A new era for mean fields:A new era for mean fields:test fields and teststest fields and tests

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

2

Out

put s

o fa

rO

utpu

t so

far

25%

3

Mean field theory is predictiveMean field theory is predictive

• Open domain with shear– Helicity is driven out of domain (Vishniac & Cho)

– Mean flow contours perpendicular to surface!

• Excitation conditions– Dependence on angular velocity

– Dependence on b.c.: symmetric vs antisymmetric

4

Best if Best if contours contours to surface to surfaceExample: convection with shear

Käpylä et al. (2008, A&A) Tobias et al. (2008, ApJ)

need small-scale helicalexhaust out of the domain,not back in on the other side

MagneticBuoyancy?

5

To prove the point: convection with To prove the point: convection with vertical shear and open b.c.svertical shear and open b.c.s

Käpylä, Korpi, & myself(2008, A&A 491, 353)

Magnetic helicity flux

Effects of b.c.s only in nonlinear regime

6

Calculate full Calculate full ijij and and ijij tensors tensors

• Imposed-field method– Convection (Brandenburg et al. 1990)

• 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)

JBUA ε

tbuε

jijjijj JB *

turbulent emf

effect and turbulentmagnetic diffusivity

7

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

8

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:

9

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

10

Kinematic Kinematic and and tt

independent of Rm (2…200)independent of Rm (2…200)

1frms3

10

rms31

0

ku

u

Sur et al. (2008, MNRAS)

1frms

231

0

31

0

ku

u

uω

11

The The xxxx and and yyyy are now the same are now the same

Brandenburg (2005, AN)

Brandenburg & Sokoloff (2002, GAFD)

12

Scale-dependenceScale-dependence

fexp)(ˆ k

'd )'()'(ˆ zzzz BB

13

Time-dependent caseTime-dependent case

21t1 )()()( kskss

0d )(ˆ)( ttes st

'd )'()'(ˆ tttt BB

14

From linear to nonlinearFrom linear to nonlinear

pqpq ab

AB

uUU

Mean and fluctuating U enter separately

Use vector potential

15

Nonlinear Nonlinear ijij and and ijij tensors tensors

jiijij

jiijij

BB

BB

ˆˆ

ˆˆ

21

21

Consistency check: consider steady state to avoid d/dt terms

0

2121121

21t1

kk

kk

Expect:

=0 (within error bars) consistency check!

16

Application to passive vector eqnApplication to passive vector eqn

ijij

ijij

jiijij

BB

BB

BB

~~

ˆˆ

1

21

21

0

cos

sin~

,

0

sin

cos

kz

kz

kz

kz

BB

BBuB ~~~

2

t

Verified by test-field method

000

0sinsincos

0sincoscosˆˆ 2

2

kzkzkz

kzkzkz

BB ji

Tilgner & Brandenburg (2008)

cf. Cattaneo & Tobias (2009)

17

tt((RRmm) dependence for B~B) dependence for B~Beqeq

(i) is small consistency(ii) 1 and 2 tend to cancel(iii) to decrease (iv) 2 is small

021t1 kk

18

est field testsest field tests

(i) = 0(ii) =(s)(iii) = (B)

021t1 kk

19

The FutureThe Future• Test fields will continue to

provide guidence• Test flows: eddy viscosity

– vorticity dynamo?

• Maxwell stresses– Turbulent flux collapse– Negative turbulent mag

presure

• Global dynamo– Shell sectors

1046 Mx2/cycle