paradigm shifts in solar dynamo modelling

Paradigm shifts in solar Paradigm shifts in solar dynamo modellingdynamo modelling

(i)(i) Magn. buoyancy, radial diff rot, & quenching Magn. buoyancy, radial diff rot, & quenching dynamo at the bottom of CZ dynamo at the bottom of CZ

(ii)(ii) Simulations: strong downward pumpingSimulations: strong downward pumping

(iii)(iii) Radial diff rot negative near surface!Radial diff rot negative near surface!

(iv)(iv) Quenching alleviated by shear-mediated helicity fluxesQuenching alleviated by shear-mediated helicity fluxes

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

2

Solar dynamos in the 1970sSolar dynamos in the 1970s

• Distributed dynamo (Roberts & Stix 1972)

– Positive alpha, negative shear– Well-defined profiles from mixing length theory

Yoshimura (1975)

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Paradigm shiftsParadigm 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)

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(ii) Positive or negative radial shear?(ii) Positive or negative radial shear?

Benevolenskaya, Hoeksema, Kosovichev, Scherrer (1999) Pulkkinen & Tuominen (1998)

nHz 473/360024360

/7.14

ds

do

o

=AZ=(180/) (1.5x107) (210-8)

=360 x 0.15 = 54 degrees!

Before helioseismologyBefore 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)Brandenburg et al. (1992)

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(iii) Quenching in mean-field theory?(iii) Quenching in mean-field theory?

• Catastrophic quenching??– ~ Rm

-1, t ~ Rm-1

– Field strength vanishingly small!?!

• Something wrong with simulations– so let’s ignore the problem

• Possible reasons:– Suppression of lagrangian chaos?– Suffocation from small-scale magnetic helicity?

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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

ylä

et a

l (20

08)

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Upcoming dynamo effort in Upcoming dynamo effort in StockholmStockholm

Soon hiring:Soon hiring:• 4 students4 students• 4 post-docs (2 now)4 post-docs (2 now)• 1 assistant professor1 assistant professor• Long-term visitorsLong-term visitors

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Built-in feedback in Parker loopBuilt-in feedback in Parker loop

031 / bjuω both for thermal/magnetic

buoyancy

JBB

T dt

d2

T

BBJ

effect produces

helical field

clockwise tilt(right handed)

left handedinternal twist

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Interpretations and predictionsInterpretations and predictions

• In closed domain: resistively slow saturation• Open domain w/o shear: low saturation

– Due to loss of LS field

– Would need loss of SS field

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

– Mean flow contours perpendicular to surface!

B

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Nonlinear stage: consistent with …Nonlinear stage: consistent with …

SSCF need

22

2ft

2SSC

2f2

1t

/1

2/

/

eqm

eqmK

BR

kt

BkR

B

FBJ

Brandenburg (2005, A

pJ)

ijjiVC UUC ,,21

ijSS

C S , BBSF

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Forced large scale dynamo with fluxesForced large scale dynamo with fluxes

geometryhere relevantto the sun

Negative current helicity:net production in northern hemisphere

SJE d2 Sje d2

1046 Mx2/cycle

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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?

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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 et al.(2008, A&A 491, 353)

Magnetic helicity flux

Effects of b.c.s only in nonlinear regime

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Lack of LS dynamos in some casesLack of LS dynamos in some cases

• LS dynamo must be excited

• SS dynamo too dominant, swamps LS field

• Dominant SS dynamo: artifact of large PrM=

1

f

rms1t

1

ff

12

31

31

1t

k

k

u

U

k

UC

k

k

kkC

CCD

u

uω

Brun, Miesch, & ToomreBrun, Miesch, & Toomre(2004, ApJ 614, 1073)(2004, ApJ 614, 1073)

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Low PrLow PrMM dynamos dynamos

with helicity do workwith helicity do work• Energy dissipation via Joule• Viscous dissipation weak• Can increase Re substantially!

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and and cyccyc in quenched state in quenched state

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2ft

2SSC

2f2

1t

/1

2/

/

eqm

eqmK

BR

kt

BkR

B

FBJ

SSC

2f2

1mt F kk

2m´ /BBJk

21tcyc k

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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

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Calculate full Calculate full ijij and and ijij tensors tensors

pqpqpqpqpqpq

tbbubuBubU

b 2

Response to arbitrary mean fields

... ,

0

0

sin

,

0

0

cos2111

kzkz

BB

pqkjijk

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:

pqpq buCalculate

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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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Time-dependent caseTime-dependent case

21t1 )()()( kskss

0d )(ˆ)( ttes st

'd )'()'(ˆ tttt BB

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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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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)

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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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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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The FutureThe Future• Models in global geometry• Realistic boundaries:

– allowing for CMEs– magnetic helicity losses

• Sunspot formation– Local conctrations– Turbulent flux collapse– Negative turbulent mag presure

• Location of dynamo– Near surface shear layer– Tachocline

1046 Mx2/cycle