1

2

3

4

5

6

Non-helical MHD at 1024Non-helical MHD at 102433 H

auge

n, B

rand

enbu

rg, &

Dob

ler

(200

3, A

pJ)

7

8

9

10

11

12

Inverse cascade of magnetic helicityInverse cascade of magnetic helicity

kqp EEE |||||| kqp HHH and

||2 pp HpE ||2 qq HqE Initial components fully helical: and

||||||2|||| qpkkqp HHkHkEHqHp

),max(||||

||||qp

HH

HqHpk

qp

qp

argument due to Frisch et al. (1975)

k is forcedto the left

13

Magnetic helicityMagnetic helicity

0 ,

BEB

t

Maxwell eqns Vector potential

AB

Uncurled induction eqn

EA

t

AEBBEBA 2t

EABEBB

AA

BBA

ttt

flux terms/2 BJBAt

14

Magnetic helicityMagnetic helicity V

VH d BA

1

2

212 H

11

d d1

SL

H SBA

2 d2

S

SA

1S

1

AB

15

Slow saturationSlow saturationB

rand

enbu

rg (

2001

, ApJ

550

, 824

)

16

Periodic boxPeriodic box, no shear, no shear: resistively limited saturation: resistively limited saturation

Significant fieldalready after

kinematicgrowth phase

followed byslow resistive

adjustment

0 bjBJ

0 baBA

0221 f

bB kk

021211 f

bB kkBlackman & Brandenburg (2002, ApJ 579, 397)

Brandenburg & SubramanianPhys. Rep. (2005, 417, 1-209)

BJBA 2dt

d

17

Magnetic helicity conservation

termssurface2 BJBA dt

d

0BJSteady state,closed box

0 bjBJ

Early times

2f

21 bB kk

0BA 0 baBA

2f1

2 / bB kk

18

Slow-down explained by Slow-down explained by magnetic helicity conservationmagnetic helicity conservation

2f

21

211 22 bBB kk

dt

dk

)(2

1

22 s211 ttkf e

k

k bB

molecular value!!

BJBA 2dt

d

19

Slow-down explained by Slow-down explained by magnetic helicity conservationmagnetic helicity conservation

20

With hyperdiffusivity

23231 f

bB kk for ordinaryhyperdiffusion

42k

Brandenburg & Sarson (2002, PRL)

21

22

23

24

25

26

27

Evidence from different simulations:Evidence from different simulations:strong fields only with helicity fluxstrong fields only with helicity flux

Convective dynamo in a boxwith shear and rotation

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

Only weak field if box is closed

Forced turbulence in domain with solar-like shear

Brandenburg (2005, ApJ 625, 539)

3-D simulations, no mean-field modeling

28

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

29

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?

30

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

Käp

ylä,

Kor

pi, B

rand

enbu

rg (

2008

, A&

A)

31

Implications of tau approximationImplications of tau approximation

1. MTA does not a priori break down at large Rm.

(Strong fluctuations of b are possible!)

2. Extra time derivative of emf

33 hyperbolic eqn, oscillatory behavior possible!

4. is not correlation time, but relaxation time

εε

JB

~

~t

3new

t

εε JB

231

31

31

~ ,

~

~ ,~

u

bjuω

with

32

Kinetic and magnetic contributionsKinetic and magnetic contributions

lKillkljijkii BuBu ~

, uBubu

lkjijkKil uu ,

~ uω ikjijkKii uu ,

~

Kij

Kij ~~

31

lMilklilijkii BbbB ~

, bbBbu

likijkMil bb ,

~ bj ijkijkMii bb ,

~

Mij

Mij ~~

31

uω

bj

33

Connection with Connection with effect: effect: writhe with writhe with internalinternal twist as by-product twist as by-product

clockwise tilt(right handed)

left handedinternal twist

031 / bjuω both for thermal/magnetic

buoyancy

JBB

T dt

d2

T

BBJ

effect produces

helical field

34

… … the same thing mathematicallythe same thing mathematically

terms)(surface 2d

d BJBA

t

BJBBA ε 22d

d

t

bjBba ε 22d

d

t

Two-scale assumption JBε t

Production of large scale helicity comes at the priceof producing also small scale magnetic helicity

031 / bjuω

35

Revised nonlinear dynamo theoryRevised nonlinear dynamo theory(originally due to Kleeorin & Ruzmaikin 1982)(originally due to Kleeorin & Ruzmaikin 1982)

BJBA 2d

d

t

Two-scale assumption

031 / bjuω

Dynamical quenching

M

eqmfM B

Rkt

2

22d

d Bε

Kleeorin & Ruzmaikin (1982)

22

20

/1

/

eqm

eqmt

BR

BR

B

BJ

Steady limit algebraic quenching:

( selectivedecay)

BJBBA ε 22d

d

t

bjBba ε 22d

d

t

JBε t

36

General formula with magnetic helicity fluxGeneral formula with magnetic helicity flux

22

2ft

2SSC

2f2

1

/1

2/

/

eqm

eqmK

BR

kt

BkR

B

FBJ

Rm also in thenumerator

37

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

38

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

39

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:

40

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

Brandenburg, R

ädler, Schrinner (2009, A&

A)

41

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

42

Scale-dependence: nonlocalityScale-dependence: nonlocality

fexp)(ˆ k

'd )'()'(ˆ zzzz BB

2die )(~)(ˆ kkzkz

cf talk by Alexander Nepomnyashchy

43

Time-dependent caseTime-dependent case

2

die)(~)(ˆ tt

tet t

0/

220

2 cos)( i1

i1)(

'd )'()'(ˆ tttt BB

Hubbard &

Brandenburg (2009, A

pJ)

44

Importance of time-dependenceImportance of time-dependence

21t1 )()()( kskss

0d )(ˆ)( ttes st

'd )'()'(ˆ tttt BB

45

From linear to nonlinearFrom linear to nonlinear

pqpq ab

AB

uUU

Mean and fluctuating U enter separately

Use vector potential

Brandenburg et al. (2008, A

pJ)

46

Nonlinear Nonlinear ijij and and ijij tensors tensors

jiijij

jiijij

BB

BB

ˆˆ

ˆˆ

21

21

Consistency check: consider steady state to avoid d3/dt terms

0

2121121

21t1

kk

kk

Expect:

3=0 (within error bars) consistency check!

47

33tt((RRmm) dependence for B~B) dependence for B~Beqeq

(i) 3 is small consistency(ii) 31 and 32 tend to cancel(iii) to decrease 3(iv) 32 is small

021t1 kk

48

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)

49

Is the field in the Sun fibril?Is the field in the Sun fibril?

Käpylä et al (2008)

with rotation without rotation

50

Takes many turnover timesTakes many turnover times

1

f

rms1t

1

ff

12

31

31

1t

k

k

u

U

k

UC

k

k

kkC

CCD

u

uω

Rm

=121, B

y, 512^3

LS dynamo not always excited

51

Deeply rooted sunspots?Deeply rooted sunspots?

• Solar activity may not be so deeply rooted• The dynamo may be a distributed one• Near-surface shear important

Hindm

an et al. (2009, ApJ)

52

Near-surface shear layerNear-surface shear layer

Benevolenskaya, Hoeksema, Kosovichev, Scherrer (1999)

53

Origin of sunspotOrigin of sunspot

Theories for shallow spots:Theories for shallow spots:(i) Collapse by suppression(i) Collapse by suppression

of turbulent heat fluxof turbulent heat flux(ii) Negative pressure effects(ii) Negative pressure effects

from <from <bbiibbjj>-<>-<uuiiuujj> vs > vs BBiiBBjj

54

Formation of flux concentrations

...... 221 Bijpjisji qBBquu

Recent work with Kleeorin &Recent work with Kleeorin &Rogachevskii (arXiv:0910.1835)Rogachevskii (arXiv:0910.1835)