• Many Similarities to
Goldreich-Sridhar
Spectrum
Inertia Range
2. Comparison with MRI in
Viscous Disks
Collaborator:
Youhei Masada (ASIAA)
MRI in Viscous Disks
• Reynolds Number
Max. Growth Rate
Critical wavelength is
unchanged by viscosity.
Characteristic Scales of Viscous MRI
Maximum Growth Rate
Reynolds Number for MRI
Masada & Sano (2008)
Characteristic Scales of Resistive MRI
Maximum Growth Rate
Sano & Miyama (1998)
Lundquist Number for MRI
Two-Dimensional Simulations
• Viscous MHD
– Radial-Vertical Plane
– Shearing Box (without
Vertical Gravity)
Nonlinear Growth of a Channel Mode
even when
Masada & Sano (2008)
Viscosity vs. Resistivity
No suppression
by Viscosity
Resistivity suppresses
the MRI when
Viscosity may enhance
the activity of MRI
Masada & Sano (2008)
Interpretation of 2D Result (Resistive MRI)
MRI Growth 
B is amplified 
L shifts longer 
Less Dissipative 
No Suppression
TIME
Sano & Miyama (1998) Masada & Sano (2008)
MRI Growth 
B is amplified 
L shifts shorter 
More Dissipative 
Resistivity could
suppress the MRI
Interpretation of 2D Result (Viscous MRI)
Critical wavelength
increases with
the field strength
for any RMRI.
TIME
Masada & Sano (2008)
MRI Growth 
B is amplified 
L shifts longer 
Less Dissipative 
No Suppression
How About Doubly Diffusive System?
Nonlinear state can be inferred from the critical
wavelength expected by the linear Analysis.
There is the minimum
of the critical wavelength
for any Pm.  SMRI,crit
Masada & Sano (2008)
Prediction of Critical Lundquist Number
Masada & Sano (2008)
Summary
• MRI turbulence with resistivity is important for the
evolution of protoplanetary disks and to understand
the saturation mechanism.
HIGH RESOLUTION STUDY
• MRI turbulence consists of small channel flows, and
their size may be related to the saturation amplitude.
• Energy spectrum at the inertia range shows the
Kolmogorov-like power index.
RESISTIVITY VS. VISCOSITY
• Resistivity can suppress the growth of MRI more
efficiently compared with viscosity.
• 2D simulation results can be understood by the
characteristics of the critical wavelength for MRI