Steven A. Balbus

Ecole Normale SupérieurePhysics Department

Paris, FranceIAS MRI Workshop

16 May 2008

The Magnetorotational Instablity: Simmering Issues and New Directions

Weak B-field in disk, before1991 (Moffatt 1978). Weak B-field in disk, after

1991 (Hawley 2000).

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are needed to see this picture.

Our conceptualization of astrophysical magnetic fields hasundergone a sea change:

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are needed to see this picture.

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/drbecome sources of instability, not just diffusive fluxes. The MRI is one of a more general class ofinstabilities (Balbus 2000, Quataert 2008).

The mechanism of the MRI is by now very familiar:

angular momentum

Schematic MRI

To rotation center

2

1

angular momentum

Schematic MRI

To rotation center

2

1

But many issues still simmer . . .

Hawley & Balbus 1992

Numerical simulations of the MRIverified 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 withmisleading 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=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, P-YL), 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 . . .

Too little, and the field becomes sterile.

January Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 8 9 10 11 12 1314 15 16 17 18 19 2021 22 23 24 25 26 2728 29 30 31

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 distinctastrophysical consequences…and some interesting possible future directions.

Given our very real computational Limitations, how can we put the MRIon 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, 441 953

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.)

86 10 12

14

16

A>H>O

1

2

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4

H>A>O

14

6 10

H>O>A

O>H>A

8 12 16

Log10 (Density cm-3)

Log10 T

PARAMETER SPACE FOR NONIDEAL MHD (Kunz & Balbus 2005)

86 10 12

14

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A>H>O

1

2

3

4

H>A>O

14

6 10

H>O>A

O>H>A

8 12 16

Log10 (Density cm-3)

Log10 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 AUTens of AU Planet forming zone?

GLOBAL PERSPECTIVE OF SOLAR NEBULA

~ 1000 AU

dead zone

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 /

Reduced Model Techniques:

(Lesaffre 2008 for parasitic modes.)

C(T)

1/A(T)

stable

unstable

Balbus & Lesaffre 2008

A parasite interpretation forthe channel eruptions (Goodman & Xu)

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

Parasitic ModesAdd a variable for parasitic amplitude (p) :

dy/dt = (1-) y - y p

dp/dt = - p + y p

dT/dt = y2 + p2 – 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), vA2 = B2/ 4

42 ( + k2)2 + (kvA)2 [ (kvA)2 +d 2/dln R] =0

|d ln /d ln R | ~ 10-6

Elsasser number = vA2 / 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]

(kvA)2max

= (1/2) |d2/d ln R| [1-(1+ 2) -1/2]

42 ( + k2)2 + (kvA)2 [ (kvA)2 +d 2/dln R] =0

r

z

r

z

r

z

Azimuthal tension

Coriolis balanceCoriolois 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• <XY>• Temporal Domain• Outflow diagnostics

NUMERICS OBSERVATIONAL PLANE

NONIDEAL MHD UN(DER)EXPLOREDDIRECTIONS