June 08 MRI Transport properties 1

MRI-driven turbulent resistivity

Pierre-Yves Longaretti (LAOG)

Geoffroy Lesur (DAMTP)

June 08 MRI Transport properties 2

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 » : Ambipolar diffusion in YSOs (Königl and coworkers) Turbulence

Angular momentum

An

gu

lar

mo

me

ntu

m

June 08 MRI Transport properties 3

Jet emitting disks (JED) vs standard accretion disks (SAD)

At given accretion rate, in JEDs w.r.t. SADs: Smaller surface densities Higher accretion velocities

Much slower protoplanet migration Dead zone moving outwards

Surface density vs radius(fixed accretion rate)

(Combet & Ferreira 08)

June 08 MRI Transport properties 4

Points of contention

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:

Quenched by magnetic pressure (Königl 04) ?

Br+ << Bz Br

+~Bz

tt Pm ~ or > R/H

Pm ~ 1 for JEDs

ejection

opening

pressure

June 08 MRI Transport properties 5

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 :

r:φ:z=2:4:1 128x128x64 Re=1600 Pm=1 Linear analysis of axisymmetric modes

June 08 MRI Transport properties 6

radius

enfo

rced

ver

tical

mag

netic

fie

ld

B0

B0 [1+ cos(2 r/L)]

3D simulations:Methodology

« shearing box »

Image ImageSimulationbox

2

2

; ;

; ;

SHJ

EbuE

SHS

TRvbbuuTR

ij

j

iijz

TrrrTzrrrr

ij

αη = function of dimensionless parameters : β, ε (and Re, Rm…)

Alternatively:B = B0 ez + ΔB0 eφ orB = B0 eφ + ΔB0 eφ

June 08 MRI Transport properties 7

3D simulations:Current and emf correlation

Remarkable linear correlation

Unexpected off-diagonal turbulent resistivity component at least in one configuration

B, ΔB along z

B along zΔB along φ

June 08 MRI Transport properties 8

3D simulations:Anisotropy (diag. component) and correlations

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

Influence of

=100 (B, B // z)=1600 (B, B // z)

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

Influence of orientation of B

=100 (B // , B // )=100 (B // z, B // )

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

Influence of orientation of B

=100 (B // z, B // z)=100 (B // z, B // )

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

/

Correlation with

=100 (B // z, B // z)=1600 (B // z, B // z)=100 (B // z, B // )=100 (B // , B // )

Collapse of β and ε dependence

?

?

Anisotropy ~ 2 to 4

Varying efficiencyof transport withvertical or azimuth.mean field

June 08 MRI Transport properties 9

Linear analysisProblem 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)

-0.5 0 0.50.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

radius

Re(

stre

amfu

nctio

n)

numerical solutionanalytic solution

-0.5 0 0.5-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

radius

Re(

stre

amfu

nctio

n) (m

inus

fun

d. m

ode)

numerical solutionanalytic solution

ε = 0.3

channel mode

June 08 MRI Transport properties 10

Linear analysisResistive transport

-0.5 0 0.5-25

-20

-15

-10

-5

0

5

10

15

20

25

- J

E

Er

Ez

ε = 0.3, channel mode

Wrong sign !

Only the channel mode has somequalitative bearing on the problem

Why is < u x B >φ so large ?

Unexpected unless direct backreactionon the MRI driving process

-0.5 0 0.5-20

-15

-10

-5

0

5

10

15

20

E

Er

Ez

Wrong behavior

ε = 0.3, kx =1 mode

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

/

=100=1600

Correlation preserved but wrong magnitude

channel mode

Nice, but…

June 08 MRI Transport properties 11

Linear transport : how ?

-0.5 0 0.5

-0.5

0

0.5

poloidal velocity - fund. mode (num.)-0.5 0 0.5

-0.5

0

0.5

poloidal velocity - corr. to fund. mode

-0.5 0 0.5

-0.5

0

0.5

poloidal magnetic field - corr. to fund. mode-0.5 0 0.5

-0.5

0

0.5

poloidal magnetic field - fund. mode

< U x B >φ =

< UzBr – UrBz > ~

Correlation betweenfundamental channelmode and its deviations

ε = 0.3, channel modeUz

1Br0

Bz1Ur

0

June 08 MRI Transport properties 12

Linear transport : why ?Origin of Ur

0Bz1 correlation

disk centralobject B B + B

Edge on

Pole on

1

2dr2/dr > 0

d/dr < 0

differentialrotation

1

2

r

z

r

1

2

1

2

magnetictorqueangular

momentum

v

-v

vr

-vr

signright andmax

0. and

)(

1

11

101

z

rr

rzrzrt

B

BB

UBUBB

June 08 MRI Transport properties 13

Summary

Efficient resistive transport: Large turbulent diffusion : ~ a few 10-2 to 0.1 Smaller than viscous diffusion (unless mean Bφ ) Radial diffusion of B ~ 3 to 4 times radial diffusion of Bz

Implications for jet-emitting disks: Anisotropy in the right direction but about an order of

magnitude too small

Open issues : What of more realistic configurations (vertical

stratification) ? Role of physical dissipation (Pm ) ?

1/81/8 8