MRI-Driven 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. 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 Importance of Resistivity 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. 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 High Resolution Resistive MHD Model 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. MRI wavelength & current thickness decreases with increasing resolution. Resolution Dependence Model 2 Box Size: L x L x L Grid: N x N x N N=128,256,512 Model 1 Box Size: L x 4L x L Grid: N x 4N x N N=32,64,128 Sano et al. in prep

14. Channel flow structures are much larger in models with a net-vertical flux. Quantitative Analysis of the Size Channel Flow Evolution Channel Flow Size (Unit Structure) Saturation Amplitude of MRI Turbulence Field Geometry with Net Vertical Field 2H x 2H x H (256 x 256 x 128) without Net Vertical Field

15. Anisotropic Turbulence Elongated by Shear Flow Weak Field Toroidal Field Dominant Vertical Azimuthal Energy Spectrum of MRI Turbulence 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. Vertical Direction Kolmogorov Spectrum Azimuthal Direction Weaker Power Steeper Decline Many Similarities to Goldreich-Sridhar Spectrum Power Spectrum at Inertia Range (2) Inertia Range

18. 2. Comparison with MRI in Viscous Disks Collaborator: Youhei Masada (ASIAA)

19. Max. Growth Rate 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. Viscous MHD Radial-Vertical Plane Shearing Box (without Vertical Gravity) Two-Dimensional Simulations Masada & Sano (2008) 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