Helicity and Helicity flux Helicity and Helicity flux during the solar cycleduring the solar cycle

Axel BrandenburgAxel Brandenburg (Nordita, Copenhagen) (Nordita, Copenhagen)Christer SandinChrister Sandin (Stockholm), & (Stockholm), &

Petri KäpyläPetri Käpylä (Freiburg+Oulu) (Freiburg+Oulu)

2

Thirty years of turbulent diffusionThirty years of turbulent diffusion

2

00

2

2d

d JBJu

BEB

t

LS magnetic energy SS magnetic energy dissipation

3

Worry: magnetic energy Worry: magnetic energy peaked at small scales??peaked at small scales??

Maron & Cowley (2001)Maron & Cowley (2001)Meneguzzi et al (1981)Meneguzzi et al (1981) Kida et al (1991)

Schekochihin et al (2003)Schekochihin et al (2003)

Conclusion until recently:magnetic energy peaked

at the resistive scale!

4

Nonhelically forced turbulenceNonhelically forced turbulence

Kazantsev spectrum Kazantsev spectrum confirmed (even for confirmed (even for =1) =1)

Spectrum remains highly Spectrum remains highly time-dependenttime-dependent

Haugen, Brandenburg, & Dobler (2003, ApJL)

5

256 processor run at 1024256 processor run at 102433

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

6

Thirty years of nonlinear dynamosThirty years of nonlinear dynamos

Brandenburg, Bigazzi, & Subramanian (2001)

3-D helical turbulence with shear

7

However, However, quenching could be in trouble! quenching could be in trouble!

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)

8

Cartesian box MHD equationsCartesian box MHD equations

JBuA

t

visc2 ln

D

DFf

BJu

sc

t

utD

lnD

AB

BJ

Induction

Equation:

Magn.Vectorpotential

Momentum andContinuity eqns

ln2312

visc SuuF

Viscous force

forcing function kk hf 0f (eigenfunction of curl)

9

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

10

Allowing for scale separationAllowing for scale separation

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

11

Helical versus nonhelical and Helical versus nonhelical and scale separationscale separation

Inverse cascade only when Inverse cascade only when scale separationscale separation

Kida et al. (1991)Kida et al. (1991)helical forcing, but no inverse cascadehelical forcing, but no inverse cascade

12

Slow saturationSlow saturationB

rand

enbu

rg (

2001

, ApJ

)

13

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

Yousef & BrandenburgA&A 407, 7 (2003)

031 / bjuω both for thermal/magnetic

buoyancy

14

MTA – the Minimal Tau ApproximationMTA – the Minimal Tau Approximation

1st aspect: replace triple correlation by quadradatic

2nd aspect: do not neglect triple correlation

3rd aspect: calculate

rather than

ub

buu

uubbuuu Similar in spirit to tau approx in EDQNM

bubuBubUb

neglected!not t

bubu t/E

'd)'( ttbu E

(Kleeorin, Mond, & Rogachevskii 1996, Blackman & Field 2002, Rädler, Kleeorin, & Rogachevskii 2003)

15

Implications of MTAImplications of MTA

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

(Strong fluctuations of b are possible!)

2. Extra time derivative of emf

hyperbolic eqn, oscillatory behavior possible!

4. is not correlation time, but relaxation time

EE

JB

~ ~

t

new

t

E

E JB2

31

31

31

~ ,

~

~ ,~

u

bjuω

with

16

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

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

17

Express in terms of Express in terms of bjBba 22

d

dE

t

M

eqmfM B

Rkt

2

22d

d BE Dynamical -quenching (Kleeorin & Ruzmaikin 1982)

22

20

/1

/

eqm

eqmt

BR

BR

B

BJ

Steady limit: consistent with

Vainshtein & Cattaneo (1992)

no additional free parameters

(algebraicquenching)

Is Is tt quenched? quenched? can be can be

checked in models with shearchecked in models with shear

031 / bjM

bjba 2fk

220

/1 eqm BR B

2m /BBJk

mkt

Also:Schmalz & Stix

(1991)

18

Full time evolutionFull time evolution

Significant fieldalready after

kinematicgrowth phase

followed byslow resistive

adjustment

0 bjBJ

0 baBA

tt quenched quenched

constant)constant)

19

Is Is tt quenched? quenched?can be in models with shearcan be in models with shear

Larger mean field

Slow growthbut short cycles:

Depends onassumption about

t-quenching!

20

Additional effect of shear

Negative shear

Positive shear

Consistent with g=3 andeq

t0t /1 Bg B

Kitchatinov et al (1996), Kleeorin & Rogachevskii (1999)Blackman & Brandenburg (2002)

21

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

22

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 LS losses

23

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)

Expected to be finite on when there is shear

Arlt & Brandenburg (2001, A&A)

Schnack et al.

24

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

• Still helically forced turbulence

• Shear driven by a friction term

• Normal field boundary condition

25

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

Negative current helicity:net production in northern hemisphere

SJE d2 Sje d21046 Mx2/cycle

26

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:

27

Where do we stand after 30 yearsWhere do we stand after 30 years

• Mean-field theory qualitatively confirmed!– Convection (e.g. Ossendrijver), forced turbulence– Alternatives (e.g. xJ and SJ effects) to be explored

• Homogeneous dynamos saturate resistively– Entirely magnetic helicity controlled

• Inhomogeneous dynamo– Open surface, equator– Current helicity flux important

• Finite if there is shear

– Avoid magnetic helicity, use current helicity