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

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

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).

The mechanism of the MRI is by now very familiar:

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

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

But many issues still simmer...

Hawley & Balbus 1992 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. WHAT TURNS OFF THE MRI? RELATION TO DYNAMOS?

The Kolmogorov picture of hydrodynamical turbulence (large scales insensitive to small scale dissipation) … MHD Turbulence  Hydro Turbulence Re=10 11 Re=10 4 …appears not to hold for MHD turbulence.

SIMMERING NUMERICAL ISSUES: 1.Is any turbulent MRI study converged? Does it ever not really matter? 2.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 ?

SIMMERING NUMERICAL ISSUES: 2.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? 3.Should we trust correlations derived from simulations (e.g. good old  )? How do we numerically separate mean quantities from their fluctuations ?

SIMMERING NUMERICAL ISSUES: 4.Does anyone know how to do a global disk simulation with finite ? 5. What aspects of a numerical simulation should we allow to be compared with observations? Too much and we will be seen to over claim...

Too little, and the field becomes sterile. January Su Mo Tu We Th Fr Sa

SIMMERING NUMERICAL ISSUES: 6.Everyone still uses Shakura-Sunyaev  theory. To what extent do direct simulations support or undermine this? Radiative transport?

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

?? Direct confrontation with observations requires care.

with no accretion, is perfectly OK. “ The results demonstrate that accretion onto black holes is fundamentally a magnetic process.” Nature 2006,

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

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

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. Why might MRI be lognormal?

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. 8.We are clearly in the Hall regime. This is never simulated, based on ONE study: Sano & Stone. Is there more? (Studies by Wardle & Salmeron.)

A>H>O H>A>O H>O>A O>H>A Log 10 (Density cm -3)  Log 10 T  PARAMETER SPACE FOR NONIDEAL MHD (Kunz & Balbus 2005)

A>H>O H>A>O H>O>A O>H>A Log 10 (Density cm -3)  Log 10 T  PARAMETER SPACE FOR NONIDEAL MHD (Kunz & Balbus 2005) PSD models

Ji et al. 2006, Nature, 444, 343

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

GLOBAL PERSPECTIVE OF SOLAR NEBULA ~ 1000 AU dead zone

dy/dt =  (T) y - A(T) y 3 dT/dt = Wy 2 - C(T) Stability criteria at fixed points: C T + 2  > 0 C T /C + A T /A >  T /  Reduced Model Techniques: (Lesaffre 2008 for parasitic modes.)

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

Balbus & Lesaffre 2008

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

Parasitic Modes Add a variable for parasitic amplitude (p) : dy/dt = (1-h) y - y p dp/dt = -  p + y p dT/dt = y 2 + p 2 – C(T) => limit cycle (acknowl.: G. Lesur)

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

MAGNETOSTROPHIC MRI (Petitdemange, Dormy, Balbus 2008)

THE MRI AT THE Petitdemange, Dormy and Balbus 2008

2  x v = (B  ) b/4  Db/Dt =  x ( v x B -   x b) Magnetostrophic MRI, in its entirety: b, v ~ exp (  t -i kz), v A 2 = B 2 / 4  4  2 (  +  k 2 ) 2 + (kv A ) 2 [ (kv A ) 2 +d  2 /dln R] =0

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

r z 

r z 

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

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