1. Catastrophic  -quenching alleviated by helicity flux and shear Axel Brandenburg (Nordita, Copenhagen) Christer Sandin (Uppsala) Collaborators: Eric G Blackman (Rochester), Kandu Subramanian (IUCAA, Pune), Petri Käpylä (Oulu)

2. 2 Theoretical framework:  model Migration direction Cycle frequency Migration away from equator Penalty to pay for  (in practice anisotropic)  meridional circulation Pouquet, Frisch, Leorat (1976)

3. Brandenburg: helicity flux and shear3 Internal twist as feedback on  (Pouquet, Frisch, Leorat 1976) How can this be used in practice? Need a closure for

4. Brandenburg: helicity flux and shear4 Example of bi-helical structure Yousef & Brandenburg (2003, A&A)

5. 5 Tilt  pol. field regeneration N-shaped (north) S-shaped (south) standard dynamo picture  internal twist as dynamo feedback Blackman & Brandenburg (2003, ApJ)

6. 6 Sigmoidal filaments (from S. Gibson)

7. Brandenburg: helicity flux and shear7 Examples of helical structures

8. 8 History of  quenching “conventional”  quenching e.g.,  ~B -3, independent of R m (Moffatt 1972, Rüdiger 1973) “catastrophic”  quenching R m –dependent (Vainshtein & Cattaneo 1972, Gruzinov & Diamond 1994-96) periodic box simulations: saturation at super-equipartition, but after resistive time (Brandenburg 2001) Dynamical quenching open domains: removal of magnetic waste by helicity flux (Blackman & Field 2000, Kleeorin et al 2000-2003) Kleeorin & Ruzmaikin (1982)

9. Brandenburg: helicity flux and shear9 Current helicity flux R m also in the numerator Advantage over magnetic helicity 1) is what enters  effect 2)Can define helicity density

10. Brandenburg: helicity flux and shear10 Full time evolution Significant field already after kinematic growth phase followed by slow resistive adjustment

11. Brandenburg: helicity flux and shear11 Helical MHD turbulence Helically forced turbulence (cyclonic events) Small & large scale field grows exponentially Past saturation: slow evolution  Explained by magnetic helicity equation

12. 12 Large 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) Periodic box with LL losses

13. 13 Significance of shear   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) Vishniac & Cho (2001, ApJ) Expected to be finite on when there is shear Arlt & Brandenburg (2001, A&A) Schnack et al.

14. Brandenburg: helicity flux and shear14 Simulating solar-like differential rotation Still helically forced turbulence Shear driven by a friction term Normal field boundary condition

15. Brandenburg: helicity flux and shear15 Impose toroidal field  measure  previously:

16. Brandenburg: helicity flux and shear16 Helicity fluxes at large and small scales Negative current helicity: net production in northern hemisphere

17. Brandenburg: helicity flux and shear17 Helical turbulence with shear and diffusive model corona B y field at periphery of box

18. Brandenburg: helicity flux and shear18 Conclusions Connection between  -effect and helicity flux  -effect produces LS (~300Mm) magnetic helicity (+ north,  south)  SS magnetic helicity as “waste” Surface losses: observed component from SS (< 30Mm) (  north, + south), about 10 46 Mx 2 /cycle  at least 30 times larger with open boundary conditions Presence of shear important Currently: include low plasma beta exterior