Helicity and Helicity and Helicity flux Helicity flux during the solar during the solar cycle cycle Axel Brandenburg Axel Brandenburg (Nordita, (Nordita, Copenhagen) Copenhagen) Christer Sandin Christer Sandin (Stockholm), & (Stockholm), & Petri Käpylä Petri Käpylä (Freiburg+Oulu) (Freiburg+Oulu)
Helicity and Helicity flux during the solar cycle. Axel Brandenburg (Nordita, Copenhagen) Christer Sandin (Stockholm), & Petri Käpylä (Freiburg+Oulu). Thirty years of turbulent diffusion. LS magnetic energy SS magnetic energy dissipation. - PowerPoint PPT Presentation
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Helicity and Helicity flux Helicity and Helicity flux during the solar cycleduring the solar cycle
Revised nonlinear dynamo theoryRevised nonlinear dynamo theory(originally due to Kleeorin & Ruzmaikin 1982)(originally due to Kleeorin & Ruzmaikin 1982)
BJBA 2d
d
t
BJBBA 22d
dE
t
bjBba 22d
dE
t
Two-scale assumption JB t E
Production of large scale helicity comes at the priceof producing also small scale magnetic helicity
17
Express in terms of Express in terms of bjBba 22
d
dE
t
M
eqmfM B
Rkt
2
22d
d BE Dynamical -quenching (Kleeorin & Ruzmaikin 1982)
22
20
/1
/
eqm
eqmt
BR
BR
B
BJ
Steady limit: consistent with
Vainshtein & Cattaneo (1992)
no additional free parameters
(algebraicquenching)
Is Is tt quenched? quenched? can be can be
checked in models with shearchecked in models with shear
031 / bjM
bjba 2fk
220
/1 eqm BR B
2m /BBJk
mkt
Also:Schmalz & Stix
(1991)
18
Full time evolutionFull time evolution
Significant fieldalready after
kinematicgrowth phase
followed byslow resistive
adjustment
0 bjBJ
0 baBA
tt quenched quenched
constant)constant)
19
Is Is tt quenched? quenched?can be in models with shearcan be in models with shear
Larger mean field
Slow growthbut short cycles:
Depends onassumption about
t-quenching!
20
Additional effect of shear
Negative shear
Positive shear
Consistent with g=3 andeq
t0t /1 Bg B
Kitchatinov et al (1996), Kleeorin & Rogachevskii (1999)Blackman & Brandenburg (2002)
21
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
22
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 LS losses
23
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 BBu 4 ,C F
Expression for current helicity flux: (first order smoothing, tau approximation)