Catastrophic Catastrophic -quenching -quenching alleviated by helicity flux alleviated by helicity flux
and shearand shear
Axel Brandenburg (Axel Brandenburg (Nordita, CopenhagenNordita, Copenhagen))
Christer Sandin (Christer Sandin (UppsalaUppsala))Collaborators: Eric G Blackman (Collaborators: Eric G Blackman (RochesterRochester),),
Kandu Subramanian (Kandu Subramanian (IUCAA, PuneIUCAA, Pune), Petri K), Petri Käpylä (äpylä (OuluOulu))
2
Theoretical framework: Theoretical framework: model model
•Migration direction
JJBBUB
tt2/1
U k
U
•Cycle frequency
Migration awayfrom equator
031 / bjuω
Penalty to pay for Penalty to pay for
(in practice anisotropic)
meridional circulation
Pouquet, Frisch, Leorat (1976)
0,,31 / pjkpjkijkip bbuu
Brandenburg: helicity flux and shear 3
Internal twist as feedback on Internal twist as feedback on (Pouquet, Frisch, Leorat 1976)(Pouquet, Frisch, Leorat 1976)
031 / bjuω
How can this be used in practice?
Need a closure for <j.b>
Brandenburg: helicity flux and shear 4
Example of bi-helical structureExample of bi-helical structure
Yousef & Brandenburg (2003, A&A)
5
Tilt Tilt pol. field regeneration pol. field regeneration
N-shaped (north)S-shaped (south)
standarddynamo picture
internal twistas dynamo feedback
Blackman & Brandenburg (2003, ApJ)
6
Sigmoidal filamentsSigmoidal filaments
(from S. Gibson)
Brandenburg: helicity flux and shear 7
Examples ofExamples ofhelical structures helical structures
8
History of History of quenching quenching
22
2SSC
2f2
1
/1
/...
eqm
eqm
BR
BRk
B
F
“conventional” quenchinge.g., ~B-3, independent of Rm
(Moffatt 1972, Rüdiger 1973)
“catastrophic” quenchingRm –dependent (Vainshtein & Cattaneo 1972,
Gruzinov & Diamond 1994-96)
22 /1/ eqmK BR B
periodic box simulations:saturation at super-equipartition,
but after resistive time(Brandenburg 2001)
Dynamical quenching
M
eqmfM B
Rkt
2
22d
d BE
open domains: removal ofmagnetic waste by helicity flux
(Blackman & Field 2000,Kleeorin et al 2000-2003)
Kleeorin & Ruzmaikin (1982)
Brandenburg: helicity flux and shear 9
Current helicity fluxCurrent helicity flux
22
2ft
2SSC
2f2
1
/1
2/
/
eqm
eqmK
BR
kt
BkR
B
BJ
F
Rm also in thenumerator
SSCt
F
cebj 2
jc
beje 2SSCF
Advantage over magnetic helicity1) <j.b> is what enters effect2) Can define helicity density
Brandenburg: helicity flux and shear 10
Full time evolutionFull time evolution
Significant fieldalready after
kinematicgrowth phase
followed byslow resistive
adjustment
0 bjBJ
0 baBA
Brandenburg: helicity flux and shear 11
Helical MHD turbulenceHelical MHD turbulence• Helically forced turbulence (cyclonic events)
• Small & large scale field grows exponentially
• Past saturation: slow evolution
Explained by magnetic helicity equation
12
Large scale vs small scale lossesLarge 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)
Periodicbox
with LL losses
13
Significance of shearSignificance of shear
• transport of helicity in k-space• Shear transport of helicity in x-space
– Mediating helicity escape ( plasmoids)
– Mediating turbulent helicity flux
kjikji BBuF 4 ,C
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.
Brandenburg: helicity flux and shear 14
Simulating solar-like differential rotation 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 Impose toroidal field measure measure
22
2SSC
2f2
1
/1
/...
eqm
eqm
BR
BRk
B
F
22
20
/1
/
eqm
eqmt
BR
BR
B
BJ
previously:
Brandenburg: helicity flux and shear 16
Helicity fluxes at large and small scalesHelicity fluxes at large and small scales
Negative current helicity:net production in northern hemisphere
Brandenburg: helicity flux and shear 17
Helical turbulence with shearHelical turbulence with shearand diffusive and diffusive modelmodel corona corona
By fieldat periphery
of box
Brandenburg: helicity flux and shear 18
ConclusionsConclusions
• 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 1046 Mx2/cycle
• at least 30 times larger with open boundary conditions Presence of shear important
• Currently: include low plasma beta exterior