Steven A. Balbus

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The Magnetorotational Instablity Simmering
Issues and New Directions
Steven A. Balbus Ecole Normale
Supérieure Physics Department Paris, France IAS
MRI Workshop 16 May 2008
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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).
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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).
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The mechanism of the MRI is by now very familiar
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Schematic MRI
?2
angular momentum
?1
To rotation center
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Schematic MRI
angular momentum
To rotation center
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But many issues still simmer . . .
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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?
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MHD Turbulence ? Hydro Turbulence
The Kolmogorov picture of hydrodynamical
turbulence (large scales insensitive to small
scale dissipation)
Re1011 Re104
appears not to hold for MHD turbulence.
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• SIMMERING NUMERICAL ISSUES
• Is any turbulent MRI study converged? Does it
  ever not really matter?
• The good old small scales dont 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 Pmltlt1
  AND Pm gtgt 1 ?

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• SIMMERING NUMERICAL ISSUES
• Does Pm sensitivity vanish when Pmgtgt1 or Pmltlt1?
  If we cant set ??0, can we ever get away with
  setting one of them to 0?
• Should we trust lt?X ?Ygt correlations derived from
  simulations (e.g. good old ?)?
• How do we numerically separate mean
  quantities from their fluctuations ?

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

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Too little, and the field becomes sterile.
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• SIMMERING NUMERICAL ISSUES
• Everyone still uses Shakura-Sunyaev ? theory. To
  what extent do direct simulations support or
  undermine this?
• Radiative transport?

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Given our very real computational Limitations,
how can we put the MRI on an observational
footing?
The MRI is not without some distinct astrophysical
consequencesand some interesting possible
future directions.
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Direct confrontation with observations requires
care.
??
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Nature 2006, 441 953
The results demonstrate that accretion onto
black holes is fundamentally a magnetic
process.
with no accretion, is perfectly OK.
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Log-normal fit to Cygnus X-1 (low/hard
state) Uttley, McHardy Vaughan (2005)
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Non-Gaussianity in numerical simulations.
Gaussian fit
Log-normal fit
(From Reynolds et al. 2008)
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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.

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• 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.)

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14
6
10
8
12
16
4
Log10 T ?
3
AgtHgtO
OgtHgtA
2
HgtAgtO
HgtOgtA
1
8
6
10
12
16
14
Log10 (Density cm-3) ?
PARAMETER SPACE FOR NONIDEAL MHD
(Kunz Balbus 2005)
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14
6
10
8
12
16
4
PSD models
Log10 T ?
3
AgtHgtO
OgtHgtA
2
HgtAgtO
HgtOgtA
1
8
6
10
12
16
14
Log10 (Density cm-3) ?
PARAMETER SPACE FOR NONIDEAL MHD
(Kunz Balbus 2005)
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Ji et al. 2006, Nature, 444, 343
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dead zone
active zone
Tens of AU ? Planet forming zone?
0.3 AU
INNER REGIONS OF SOLAR NEBULA
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dead zone
1000 AU
GLOBAL PERSPECTIVE OF SOLAR NEBULA
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Reduced Model Techniques
dy/dt ?(T) y - A(T) y3 dT/dt Wy2 -
C(T) Stability criteria at fixed points CT 2
? gt 0 CT/C AT/A gt ?T /?
(Lesaffre 2008 for parasitic modes.)
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C(T)
stable
unstable
1/A(T)
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Balbus Lesaffre 2008
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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)

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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)
• gt limit cycle (acknowl. G. Lesur)

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Reduced Model Results
T
y
p
dotted
Solid T Dashed y Dotted p
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MAGNETOSTROPHIC MRI
(Petitdemange, Dormy, Balbus 2008)
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THE MRI AT THE
Petitdemange, Dormy and Balbus 2008
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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
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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
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z
?
r
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z
?
r
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z
?
r
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SUMMARY

• Dissipation. Local?
• Large scale structure
• Ouflows
• Dynamo connection
• Role of geometry

• Radiation
• ltXYgt
• Temporal Domain
• Outflow diagnostics

NUMERICS
OBSERVATIONAL PLANE

• Nonideal MHD, dust
• Dead zones
• Global accretion struc.
• Planets in MRI turb.

• Reduced Models
• Nontraditional applications
• Scalar Diffusion

UN(DER)EXPLORED DIRECTIONS
NONIDEAL MHD