Catastrophic Catastrophic -quenching -quenching alleviated by helicity flux alleviated by helicity flux

and shearand shear

Axel Brandenburg (Axel Brandenburg (Nordita, CopenhagenNordita, Copenhagen))

Christer Sandin (Christer Sandin (UppsalaUppsala))Collaborators: Eric G Blackman (Collaborators: Eric G Blackman (RochesterRochester),),

Kandu Subramanian (Kandu Subramanian (IUCAA, PuneIUCAA, Pune), Petri K), Petri Käpylä (äpylä (OuluOulu))

2

Theoretical framework: Theoretical framework: model model

•Migration direction

JJBBUB

tt2/1

U k

U

•Cycle frequency

Migration awayfrom equator

031 / bjuω

Penalty to pay for Penalty to pay for

(in practice anisotropic)

meridional circulation

Pouquet, Frisch, Leorat (1976)

0,,31 / pjkpjkijkip bbuu

Brandenburg: helicity flux and shear 3

Internal twist as feedback on Internal twist as feedback on (Pouquet, Frisch, Leorat 1976)(Pouquet, Frisch, Leorat 1976)

031 / bjuω

How can this be used in practice?

Need a closure for <j.b>

Brandenburg: helicity flux and shear 4

Example of bi-helical structureExample of bi-helical structure

Yousef & Brandenburg (2003, A&A)

5

Tilt Tilt pol. field regeneration pol. field regeneration

N-shaped (north)S-shaped (south)

standarddynamo picture

internal twistas dynamo feedback

Blackman & Brandenburg (2003, ApJ)

6

Sigmoidal filamentsSigmoidal filaments

(from S. Gibson)

Brandenburg: helicity flux and shear 7

Examples ofExamples ofhelical structures helical structures

8

History of History of quenching quenching

22

2SSC

2f2

1

/1

/...

eqm

eqm

BR

BRk

B

F

“conventional” quenchinge.g., ~B-3, independent of Rm

(Moffatt 1972, Rüdiger 1973)

“catastrophic” quenchingRm –dependent (Vainshtein & Cattaneo 1972,

Gruzinov & Diamond 1994-96)

22 /1/ eqmK BR B

periodic box simulations:saturation at super-equipartition,

but after resistive time(Brandenburg 2001)

Dynamical quenching

M

eqmfM B

Rkt

2

22d

d BE

open domains: removal ofmagnetic waste by helicity flux

(Blackman & Field 2000,Kleeorin et al 2000-2003)

Kleeorin & Ruzmaikin (1982)

Brandenburg: helicity flux and shear 9

Current helicity fluxCurrent helicity flux

22

2ft

2SSC

2f2

1

/1

2/

/

eqm

eqmK

BR

kt

BkR

B

BJ

F

Rm also in thenumerator

SSCt

F

cebj 2

jc

beje 2SSCF

Advantage over magnetic helicity1) <j.b> is what enters effect2) Can define helicity density

Brandenburg: helicity flux and shear 10

Full time evolutionFull time evolution

Significant fieldalready after

kinematicgrowth phase

followed byslow resistive

adjustment

0 bjBJ

0 baBA

Brandenburg: helicity flux and shear 11

Helical MHD turbulenceHelical MHD turbulence• Helically forced turbulence (cyclonic events)

• Small & large scale field grows exponentially

• Past saturation: slow evolution

Explained by magnetic helicity equation

12

Large scale vs small scale lossesLarge scale vs small scale losses

Numerical experiment:remove field for k>4

every 1-3 turnover times(Brandenburg et al. 2002)

Small scale losses (artificial) higher saturation level still slow time scale

Diffusive large scale losses: lower saturation level

(Brandenburg & Dobler 2001)

Periodicbox

with LL losses

13

Significance of shearSignificance of shear

• transport of helicity in k-space• Shear transport of helicity in x-space

– Mediating helicity escape ( plasmoids)

– Mediating turbulent helicity flux

kjikji BBuF 4 ,C

Expression for current helicity flux: (first order smoothing, tau approximation)

Vishniac & Cho (2001, ApJ)

Expected to be finite on when there is shear

Arlt & Brandenburg (2001, A&A)

Schnack et al.

Brandenburg: helicity flux and shear 14

Simulating solar-like differential rotation Simulating solar-like differential rotation

• Still helically forced turbulence

• Shear driven by a friction term

• Normal field boundary condition

Brandenburg: helicity flux and shear 15

Impose toroidal field Impose toroidal field measure measure

22

2SSC

2f2

1

/1

/...

eqm

eqm

BR

BRk

B

F

22

20

/1

/

eqm

eqmt

BR

BR

B

BJ

previously:

Brandenburg: helicity flux and shear 16

Helicity fluxes at large and small scalesHelicity fluxes at large and small scales

Negative current helicity:net production in northern hemisphere

Brandenburg: helicity flux and shear 17

Helical turbulence with shearHelical turbulence with shearand diffusive and diffusive modelmodel corona corona

By fieldat periphery

of box

Brandenburg: helicity flux and shear 18

ConclusionsConclusions

• Connection between -effect and helicity flux• -effect produces LS (~300Mm) magnetic helicity

(+ north, south) SS magnetic helicity as “waste”

• Surface losses: observed component from SS (< 30Mm) ( north, + south), about 1046 Mx2/cycle

• at least 30 times larger with open boundary conditions Presence of shear important

• Currently: include low plasma beta exterior