MRIDriven Turbulence with Resistivity - PowerPoint PPT Presentation

Title: MRIDriven Turbulence with Resistivity


1
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
2
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

3
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)
4
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
5
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)
6
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
7
1. Small-Scale Structures in MRI-Driven Turbulence

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

8
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
9
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
10
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
11
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.
12
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
13
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
14
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
15
Energy Spectrum of MRI Turbulence

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

Sano et al. in prep
16
Power Spectrum at Inertia Range (1)
MRI Active Range
Best Fit of the Power
Dissipation Dominant Range
Inertia Range
Sano et al. in prep
17
Power Spectrum at Inertia Range (2)

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

Inertia Range
18
2. Comparison with MRI in Viscous Disks

• Collaborator
• Youhei Masada (ASIAA)

19
MRI in Viscous Disks

• Reynolds Number

Critical wavelength is unchanged by viscosity.
20
Characteristic Scales of Viscous MRI
Maximum Growth Rate
Reynolds Number for MRI
Masada Sano (2008)
21
Characteristic Scales of Resistive MRI
Maximum Growth Rate
Sano Miyama (1998)
Lundquist Number for MRI
22
Two-Dimensional Simulations
Masada Sano (2008)

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

Nonlinear Growth of a Channel Mode even when
23
Viscosity vs. Resistivity
No suppression by Viscosity
Resistivity suppresses the MRI when
Viscosity may enhance the activity of MRI
Masada Sano (2008)
24
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)
25
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)
26
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)
27
Prediction of Critical Lundquist Number
Masada Sano (2008)
28
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