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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)

LS magnetic energy SS magnetic energy dissipation

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)

Haugen, Brandenburg, & Dobler (2003, ApJL)

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

Spectrum remains highly time-dependent

-3/2

slope?

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

Result: not peaked at resistive scale Kolmogov scaling!

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

3-D helical turbulence with shear

Brandenburg, Bigazzi, & Subramanian (2001)

“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)

Magn.

Vector

potential

Induction

Equation:

Momentum and

Continuity eqns

Viscous force

forcing function

(eigenfunction of curl)

- Helically forced turbulence (cyclonic events)
- Small & large scale field grows exponentially
- Past saturation: slow evolution
Explained by magnetic helicity equation

Position of the peak compatible with

No inverse cascade in kinematic regime

Decomposition in terms of Chandrasekhar-Kendall-Waleffe functions

Kida et al. (1991)

helical forcing, but no inverse cascade

Inverse cascade only when scale separation

Brandenburg (2001, ApJ)

clockwise tilt

(right handed)

W

left handed

internal twist

Yousef & Brandenburg

A&A 407, 7 (2003)

both for thermal/magnetic buoyancy

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)

- 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

Two-scale assumption

Production of large scale helicity comes at the price

of producing also small scale magnetic helicity

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

(ht quenched

constant)

Significant field

already after

kinematic

growth phase

followed by

slow resistive

adjustment

Larger mean field

Slow growth

but short cycles:

Depends on

assumption about

ht-quenching!

Negative

shear

Positive

shear

Consistent with g=3 and

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

Blackman & Brandenburg (2002)

- Advantage over magnetic helicity
- <j.b> is what enters a effect
- Can define helicity density

Rm also in the

numerator

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)

- 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)

- Still helically forced turbulence
- Shear driven by a friction term
- Normal field boundary condition

Negative current helicity:

net production in northern hemisphere

1046 Mx2/cycle

previously:

- 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