Helicity and helicity flux during the solar cycle
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Helicity and Helicity flux during the solar cycle. Axel Brandenburg (Nordita, Copenhagen) Christer Sandin (Stockholm), & Petri Käpylä (Freiburg+Oulu). Thirty years of turbulent diffusion. LS magnetic energy SS magnetic energy  dissipation.

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Helicity and Helicity flux during the solar cycle

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Helicity and Helicity flux during the solar cycle

Axel Brandenburg (Nordita, Copenhagen)

Christer Sandin (Stockholm), &

Petri Käpylä (Freiburg+Oulu)

Thirty years of turbulent diffusion

LS magnetic energy SS magnetic energy  dissipation

Worry: magnetic energy peaked at small scales??

Meneguzzi et al (1981)

Kida et al (1991)

Maron & Cowley (2001)

Conclusion until recently:

magnetic energy peaked

at the resistive scale!

Schekochihin et al (2003)

Nonhelically forced turbulence

Haugen, Brandenburg, & Dobler (2003, ApJL)

Kazantsev spectrum confirmed (even for n/h=1)

Spectrum remains highly time-dependent

256 processor run at 10243



Haugen et al. (2003, ApJ 597, L141)

Result: not peaked at resistive scale  Kolmogov scaling!

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

Thirty years of nonlinear dynamos

3-D helical turbulence with shear

Brandenburg, Bigazzi, & Subramanian (2001)

However, a quenching could be in trouble!

“catastrophic” a quenching

Rm –dependent

(Vainshtein & Cattaneo 1972,

Gruzinov & Diamond 1994-96)

“conventional” a quenching

e.g., a~B-3, independent of Rm

(Moffatt 1972, Rüdiger 1973)

periodic box simulations:

saturation at super-equipartition,

but after resistive time

(Brandenburg 2001)

open domains: removal of

magnetic waste by helicity flux

(Blackman & Field 2000,

Kleeorin et al 2000-2003)

Dynamical quenching

Kleeorin & Ruzmaikin (1982)

Cartesian box MHD equations






Momentum and

Continuity eqns

Viscous force

forcing function

(eigenfunction of curl)

Helical MHD turbulence

  • Helically forced turbulence (cyclonic events)

  • Small & large scale field grows exponentially

  • Past saturation: slow evolution

     Explained by magnetic helicity equation

Allowing for scale separation

Position of the peak compatible with

No inverse cascade in kinematic regime

Decomposition in terms of Chandrasekhar-Kendall-Waleffe functions

Helical versus nonhelical and scale separation

Kida et al. (1991)

helical forcing, but no inverse cascade

Inverse cascade only when scale separation

Slow saturation

Brandenburg (2001, ApJ)

Connection with a effect: writhe with internal twist as by-product

clockwise tilt

(right handed)


 left handed

internal twist

Yousef & Brandenburg

A&A 407, 7 (2003)

both for thermal/magnetic buoyancy

MTA – the Minimal Tau Approximation

1st aspect: replace triple correlation by quadradatic

2nd aspect: do not neglect triple correlation

3rd aspect: calculate

rather than

Similar in spirit to tau approx in EDQNM 

(Kleeorin, Mond, & Rogachevskii 1996, Blackman & Field 2002,

Rädler, Kleeorin, & Rogachevskii 2003)

Implications of MTA

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

    (Strong fluctuations of b are possible!)

  • Extra time derivative of emf

  •  hyperbolic eqn, oscillatory behavior possible!

  • t is not correlation time, but relaxation time


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

Two-scale assumption

Production of large scale helicity comes at the price

of producing also small scale magnetic helicity

Express in terms of a

 Dynamical a-quenching (Kleeorin & Ruzmaikin 1982)


Schmalz & Stix


no additional free parameters

Steady limit:

consistent with

Vainshtein & Cattaneo (1992)



Is ht quenched?  can be

checked in models with shear

(ht quenched


Full time evolution

Significant field

already after


growth phase

followed by

slow resistive


Is ht quenched?can be in models with shear

Larger mean field

Slow growth

but short cycles:

Depends on

assumption about


Additional effect of shear





Consistent with g=3 and

Kitchatinov et al (1996), Kleeorin & Rogachevskii (1999)

Blackman & Brandenburg (2002)

Current helicity flux

  • Advantage over magnetic helicity

  • <j.b> is what enters a effect

  • Can define helicity density

Rm also in the


Large scale vs small scale losses

Diffusive large scale losses:

 lower saturation level

(Brandenburg & Dobler 2001)



with LS losses

Small scale losses (artificial)

 higher saturation level

 still slow time scale

Numerical experiment:

remove field for k>4

every 1-3 turnover times

(Brandenburg et al. 2002)

Significance of shear

  • a transport of helicity in k-space

  • Shear  transport of helicity in x-space

    • Mediating helicity escape ( plasmoids)

    • Mediating turbulent helicity flux

Expression for current helicity flux:

(first order smoothing, tau approximation)

Schnack et al.

Vishniac & Cho (2001, ApJ)

Expected to be finite on when there is shear

Arlt & Brandenburg (2001, A&A)

Simulating solar-like differential rotation

  • Still helically forced turbulence

  • Shear driven by a friction term

  • Normal field boundary condition

Helicity fluxes at large and small scales

Negative current helicity:

net production in northern hemisphere

1046 Mx2/cycle

Impose toroidal field  measure a


Where do we stand after 30 years

  • Mean-field theory qualitatively confirmed!

    • Convection (e.g. Ossendrijver), forced turbulence

    • Alternatives (e.g. WxJ 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

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