Large Scale Dynamo Large Scale Dynamo Action in MRI DisksAction in MRI Disks

Role of stratificationRole of stratification

Dynamo cyclesDynamo cycles

Mean-field interpretationMean-field interpretation

Incoherent alpha-shear dynamoIncoherent alpha-shear dynamo

Axel Brandenburg (Axel Brandenburg (Nordita, StockholmNordita, Stockholm))

2

Unstratified MRI turbulenceUnstratified MRI turbulence

5123

w/o hypervisc.t = 60 = 2 orbits

No large scale field(i) Too short?(ii) No stratification?

3

Vorticity and DensityVorticity and Density

See poster by Tobi Heinemann on density wave excitation!

4

Small scales dominateSmall scales dominate

5

Animated spectraAnimated spectra

Red EM

Black EK

6

Old stratified runsOld stratified runs

Brandenburg et al. (1995)

<By >

7

Different boundary conditionsDifferent boundary conditions

0, zzyx BBB

0,, zzyzx BBB

8

Tall disk withTall disk withpotential field potential field

b.c.b.c.

2x 2 x 8

9

z-t diagramz-t diagram

10

Longer and biggerLonger and bigger

Parity not always well defined

11

Mean-field interpretationMean-field interpretation

• Correlation method– MRI accretion discs (Brandenburg & Sokoloff 2002)– Galactic turbulence (Kowal et al. 2005, 2006)

• Test field method– Stationary geodynamo (Schrinner et al. 2005, 2007)– Shear flow turbulence (Brandenburg 2005)

JBUA ε

tbuε

jijjijj JB *

turbulent emf

effect and turbulentaagnetic diffusivity

12

Calculate full Calculate full ijij and and ijij tensors tensors

JBUA

t

JbuBUA

t

jbubuBubUa

t

pqpqpqpqpqpq

tjbubuBubU

a

Original equation (uncurled)

Mean-field equation

fluctuations

Response to arbitrary mean fields

13

Test fieldsTest fields

0

sin

0

,

0

cos

0

0

0

sin

,

0

0

cos

2212

2111

kzkz

kzkz

BB

BBpqkjijk

pqjij

pqj BB ,

kzkkz

kzkkz

cossin

sincos

1131121

1

1131111

1

21

1

111

113

11

cossin

sincos

kzkz

kzkz

k

213223

113123

*22

*21

*12

*11

Example:

14

Validation: Roberts flowValidation: Roberts flowpqpqpqpqpq

pq

tjbubuBubU

a

1frmsm3

1t

rmsm31

-kuR

uR

SOCA

ykxk

ykxk

ykxk

uU

yx

yx

yx

rms

coscos2

cossin

sincos

2/fkkk yx

1frms3

1t0

rms31

0

-ku

u

normalize

SOCA result

15

(i) Turbulence: kinematic (i) Turbulence: kinematic and and tt independent of Rm independent of Rm

1frms3

10

rms31

0

ku

u

Galloway-Proctor:

16

Full alpha tensor for MRIFull alpha tensor for MRI

yy negative, as before (Brandenburg et al. 1995)

17

Full eta tensorFull eta tensor

xx and xx the same and positive (new)

yx always positive (new)

18

(ii) Shear turbulence(ii) Shear turbulence

JJμJδε t

0

0

μ

*21

*12

t

*12

21tt

*21

21t

1

k

S

k

Growth rate

Use S<0, so need negative *21 for dynamo

19

Case with just shearCase with just shear

Similar to G. Lesur’s plot of yesterday!

20

Dependence on Sh and RmDependence on Sh and Rm

Againyx always positive

21

Gaussian fluctuationsGaussian fluctuations

22

Fluctuations of Fluctuations of ijij and and ijij

Incoherent effect(Vishniac & Brandenburg 1997,Sokoloff 1997, Silantev 2000,Proctor 2007)

23

Onset and saturation of Onset and saturation of incoherent alpha-shear incoherent alpha-shear

dynamosdynamos

24

ConclusionsConclusions

• Cycles w/ stratification• Test field method robust

– Even when small scale dynamo

– 0 and t0

• Rotation and shear: *ij

– WxJ not (yet?) excited– Incoherent works