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Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France IAS MRI Workshop

The Magnetorotational Instablity: Simmering Issues and New Directions. Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France IAS MRI Workshop 16 May 2008. Our conceptualization of astrophysical magnetic fields has undergone a sea change:.

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Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France IAS MRI Workshop

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  1. The Magnetorotational Instablity: Simmering Issues and New Directions Steven A. Balbus Ecole Normale Supérieure Physics Department Paris,France IAS MRI Workshop 16 May 2008

  2. Our conceptualization of astrophysical magnetic fields has undergone a sea change: Weak B-field in disk, before 1991 (Moffatt 1978). Weak B-field in disk, after 1991 (Hawley 2000).

  3. The MAGNETOROTATIONAL INSTABILITY (MRI) has taught us that weak magnetic fields are not simply sheared out in differentially rotating flows. The presence of B leads to a breakdown of laminar rotation into turbulence. More generally, free energy gradients dT/dr, d/dr become sources of instability, not just diffusive fluxes. The MRI is one of a more general class of instabilities (Balbus 2000, Quataert 2008).

  4. The mechanism of the MRI is by now very familiar:

  5. Schematic MRI 2 angular momentum 1 To rotation center

  6. 2 1 Schematic MRI angular momentum To rotation center

  7. But many issues still simmer . . .

  8. Numerical simulations of the MRI verified enhanced turbulent angular momentum transport. This was seen in both local (shearing box) and global runs. But the simulation of a turbulent fluid is an art, and fraught with misleading traps for the unwary. Hawley & Balbus 1992 WHAT TURNS OFF THE MRI? RELATION TO DYNAMOS?

  9. MHD Turbulence  Hydro Turbulence The Kolmogorov picture of hydrodynamical turbulence (large scales insensitive to small scale dissipation) … Re=1011 Re=104 …appears not to hold for MHD turbulence.

  10. SIMMERING NUMERICAL ISSUES: Is any turbulent MRI study converged? Does it ever not really matter? The good old “small scales don’t matter” days are gone. The magnetic Prandtl number Pm=/ has an unmistakable effect on MHD turbulence (AS, SF, GL, P-YL), fluctuations and coherence increase with Pm (at fixed Re or Rm). Disks with Pm<<1 AND Pm >> 1 ?

  11. SIMMERING NUMERICAL ISSUES: Does Pm sensitivity vanish when Pm>>1 or Pm<<1? If we can’t set ==0, can we ever get away with setting one of them to 0? Should we trust <X Y> correlations derived from simulations (e.g. good old )? How do we numerically separate mean quantities from their fluctuations ?

  12. SIMMERING NUMERICAL ISSUES: Does anyone know how to do a global disk simulation with finite <BZ> ? What aspects of a numerical simulation should we allow to be compared with observations? Too much and we will be seen to over claim . . .

  13. January Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20 22 23 24 25 26 27 28 29 30 31 Too little, and the field becomes sterile.

  14. SIMMERING NUMERICAL ISSUES: Everyone still uses Shakura-Sunyaev  theory. To what extent do direct simulations support or undermine this? Radiative transport?

  15. Given our very real computational Limitations, how can we put the MRI on an observational footing? The MRI is not without some distinct astrophysical consequences…and some interesting possible future directions.

  16. Direct confrontation with observations requires care. ??

  17. Nature 2006, 441 953 “The results demonstrate that accretion onto black holes is fundamentally amagnetic process.” with no accretion, is perfectly OK.

  18. Log-normal fit to Cygnus X-1 (low/hard state) Uttley, McHardy & Vaughan (2005)

  19. Non-Gaussianity in numerical simulations. Gaussian fit Log-normal fit (From Reynolds et al. 2008)

  20. Why might MRI be lognormal? • Numerically, MRI exhibits linear local exponential growth, abruptly terminated when fluid elements are mixed. • Lifetime of linear growth is a random gaussian (symmetric bell-shaped) variable, t. • Local amplitudes of fields grow like exp(at), then themalized and radiated; responsible for luminosity. • If t is a gaussian random variable, then exp(at) is a lognormal random variable.

  21. SIMMERING NUMERICAL ISSUES: 7. Protostellar disks are one of the most imortant MRI challenges, and perhaps the most difficult. (Nonideal MHD, dust, molecules, nonthermal ionization…) Global problem, passive scalar diffusion. We are clearly in the Hall regime. This is never simulated, based on ONE study: Sano & Stone. Is there more? (Studies by Wardle & Salmeron.)

  22. 6 8 10 12 14 16 4 Log10 T  3 A>H>O O>H>A 2 H>A>O H>O>A 1 6 8 10 12 14 16 Log10 (Density cm-3)  PARAMETER SPACE FOR NONIDEAL MHD (Kunz & Balbus 2005)

  23. 6 8 10 12 14 16 4 PSD models Log10 T  3 A>H>O O>H>A 2 H>A>O H>O>A 1 6 8 10 12 14 16 Log10 (Density cm-3)  PARAMETER SPACE FOR NONIDEAL MHD (Kunz & Balbus 2005)

  24. Ji et al. 2006, Nature, 444, 343

  25. “dead zone” active zone Tens of AU  Planet forming zone? ~ 0.3 AU INNER REGIONS OF SOLAR NEBULA

  26. dead zone ~ 1000 AU GLOBAL PERSPECTIVE OF SOLAR NEBULA

  27. Reduced Model Techniques: dy/dt = (T) y - A(T) y3 dT/dt = Wy2 - C(T) Stability criteria at fixed points: CT + 2  > 0 CT/C + AT/A > T / (Lesaffre 2008 for parasitic modes.)

  28. C(T) stable unstable 1/A(T)

  29. Balbus & Lesaffre 2008

  30. A parasite interpretation forthe channel eruptions (Goodman & Xu) • Energy is found either in channel flow or in parasites • Temperature peaks lag (due to finite radiative cooling) • Parasites grow only when channel flow grows non-linear • Rate of growth increases with channel amplitude (as predicted by Goodman & Xu 1998)

  31. Parasitic Modes Add a variable for parasitic amplitude (p) : dy/dt = (1-h) y - y p dp/dt = - p + y p dT/dt = y2 + p2 – C(T) => limit cycle (acknowl.: G. Lesur)

  32. Reduced Model Results T y “dotted” p Solid =T Dashed= y Dotted = p

  33. MAGNETOSTROPHIC MRI (Petitdemange, Dormy, Balbus 2008)

  34. THE MRI AT THE Petitdemange, Dormy and Balbus 2008

  35. Magnetostrophic MRI, in its entirety: 2  x v = (B •) b/4 Db/Dt =  x ( v x B -   x b) b, v ~ exp (t -i kz), vA2 = B2/ 4 42 ( + k2)2 + (kvA)2 [ (kvA)2 +d 2/dln R] =0

  36. Magnetostrophic MRI |d ln  /d ln R | ~ 10-6 Elsasser number  = vA2 / 2 ~ 1 (must be order unity for k to “fit in.”) max = (1/2) |d/d ln R| /[1+(1+ 2)1/2] (kvA)2max= (1/2) |d2/d ln R| [1-(1+ 2) -1/2] 42 ( + k2)2 + (kvA)2 [ (kvA)2 +d 2/dln R] =0

  37. z  r

  38. z  r

  39. Azimuthal tension Coriolis balance Coriolois from more radial flow z  r

  40. SUMMARY: • Dissipation. Local? • Large scale structure • Ouflows • Dynamo connection • Role of geometry • Radiation • <XY> • Temporal Domain • Outflow diagnostics NUMERICS OBSERVATIONALPLANE • Nonideal MHD, dust • Dead zones • Global accretion struc. • Planets in MRI turb. • Reduced Models • Nontraditional applications • Scalar Diffusion UN(DER)EXPLORED DIRECTIONS NONIDEAL MHD

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