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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Weak B-field in disk, before
1991 (Moffatt 1978).
Weak B-field in disk, after
1991 (Hawley 2000).
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:
Schematic MRI
2
angular momentum
1
To rotation center
Schematic MRI
2
angular momentum
1
To rotation center
But many issues still simmer . . .
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?
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.
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, PYL), fluctuations and coherence increase
with Pm (at fixed Re or Rm). Disks with
Pm<<1 AND 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 <X Y> 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 <BZ> ?
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 . . .
January
Su Mo
1
7 8
14 15
21 22
28 29
Tu We Th Fr
2
3
4 5
9 10 11 12
16 17 18 19
23 24 25 26
30 31
Sa
6
13
20
27
Too little, and the field becomes sterile.
SIMMERING NUMERICAL ISSUES:
6. Everyone still uses Shakura-Sunyaev 
theory. To what extent do direct
simulations support or undermine this?
Radiative transport?
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.
Direct confrontation with observations requires care.
??
Nature 2006, 441 953
“The results demonstrate that accretion onto
black holes is fundamentally a magnetic
process.”
with no accretion,
is perfectly OK.
Log-normal fit to Cygnus X-1
(low/hard state)
Uttley, McHardy & Vaughan (2005)
Non-Gaussianity in numerical simulations.
Log-normal fit
Gaussian fit
(From Reynolds et al. 2008)
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.
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.)
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)
6
8
10
4
Log10 T 
12
14
16
PSD models
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)
Ji et al. 2006, Nature, 444, 343
active zone
~ 0.3 AU
“dead zone”
Tens of AU 
Planet forming zone?
INNER REGIONS OF SOLAR NEBULA
dead zone
~ 1000 AU
GLOBAL PERSPECTIVE OF SOLAR NEBULA
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.)
C(T)
stable
unstable
1/A(T)
Balbus & Lesaffre 2008
A parasite interpretation for
the 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 nonlinear
• 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 = y2 + p2 – C(T)
=> limit cycle (acknowl.: G. Lesur)
Reduced Model Results
T
y
“dotted”
Solid =T Dashed= y Dotted = p
p
MAGNETOSTROPHIC MRI
(Petitdemange, Dormy, Balbus 2008)
THE MRI AT THE
Petitdemange, Dormy and Balbus 2008
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
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
z

r
z

r
z

Azimuthal tension
r
Coriolois from more radial flow
Coriolis balance
SUMMARY:
• Dissipation. Local?
• Large scale structure
• Ouflows
• Dynamo connection
• Role of geometry
NUMERICS
• Nonideal MHD, dust
• Dead zones
• Global accretion struc.
• Planets in MRI turb.
NONIDEAL MHD
• Radiation
• <XY>
• Temporal Domain
• Outflow diagnostics
OBSERVATIONAL PLANE
• Reduced Models
• Nontraditional applications
• Scalar Diffusion
UN(DER)EXPLORED
DIRECTIONS