MRI-driven turbulent resistivity
Pierre-Yves Longaretti (LAOG)
Geoffroy Lesur (DAMTP)
June 08
MRI Transport properties
1
Angular momentum
Turbulent resistivity and ejection

Standard accretion disk (non-existent or weak ejection):


Outwards transport. Requires « anomalous viscosity »
Jet-emitting disk (strong ejection, requires β~1 and PmT ~1):

Vertical transport. Requires « anomalous resistivity » :


June 08
Angular momentum
Ambipolar diffusion in YSOs (Königl and coworkers)
Turbulence
MRI Transport properties
2
Jet emitting disks (JED) vs standard
accretion disks (SAD)
Surface density vs radius
(fixed accretion rate)
(Combet & Ferreira 08)

At given accretion rate, in JEDs w.r.t. SADs:




June 08
Smaller surface densities
Higher accretion velocities
Much slower protoplanet migration
Dead zone moving outwards
MRI Transport properties
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Points of contention
opening
Br+ << Bz
Br+~Bz
tn ~ th  Pm ~ or > R/H
ejection
pressure

Pm ~ 1 for JEDs

LPP 94a: advection of flux by the disk conflicts with ejection
requirement:


Relevance of initial conditions (Br~Bz on td due to collapse) ?
LPP94b, Cao & Spruit 02: ejection instability:

June 08
Quenched by magnetic pressure (Königl 04) ?
MRI Transport properties
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What do we want to know ?

Turbulent resistivity = correlation between the emf
and J :



Is it present ?
If so, why and what is the resulting « η »?
Weapons:

3D MHD shearing box simulations :


June 08
r:φ:z=2:4:1
128x128x64
Re=1600
Pm=1
Linear analysis of axisymmetric modes
MRI Transport properties
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3D simulations:
Methodology
B
0
B [1+ cos(2 r/L)]
Image
Simulation
box
Image
0
enforced vertical magnetic field
« shearing box »
Alternatively:
B = B0 ez + ΔB0 eφ or
B = B0 eφ + ΔB0 eφ
radius
Rr  Tr  ur u  br b  z ; vT n r
Rr  Tr
S
; n 

h
 
E
E  u  b  z ; hij   i ; hij  ij 2
Jj
SH
nT
SH 2
αη = function of dimensionless parameters : β, ε (and Re, Rm…)
June 08
MRI Transport properties
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3D simulations:
Current and emf correlation
B along z
ΔB along φ
B, ΔB along z

Remarkable linear correlation

Unexpected off-diagonal turbulent resistivity component at
least in one configuration
June 08
MRI Transport properties
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3D simulations:
Anisotropy (diag. component) and correlations
Influence of 
Influence of orientation of B
0.05
0.1
=100 (B, B // z)
=1600 (B, B // z)
0.04
0.08
0.03
=100 (B // , B // )
h
h
0.06
0.02
?
0.01
0
0
=100 (B // z, B // )
0.04
?
0.02
0.2
0.4
0.6
0.8
0
0
1

0.2
Influence of orientation of B
0.1
2
Varying efficiency
0.4
0.6
1
of 0.8
transport
with

vertical or azimuth.
Correlation with n
mean field
0.08
1.5
0.06
=100 (B // z, B // z)
=100 (B // z, B // z)
=1600 (B // z, B // z)
h /n
h
=100 (B // z, B // )
0.04
0.2
0.4
0.6

June 08
=100 (B // , B // )
0.5
0.02
0
0
=100 (B // z, B // )
1
0.8
1
0
0
0.2
0.4
0.6
0.8
1

MRI Transport properties
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Linear analysis
Problem formulation



Interest  recurrence of channel mode in 3D simulations
Axisymmetric modes, incompressible motions  reduced to second
order equation for the poloidal velocity stream function
Analytic solution through an expansion in ε = ΔB/B (B, ΔB // z)
1.25
1.2
0.2
ε = 0.3
0.15
Re(streamfunction) (minus fund. mode)
1.15
Re(streamfunction)
1.1
1.05
1
0.95
0.9
0.85
0.8
-0.5
June 08
channel mode
numerical solution
analytic solution
0
radius
0.5
0.1
0.05
0
-0.05
-0.1
-0.15
numerical solution
-0.2
-0.25
-0.5
MRI Transport properties
analytic solution
0
radius
0.5
9
Linear analysis
Resistive transport
2
25
-h

20
J
channel mode

E

15
1.5
E
r
5
Nice, but…
E
z
h /n
10
0
Correlation preserved
but wrong magnitude
1
-5
-10
0.5
Wrong sign !
-20
-25
-0.5
=100
=1600
-15
ε = 0.3, channel mode
0
0.5
0
0
0.1
0.2
0.3
0.4
0.5

20
E
15
ε = 0.3, kx =1 mode

Only the channel mode has some
qualitative bearing on the problem
E
r
E
10
z
5
Why is < u x B >φ so large ?
Unexpected unless direct backreaction
on the MRI driving process
0
-5
-10
-15
-20
-0.5
June 08
Wrong behavior
0
0.5
MRI Transport properties
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Linear transport : how ?
0.5
0.5
0
0
-0.5
-0.5
-0.5
0
0.5
poloidal velocity - fund. mode (num.)
-0.5
0
0.5
poloidal velocity - corr. to fund. mode
ε = 0.3, channel mode
0.5
Bz1Ur0
0
0.5
Correlation between
fundamental channel
mode and its deviations
0
-0.5
-0.5
-0.5
0
0.5
poloidal magnetic field - corr. to fund. mode
June 08
Uz1Br0
< U x B >φ =
< UzBr – UrBz > ~
-0.5
0
0.5
poloidal magnetic field - fund. mode
MRI Transport properties
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Linear transport : why ?
Origin of Ur0Bz1 correlation
 t Br1  B zU r0 (  B zU r1 )
 r Br1 and
.B1  0
Bz1 max and right sign
June 08
MRI Transport properties
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Summary

Efficient resistive transport:




Implications for jet-emitting disks:


Large turbulent diffusion : h ~ a few 10-2 to 0.1
Smaller than viscous diffusion (unless mean Bφ )
Radial diffusion of Bh ~ 3 to 4 times radial diffusion of Bz
Anisotropy in the right direction but h about an order of
magnitude too small
Open issues :


What of more realistic configurations (vertical stratification) ?
Role of physical dissipation (Pm ) ?
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June 08
MRI Transport properties
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