MRIDriven Turbulence with Resistivity

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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
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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

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Importance of Resistivity

• Protoplanetary Disks
• Resistivity gtgt 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)
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MRI in Resistive Disks

• Resistivity ? Lundquist Number

Sano Miyama (1999)
Sano Stone (2002)
Linear Dispersion Relation
Nonlinear Saturation Amplitude of the Stress
3D Shearing Box Simulations with Ohmic Dissipation
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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)
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Saturation Amplitude of MRI

• Importance of Net Magnetic Flux
• Veritcal or Toroidal Flux
• Resolution Dependence ? Higher Resolution

Net Bz
Net By
Preliminary Result
Pessah et al. (2007)
Resolution Dependence of Saturated Stress in
Uniform By Runs
Stronger Initial By
Weaker By
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1. Small-Scale Structures in MRI-Driven Turbulence

• Collaborators
• Shuichiro Inutsuka (Kyoto)
• Takeru K. Suzuki (Tokyo)

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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

2563
Stress
5123
Azimuthal Component of Magnetic Field in
Radial-Vertical Plane at 90 orbits
Orbit
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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
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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
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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.
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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
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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 N32,64,128
Model 2 Box Size L x L x L Grid N x N x
N N128,256,512
Sano et al. in prep
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Field Geometry
with Net Vertical Field

• Channel flow structures are much larger in models
  with a net-vertical flux.
• Quantitative Analysis of the Size
• Channel Flow Evolution

2H x 2H x H (256 x 256 x 128)
without Net Vertical Field
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Energy Spectrum of MRI Turbulence

• Anisotropic Turbulence
• Elongated by Shear Flow
• Weak Field
• Toroidal Field Dominant
• Vertical
• Azimuthal

Sano et al. in prep
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Power Spectrum at Inertia Range (1)
MRI Active Range
Best Fit of the Power
Dissipation Dominant Range
Inertia Range
Sano et al. in prep
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Power Spectrum at Inertia Range (2)

• Vertical Direction
• Kolmogorov Spectrum
• Azimuthal Direction
• Weaker Power
• Steeper Decline
• Many Similarities to Goldreich-Sridhar Spectrum

Inertia Range
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2. Comparison with MRI in Viscous Disks

• Collaborator
• Youhei Masada (ASIAA)

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MRI in Viscous Disks

• Reynolds Number

Critical wavelength is unchanged by viscosity.
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Characteristic Scales of Viscous MRI
Maximum Growth Rate
Reynolds Number for MRI
Masada Sano (2008)
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Characteristic Scales of Resistive MRI
Maximum Growth Rate
Sano Miyama (1998)
Lundquist Number for MRI
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Two-Dimensional Simulations
Masada Sano (2008)

• Viscous MHD
• Radial-Vertical Plane
• Shearing Box (without Vertical Gravity)

Nonlinear Growth of a Channel Mode even when
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Viscosity vs. Resistivity
No suppression by Viscosity
Resistivity suppresses the MRI when
Viscosity may enhance the activity of MRI
Masada Sano (2008)
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Interpretation of 2D Result (Resistive MRI)
MRI Growth ? B is amplified ? L shifts longer
? Less Dissipative ? No Suppression
MRI Growth ? B is amplified ? L shifts shorter
? More Dissipative ? Resistivity could suppress
the MRI
TIME
Sano Miyama (1998) Masada Sano (2008)
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Interpretation of 2D Result (Viscous MRI)
Critical wavelength increases with the field
strength for any RMRI.
MRI Growth ? B is amplified ? L shifts longer
? Less Dissipative ? No Suppression
TIME
Masada Sano (2008)
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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)
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Prediction of Critical Lundquist Number
Masada Sano (2008)
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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