1. Models of Turbulent Angular Momentum Transport Beyond the  Parameterization Martin Pessah Institute for Advanced Study Workshop on Saturation and Transport Properties of MRI-Driven Turbulence - IAS- June 16, 2008 C.K. Chan - ITC, Harvard D. Psaltis - University. of Arizona

2. Timescales and Lengthscales in Accretion Disks 1Gyr 1yr 1sec 0.1AU1AU1000AU Mean Field Models BLR Inside Horizon Numerical Simulations Viscous Orbital Saturation BH Growth BH Variability Hubble Time 10 7 M o Disk Spectrum

3. Standard Accretion Disk Model This prescription dictates the structure of the accretion disk Angular momentum transport in turbulent media Shakura & Sunyaev (‘70s) Angular momentum transport: Turbulence and magnetic fields

4. Standard DiskMRI Disk MRI

5. The need to go beyond  - viscosity Issues… *Dynamics:  is not constant *Causality: Information propagates across sonic points (Stress reacts instantly to changes in the flow) *Stability criterion: Standard model works always (Stresses only for MRI-unstable disks) *”Viscosity”: MRI does not behave like Newtonian viscosity

6. Long-Term Evolution of MRI MRI drives MHD turbulence at a well defined scale which depends only on the magnetic field and the shear small scales large scales

7. MRI Accretion Disks in Fourier Space k “Stress Power” MRI DissipationParker Mode interactions Parasitic Instabilities small scales large scales (Talk by C.K. Chan later today)

8. The Closure Problem in MHD Turbulence We can derive an equation for angular momentum conservation Need for a CLOSURE model beyond alpha!! which involves 2nd order correlations… … but they involve 3rd order correlations… We can derive equations for 2nd order correlations… We can derive equations for 3nd order correlations… (Closure models with no mean fields; papers by Kato et al 90’s; Ogilvie 2003)

9. Stress Modeling with No Mean Fields Kato & Yoshizawa, 1995 Pressure-strain tensor, tends to isotropize the turbulence

10. Stress Modeling with No Mean Fields Ogilvie 2003 C2 return to isotropy in decaying turbulence C3, C4 energy transfer C1, C5 energy dissip.

11. New correlation drives the growth of Reynolds & Maxwell stresses A Model for Turbulent MRI-driven Stresses

12. W in the Turbulent State W tensor dynamically important! -W r  -MaxwellReynolds

13. The model produces initial exponential growth and leads to stresses with ratios in agreement with the saturated regime The parameter  controls the ratio between Reynolds & Maxwell stresses Pessah, Chan, & Psaltis, 2006a  =0.3 Calibration of MRI Saturation with Simulations

14. Pessah, Chan, & Psaltis, 2006b Hawley et al ‘95 Calibration of MRI Saturation with Simulations A physically motivated model for angular momentum transport Keplerian disks that incorporates the MRI The parameter  controls the level at which the magnetic energy saturates  =11.3

15. Shakura & Sunyaev, 1973 Pessah, Chan, & Psaltis, 2008 Shakura & Sunyaev, 1973 Pessah, Chan, & Psaltis, 2006b Pessah, Chan, & Psaltis, 2008 Shakura & Sunyaev, 1973 Pessah, Chan, & Psaltis, 2006b Pessah, Chan, & Psaltis, 2008 A Local Model for Angular Momentum Transport in Turbulent Magnetized Disks A model for MRI-driven angular momentum transport in agreement with numerical simulations Pessah, Chan, & Psaltis, 2006b Simulations by Hawley et al. 1995

16. Scaling in MRI Simulations Pessah, Chan, & Psaltis, 2007; Sano et al. 2004 Deeper understanding of scalings? Non-zero B z (few % B eq) Magnetic pressure must be important in setting H Spectral hardening (Blaes et al. 2006) (Talk by S. Fromang later today)

17. Viscous, Resistive MRI Pessah & Chan, 2008 Pm<<1 Pm>>1 (Various limits studied by Sano et al., Lesaffre &Balbus, Lesur & Longaretti, … )

18. Viscous, Resistive MRI: Stresses & Energy * Lowest ratio magnetic-to-kinetic: Ideal MHD * Linear MRI effective at low/high Pm as long as Re large * Need higher Re to drive MHD turbulence at low Pm (Lesur & Longaretti, 2007; Fromang et al., 2007) Pm<<1 Pm>>1 Pm<<1 Pm>>1

19. Viscous, Resistive MRI Modes Pm=1

20. Viscous, Resistive MRI Modes Ideal MHD khkh Parasitic Instabilities sensitive to viscosity and resistivity (Goodman & Xu, 1994)

21. Viscous, Resistive MRI Modes Pessah & Chan, 2008 Pm<<1 Pm>>1

22. Viscous, Resistive Parasitic Instabilities PRELIMINARY (with J. Goodman, in prep.) Parasitic Instabilities seem to be weaker at high Pm Hint for difficulties to drive MRI at low Pm? Can the destruction of primary MRI modes by parasitic instabilities explain (Re, Pm) dependencies? 1ow Pm high Pm