1. Catastrophic a-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)
Invited talk given in Cambridge, September 14, 2004.

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

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

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

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

6. Sigmoidal filaments
(from S. Gibson)

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

8. History of a quenching “catastrophic” a quenching Rm –dependent
(Vainshtein & Cattaneo 1972,
Gruzinov & Diamond )
“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 )
Dynamical quenching
Kleeorin & Ruzmaikin (1982)

9. Current helicity flux Advantage over magnetic helicity
<j.b> is what enters a effect
Can define helicity density
Rm also in the
numerator
Brandenburg: helicity flux and shear

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

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

12. Large scale vs small scale losses
Diffusive large scale losses:
 lower saturation level
(Brandenburg & Dobler 2001)
Periodic
box
with LL 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)

13. 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)

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

15. Impose toroidal field  measure a
previously:
Brandenburg: helicity flux and shear

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

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

18. Brandenburg: helicity flux and shear
Conclusions
Connection between a-effect and helicity flux
a-effect produces LS (~300Mm) magnetic helicity
(+ north, - south)  SS magnetic helicity as “waste”
Surface losses: observed component from SS (< 30Mm)
(- north, + south), about 1046 Mx2/cycle
a at least 30 times larger with open boundary conditions
Presence of shear important
Currently: include low plasma beta exterior
Brandenburg: helicity flux and shear