The Test-field MethodThe Test-field Method

2

Output so farOutput so far

3

Simulations showing large-scale fieldsSimulations showing large-scale fieldsHelical turbulence (By) Helical shear flow turb.

Convection with shear Magneto-rotational Inst.

1t

21t

kc

k

Käp

yla

et a

l (20

08)

4

Low PrLow PrMM dynamos dynamos

Sun PrM==10-5

Here: non-helicallyforced turbulence

Schekochihinet al (2005)

k

Helical turbulence

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Upcoming dynamo effort in Upcoming dynamo effort in StockholmStockholm

Soon hiring:Soon hiring:• 4 students4 students• 3 post-docs3 post-docs• 1 assistant professor1 assistant professor• Long-term visitorsLong-term visitors

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Calculate full Calculate full ijij and and ijij tensors tensors

• 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 t

buε

jijjijj JB *

turbulent emf

effect and turbulentaagnetic diffusivity

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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

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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:

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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

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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

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Non-helical with rotationNon-helical with rotation

Rotational quenching, finite >0 for >0

JJδε t

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Shear turbulenceShear 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

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Dependence on Sh and RmDependence on Sh and Rm

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Direct simulationsDirect simulations

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Fluctuations of Fluctuations of ijij and and ijij

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

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From linear to nonlinearFrom linear to nonlinear

pqpq ab

AB

uUU

Mean and fluctuating U enter separately

Use vector potential

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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!

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RRmm dependence for B~B dependence for B~Beqeq

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

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Earlier results on Earlier results on t t quenching quenching

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2ft

2SSC

2f2

1t

/1

2/

/

eqm

eqmK

BR

kt

BkR

B

FBJ

mt2

t / : kRm BBJ

1/ ,3 ,/1

ttt0

t

gBg eqB

Yousef et al.(2003, A&A)

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Nonlinear Nonlinear ijij and and ijij tensors tensors

0~ˆˆ

ˆˆ21

jji

jiijij

BBB

BB

Another consistency check: passive vector equation

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Solar paradigm shiftsSolar paradigm shiftsi) 1980: magnetic buoyancy (Spiegel & Weiss)

overshoot layer dynamos

ii) 1985: helioseismology: d/dr > 0 dynamo dilema, flux transport dynamos

iii) 1992: catastrophic -quenching Rm-1 (Vainshtein & Cattaneo) Parker’s interface dynamo Backcock-Leighton mechanism

(i) Is magnetic buoyancy a problem?(i) Is magnetic buoyancy a problem?

Stratified dynamo simulation in 1990Expected strong buoyancy losses,but no: downward pumping Tobias et al. (2001)

(ii) Before helioseismology(ii) Before helioseismology• Angular velocity (at 4o latitude):

– very young spots: 473 nHz– oldest spots: 462 nHz– Surface plasma: 452 nHz

• Conclusion back then:– Sun spins faster in deaper convection zone– Solar dynamo works with d/dr<0: equatorward migr

Yoshimura (1975) Thompson et al. (2003) Benevolenskaya et al. (1998)

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Revisit paradigm shiftsRevisit paradigm shiftsi) 1980: magnetic buoyancy

counteracted by pumping

ii) 1985: helioseismology: d/dr > 0 negative gradient in near-surface shear layer

iii) 1992: catastrophic -quenching overcome by helicity fluxes in the Sun: by coronal mass ejections

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ConclusionConclusion • 11 yr cycle• Dyamo (SS vs LS)• Problems

– -quenching– slow saturation

• Solution– Modern -effect theory– j.b contribution– Magnetic helicity fluxes

• Location of dynamo– Distrubtion, shaped by– near-surface shear

1046 Mx2/cycle