Magnetic helicity: why is it so important and how to get rid of it
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Magnetic helicity:Magnetic helicity:why is it so important and why is it so important and
how to get rid of ithow to get rid of itAxel BrandenburgAxel Brandenburg (Nordita, Copenhagen) (Nordita, Copenhagen)
Kandaswamy Subramanian Kandaswamy Subramanian (Pune)(Pune)
Brandenburg (2001, ApJ 550, 824; 2005, ApJ 625, 539)Brandenburg & Subramanian (2005, Phys. Rep., astro-ph/0405052)
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Magnetic helicity Magnetic helicity V
VH d BA
1
2
212 H
11
d d1
SL
H SBA
2 d2
S
SA
1S
1
AB
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Magnetic helicity conservationMagnetic helicity conservation
2221
d
dJBJuB
t
0d
d 2/12/121 BJBBuBA
t
kkBJ 2/1How J diverges as 0
Ideal limit and ideal case similar!
2/112/1221
d
d uωωfuωt
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Inverse cascade of magnetic helicityInverse cascade of magnetic helicity
kqp EEE |||||| kqp HHH and
||2 pp HpE ||2 qq HqE Initial components fully helical: and
||||||2|||| qpkkqp HHkHkEHqHp
),max(||||
||||qp
HH
HqHpk
qp
qp
Pouquet, Frisch, & Leorat (1976)
k is forcedto the left
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Production of LS helicityProduction of LS helicity
0 baBABA
bjforcing produces uω baand
bBB jJJ aAA
Yousef & BrandenburgA&A 407, 7 (2003)
But no net helicity production
baBA therefore:
alpha effect
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LS dynamosLS dynamos• Difference to SS dynamos
– Field at scale of turbulence
– The small PrM problem
• Mechanisms for producing LS fields– Field at scale larger than that of turbulence
– Alpha effect (requires helicity)
– Shear-current of WxJ effect
– Others: incoherent alpha, Vishniac-Cho effect, + perhaps other effects
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Cartesian box MHD equationsCartesian box MHD equations
JBuA
t
visc2 ln
D
DFf
BJu
sct
utD
lnD
AB
BJ
Induction
Equation:
Magn.Vectorpotential
Momentum andContinuity eqns
ln2312
visc SuuF
Viscous force
forcing function kk hf 0f (eigenfunction of curl)
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(i) Small scale dynamos(i) Small scale dynamosSmall PrM: stars and discs around NSs and YSOs
Here: non-helicallyforced turbulence
Schekochihinet al (2005)ApJ 625, 115L
k
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256 processor run at 1024256 processor run at 102433 at Pr at PrMM=1=1
Result: not peaked at resistive scale Kolmogov scaling!
Haugen et al. (2003, A
pJ 597, L141)
-3/2slope?
instead: kpeak~Rm,crit1/2 kf ~ 6kf
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(ii) Large scale dynamos: 2 different geometries(ii) Large scale dynamos: 2 different geometries
• Helically forced turbulence (cyclonic events)
• Small & large scale field grows exponentially
• Past saturation: slow evolution
Explained by magnetic helicity equation
(a) Periodic box, no shear (b) open box, w/ shear
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Scale separation: inverse cascadeScale separation: inverse cascade
No inverse cascade in No inverse cascade in kinematic regimekinematic regime
Decomposition in terms of Decomposition in terms of Chandrasekhar-Kendall-Waleffe functionsChandrasekhar-Kendall-Waleffe functions
00kkkkkkk hhhA aaa
t2
peakk
Position of the peak compatible with
0
sin
cos
z
z
B LS field: force-free Beltrami
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Time dependence: slow saturationTime dependence: slow saturationB
rand
enbu
rg (
2001
, ApJ
550
, 824
)
t2
peakk
Position of the peak compatible with
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Connection with Connection with effect: effect: writhe with writhe with internalinternal twist as by-product twist as by-product
clockwise tilt(right handed)
left handedinternal twist
031 / bjuω both for thermal/magnetic
buoyancy
JBB
T dt
d2
T
BBJ
effect produces
helical field
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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
031 / bjuω
Dynamical quenching
M
eqmfM B
Rkt
2
22d
d BE
Kleeorin & Ruzmaikin (1982)
22
20
/1
/
eqm
eqmt
BR
BR
B
BJ
Steady limit algebraic quenching:
( selectivedecay)
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Dynamo growth & saturationDynamo growth & saturation
Significant fieldalready after
kinematicgrowth phase
followed byslow resistive
adjustment
0 bjBJ
0 baBA
0221 f
bB kk
021211 f
bB kk
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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, AN 323 99)
Small scale losses (artificial) higher saturation level still slow time scale
Diffusive large scale losses: lower saturation level
Brandenburg & Dobler (2001 A&A 369, 329)
Periodicbox
with LS losses
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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
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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)
Vishniac & Cho (2001, ApJ 550, 752)Subramanian & Brandenburg (2004, PRL 93, 20500)
Expected to be finite on when there is shear
Arlt & Brandenburg (2001, A&A 380, 359)
Schnack et al.
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(ii) Forced LS dynamo with (ii) Forced LS dynamo with nono stratification stratification
geometryhere relevantto the sun
no helicity, e.g.
azimuthallyaveraged
neg helicity(northern hem.)
...21
JWBB
a
t
Rogachevskii & Kleeorin (2003, 2004)
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ConclusionsConclusions• Shearflow turbulence: likely to produce LS field
– even w/o stratification (WxJ effect, similar to Rädler’s xJ effect)
• Stratification: can lead to effect– modify WxJ effect– but also instability of its own
• SS dynamo not obvious at small Pm• Application to the sun?
– distributed dynamo can produce bipolar regions– perhaps not so important?– solution to quenching problem? No: M even from WxJ effect
1046 Mx2/cycle