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

Helicity and Helicity flux during the solar cycle

Axel Brandenburg (Nordita, Copenhagen)

Christer Sandin (Stockholm), &

Petri Käpylä (Freiburg+Oulu)


Thirty years of turbulent diffusion

Thirty years of turbulent diffusion

LS magnetic energy SS magnetic energy  dissipation


Worry magnetic energy peaked at small scales

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

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 1024 3

256 processor run at 10243

-3/2

slope?

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

Thirty years of nonlinear dynamos

3-D helical turbulence with shear

Brandenburg, Bigazzi, & Subramanian (2001)


However a quenching could be in trouble

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

Cartesian box MHD equations

Magn.

Vector

potential

Induction

Equation:

Momentum and

Continuity eqns

Viscous force

forcing function

(eigenfunction of curl)


Helical mhd turbulence

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

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

Helical versus nonhelical and scale separation

Kida et al. (1991)

helical forcing, but no inverse cascade

Inverse cascade only when scale separation


Slow saturation

Slow saturation

Brandenburg (2001, ApJ)


Connection with a effect writhe with internal twist as by product

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

clockwise tilt

(right handed)

W

 left handed

internal twist

Yousef & Brandenburg

A&A 407, 7 (2003)

both for thermal/magnetic buoyancy


Mta the minimal tau approximation

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

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

with


Revised nonlinear dynamo theory originally due to kleeorin ruzmaikin 1982

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

Express in terms of a

 Dynamical a-quenching (Kleeorin & Ruzmaikin 1982)

Also:

Schmalz & Stix

(1991)

no additional free parameters

Steady limit:

consistent with

Vainshtein & Cattaneo (1992)

(algebraic

quenching)

Is ht quenched?  can be

checked in models with shear


Full time evolution

(ht quenched

constant)

Full time evolution

Significant field

already after

kinematic

growth phase

followed by

slow resistive

adjustment


Is h t quenched can be in models with shear

Is ht quenched?can be in models with shear

Larger mean field

Slow growth

but short cycles:

Depends on

assumption about

ht-quenching!


Additional effect of shear

Additional effect of shear

Negative

shear

Positive

shear

Consistent with g=3 and

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

Blackman & Brandenburg (2002)


Current helicity flux

Current helicity flux

  • Advantage over magnetic helicity

  • <j.b> is what enters a effect

  • Can define helicity density

Rm also in the

numerator


Large scale vs small scale losses

Large scale vs small scale losses

Diffusive large scale losses:

 lower saturation level

(Brandenburg & Dobler 2001)

Periodic

box

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

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

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

Helicity fluxes at large and small scales

Negative current helicity:

net production in northern hemisphere

1046 Mx2/cycle


Impose toroidal field measure a

Impose toroidal field  measure a

previously:


Where do we stand after 30 years

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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