Helicity and Helicity flux during the solar cycle

1. Helicity and Helicity flux during the solar cycle Axel Brandenburg (Nordita, Copenhagen) Christer Sandin (Stockholm), & Petri Käpylä (Freiburg+Oulu)

2. Thirty years of turbulent diffusion LS magnetic energy SS magnetic energy  dissipation

3. 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)

4. Nonhelically forced turbulence Haugen, Brandenburg, & Dobler (2003, ApJL) Kazantsev spectrum confirmed (even for n/h=1) Spectrum remains highly time-dependent

5. 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

6. Thirty years of nonlinear dynamos 3-D helical turbulence with shear Brandenburg, Bigazzi, & Subramanian (2001)

7. 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)

8. Cartesian box MHD equations Magn. Vector potential Induction Equation: Momentum and Continuity eqns Viscous force forcing function (eigenfunction of curl)

9. Helical 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 separation Position of the peak compatible with No inverse cascade in kinematic regime Decomposition in terms of Chandrasekhar-Kendall-Waleffe functions

11. Helical versus nonhelical and scale separation Kida et al. (1991) helical forcing, but no inverse cascade Inverse cascade only when scale separation

12. Slow saturation Brandenburg (2001, ApJ)

13. 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

14. 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)

15. 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

16. 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

17. 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

18. (ht quenched constant) Full time evolution Significant field already after kinematic growth phase followed by slow resistive adjustment

19. Is ht quenched?can be in models with shear Larger mean field Slow growth but short cycles: Depends on assumption about ht-quenching!

20. Additional effect of shear Negative shear Positive shear Consistent with g=3 and Kitchatinov et al (1996), Kleeorin & Rogachevskii (1999) Blackman & Brandenburg (2002)

21. Current helicity flux • Advantage over magnetic helicity • <j.b> is what enters a effect • Can define helicity density Rm also in the numerator

22. 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)

23. 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)

24. 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 scales Negative current helicity: net production in northern hemisphere 1046 Mx2/cycle

26. Impose toroidal field  measure a previously:

27. 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