Magnetic helicity:Magnetic helicity:why is it so important and why is it so important and

how to get rid of ithow to get rid of itAxel BrandenburgAxel Brandenburg (Nordita, Copenhagen) (Nordita, Copenhagen)

Kandaswamy Subramanian Kandaswamy Subramanian (Pune)(Pune)

Brandenburg (2001, ApJ 550, 824; 2005, ApJ 625, 539)Brandenburg & Subramanian (2005, Phys. Rep., astro-ph/0405052)

2

Magnetic helicity Magnetic helicity V

VH d BA

1

2

212 H

11

d d1

SL

H SBA

2 d2

S

SA

1S

1

AB

3

Magnetic helicity conservationMagnetic helicity conservation

2221

d

dJBJuB

t

0d

d 2/12/121 BJBBuBA

t

kkBJ 2/1How J diverges as 0

Ideal limit and ideal case similar!

2/112/1221

d

d uωωfuωt

4

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

Pouquet, Frisch, & Leorat (1976)

k is forcedto the left

5

Production of LS helicityProduction of LS helicity

0 baBABA

bjforcing produces uω baand

bBB jJJ aAA

Yousef & BrandenburgA&A 407, 7 (2003)

But no net helicity production

baBA therefore:

alpha effect

6

LS dynamosLS dynamos• Difference to SS dynamos

– Field at scale of turbulence

– The small PrM problem

• Mechanisms for producing LS fields– Field at scale larger than that of turbulence

– Alpha effect (requires helicity)

– Shear-current of WxJ effect

– Others: incoherent alpha, Vishniac-Cho effect, + perhaps other effects

7

Cartesian box MHD equationsCartesian box MHD equations

JBuA

t

visc2 ln

D

DFf

BJu

sct

utD

lnD

AB

BJ

Induction

Equation:

Magn.Vectorpotential

Momentum andContinuity eqns

ln2312

visc SuuF

Viscous force

forcing function kk hf 0f (eigenfunction of curl)

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(i) Small scale dynamos(i) Small scale dynamosSmall PrM: stars and discs around NSs and YSOs

Here: non-helicallyforced turbulence

Schekochihinet al (2005)ApJ 625, 115L

k

9

256 processor run at 1024256 processor run at 102433 at Pr at PrMM=1=1

Result: not peaked at resistive scale Kolmogov scaling!

Haugen et al. (2003, A

pJ 597, L141)

-3/2slope?

instead: kpeak~Rm,crit1/2 kf ~ 6kf

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(ii) Large scale dynamos: 2 different geometries(ii) Large scale dynamos: 2 different geometries

• Helically forced turbulence (cyclonic events)

• Small & large scale field grows exponentially

• Past saturation: slow evolution

Explained by magnetic helicity equation

(a) Periodic box, no shear (b) open box, w/ shear

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Scale separation: inverse cascadeScale separation: inverse cascade

No inverse cascade in No inverse cascade in kinematic regimekinematic regime

Decomposition in terms of Decomposition in terms of Chandrasekhar-Kendall-Waleffe functionsChandrasekhar-Kendall-Waleffe functions

00kkkkkkk hhhA aaa

t2

peakk

Position of the peak compatible with

0

sin

cos

z

z

B LS field: force-free Beltrami

Time dependence: slow saturationTime dependence: slow saturationB

rand

enbu

rg (

2001

, ApJ

550

, 824

)

t2

peakk

Position of the peak compatible with

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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

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

BJBA 2d

d

t

BJBBA 22d

dE

t

bjBba 22d

dE

t

Two-scale assumption

JB t E

031 / bjuω

Dynamical quenching

M

eqmfM B

Rkt

2

22d

d BE

Kleeorin & Ruzmaikin (1982)

22

20

/1

/

eqm

eqmt

BR

BR

B

BJ

Steady limit algebraic quenching:

( selectivedecay)

15

Dynamo growth & saturationDynamo growth & saturation

Significant fieldalready after

kinematicgrowth phase

followed byslow resistive

adjustment

0 bjBJ

0 baBA

0221 f

bB kk

021211 f

bB kk

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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, AN 323 99)

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

Diffusive large scale losses: lower saturation level

Brandenburg & Dobler (2001 A&A 369, 329)

Periodicbox

with LS losses

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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

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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 BBu 4 ,C F

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

Vishniac & Cho (2001, ApJ 550, 752)Subramanian & Brandenburg (2004, PRL 93, 20500)

Expected to be finite on when there is shear

Arlt & Brandenburg (2001, A&A 380, 359)

Schnack et al.

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(ii) Forced LS dynamo with (ii) Forced LS dynamo with nono stratification stratification

geometryhere relevantto the sun

no helicity, e.g.

azimuthallyaveraged

neg helicity(northern hem.)

...21

JWBB

a

t

Rogachevskii & Kleeorin (2003, 2004)

April 21, 2023 20

ConclusionsConclusions• Shearflow turbulence: likely to produce LS field

– even w/o stratification (WxJ effect, similar to Rädler’s xJ effect)

• Stratification: can lead to effect– modify WxJ effect– but also instability of its own

• SS dynamo not obvious at small Pm• Application to the sun?

– distributed dynamo can produce bipolar regions– perhaps not so important?– solution to quenching problem? No: M even from WxJ effect

1046 Mx2/cycle