non-helical mhd at 1024 3
DESCRIPTION
Non-helical MHD at 1024 3. Haugen, Brandenburg, & Dobler (2003, ApJ). Inverse cascade of magnetic helicity. argument due to Frisch et al. (197 5 ). and. Initial components fully helical:. and. k is forced to the left. Magnetic helicity. Maxwell eqns. Vector potential. - PowerPoint PPT PresentationTRANSCRIPT
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Non-helical MHD at 1024Non-helical MHD at 102433 H
auge
n, B
rand
enbu
rg, &
Dob
ler
(200
3, A
pJ)
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8
9
10
11
12
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
argument due to Frisch et al. (1975)
k is forcedto the left
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Magnetic helicityMagnetic helicity
0 ,
BEB
t
Maxwell eqns Vector potential
AB
Uncurled induction eqn
EA
t
AEBBEBA 2t
EABEBB
AA
BBA
ttt
flux terms/2 BJBAt
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Magnetic helicityMagnetic helicity V
VH d BA
1
2
212 H
11
d d1
SL
H SBA
2 d2
S
SA
1S
1
AB
15
Slow saturationSlow saturationB
rand
enbu
rg (
2001
, ApJ
550
, 824
)
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Periodic boxPeriodic box, no shear, no shear: resistively limited saturation: resistively limited saturation
Significant fieldalready after
kinematicgrowth phase
followed byslow resistive
adjustment
0 bjBJ
0 baBA
0221 f
bB kk
021211 f
bB kkBlackman & Brandenburg (2002, ApJ 579, 397)
Brandenburg & SubramanianPhys. Rep. (2005, 417, 1-209)
BJBA 2dt
d
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Magnetic helicity conservation
termssurface2 BJBA dt
d
0BJSteady state,closed box
0 bjBJ
Early times
2f
21 bB kk
0BA 0 baBA
2f1
2 / bB kk
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Slow-down explained by Slow-down explained by magnetic helicity conservationmagnetic helicity conservation
2f
21
211 22 bBB kk
dt
dk
)(2
1
22 s211 ttkf e
k
k bB
molecular value!!
BJBA 2dt
d
19
Slow-down explained by Slow-down explained by magnetic helicity conservationmagnetic helicity conservation
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With hyperdiffusivity
23231 f
bB kk for ordinaryhyperdiffusion
42k
Brandenburg & Sarson (2002, PRL)
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22
23
24
25
26
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Evidence from different simulations:Evidence from different simulations:strong fields only with helicity fluxstrong fields only with helicity flux
Convective dynamo in a boxwith shear and rotation
Käpylä, Korpi, Brandenburg(2008, A&A 491, 353)
Only weak field if box is closed
Forced turbulence in domain with solar-like shear
Brandenburg (2005, ApJ 625, 539)
3-D simulations, no mean-field modeling
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Nonlinear stage: consistent with …Nonlinear stage: consistent with …
SSCF need
22
2ft
2SSC
2f2
1t
/1
2/
/
eqm
eqmK
BR
kt
BkR
B
FBJ
Brandenburg (2005, A
pJ)
ijjiVC UUC ,,21
ijSS
C S , BBSF
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Best if Best if contours contours to surface to surfaceExample: convection with shear
Käpylä et al. (2008, A&A) Tobias et al. (2008, ApJ)
need small-scale helicalexhaust out of the domain,not back in on the other side
MagneticBuoyancy?
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To prove the point: convection with To prove the point: convection with vertical shear and open b.c.svertical shear and open b.c.s
Käpylä, Korpi, & myself(2008, A&A 491, 353)
Magnetic helicity flux
Effects of b.c.s only in nonlinear regime
Käp
ylä,
Kor
pi, B
rand
enbu
rg (
2008
, A&
A)
31
Implications of tau approximationImplications of tau approximation
1. MTA does not a priori break down at large Rm.
(Strong fluctuations of b are possible!)
2. Extra time derivative of emf
33 hyperbolic eqn, oscillatory behavior possible!
4. is not correlation time, but relaxation time
εε
JB
~
~t
3new
t
εε JB
231
31
31
~ ,
~
~ ,~
u
bjuω
with
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Kinetic and magnetic contributionsKinetic and magnetic contributions
lKillkljijkii BuBu ~
, uBubu
lkjijkKil uu ,
~ uω ikjijkKii uu ,
~
Kij
Kij ~~
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lMilklilijkii BbbB ~
, bbBbu
likijkMil bb ,
~ bj ijkijkMii bb ,
~
Mij
Mij ~~
31
uω
bj
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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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… … the same thing mathematicallythe same thing mathematically
terms)(surface 2d
d BJBA
t
BJBBA ε 22d
d
t
bjBba ε 22d
d
t
Two-scale assumption JBε t
Production of large scale helicity comes at the priceof producing also small scale magnetic helicity
031 / bjuω
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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
Two-scale assumption
031 / bjuω
Dynamical quenching
M
eqmfM B
Rkt
2
22d
d Bε
Kleeorin & Ruzmaikin (1982)
22
20
/1
/
eqm
eqmt
BR
BR
B
BJ
Steady limit algebraic quenching:
( selectivedecay)
BJBBA ε 22d
d
t
bjBba ε 22d
d
t
JBε t
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General formula with magnetic helicity fluxGeneral formula with magnetic helicity flux
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2ft
2SSC
2f2
1
/1
2/
/
eqm
eqmK
BR
kt
BkR
B
FBJ
Rm also in thenumerator
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Mean field theory is predictiveMean field theory is predictive
• Open domain with shear– Helicity is driven out of domain (Vishniac & Cho)
– Mean flow contours perpendicular to surface!
• Excitation conditions– Dependence on angular velocity
– Dependence on b.c.: symmetric vs antisymmetric
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Calculate full Calculate full ijij and and ijij tensors tensors
JBUA
t
JbuBUA
t
jbubuBubUa
t
pqpqpqpqpqpq
tjbubuBubU
a
Original equation (uncurled)
Mean-field equation
fluctuations
Response to arbitrary mean fields
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Test fieldsTest fields
0
sin
0
,
0
cos
0
0
0
sin
,
0
0
cos
2212
2111
kzkz
kzkz
BB
BBpqkjijk
pqjij
pqj BB ,
kzkkz
kzkkz
cossin
sincos
1131121
1
1131111
1
21
1
111
113
11
cossin
sincos
kzkz
kzkz
k
213223
113123
*22
*21
*12
*11
Example:
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Validation: Roberts flowValidation: Roberts flowpqpqpqpqpq
pq
tjbubuBubU
a
1frmsm3
1t
rmsm31
-kuR
uR
SOCA
ykxk
ykxk
ykxk
uU
yx
yx
yx
rms
coscos2
cossin
sincos
2/fkkk yx
1frms3
1t0
rms31
0
-ku
u
normalize
SOCA result
Brandenburg, R
ädler, Schrinner (2009, A&
A)
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Kinematic Kinematic and and tt
independent of Rm (2…200)independent of Rm (2…200)
1frms3
10
rms31
0
ku
u
Sur et al. (2008, MNRAS)
1frms
231
0
31
0
ku
u
uω
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Scale-dependence: nonlocalityScale-dependence: nonlocality
fexp)(ˆ k
'd )'()'(ˆ zzzz BB
2die )(~)(ˆ kkzkz
cf talk by Alexander Nepomnyashchy
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Time-dependent caseTime-dependent case
2
die)(~)(ˆ tt
tet t
0/
220
2 cos)( i1
i1)(
'd )'()'(ˆ tttt BB
Hubbard &
Brandenburg (2009, A
pJ)
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Importance of time-dependenceImportance of time-dependence
21t1 )()()( kskss
0d )(ˆ)( ttes st
'd )'()'(ˆ tttt BB
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From linear to nonlinearFrom linear to nonlinear
pqpq ab
AB
uUU
Mean and fluctuating U enter separately
Use vector potential
Brandenburg et al. (2008, A
pJ)
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Nonlinear Nonlinear ijij and and ijij tensors tensors
jiijij
jiijij
BB
BB
ˆˆ
ˆˆ
21
21
Consistency check: consider steady state to avoid d3/dt terms
0
2121121
21t1
kk
kk
Expect:
3=0 (within error bars) consistency check!
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33tt((RRmm) dependence for B~B) dependence for B~Beqeq
(i) 3 is small consistency(ii) 31 and 32 tend to cancel(iii) to decrease 3(iv) 32 is small
021t1 kk
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Application to passive vector eqnApplication to passive vector eqn
ijij
ijij
jiijij
BB
BB
BB
~~
ˆˆ
1
21
21
0
cos
sin~
,
0
sin
cos
kz
kz
kz
kz
BB
BBuB ~~~
2
t
Verified by test-field method
000
0sinsincos
0sincoscosˆˆ 2
2
kzkzkz
kzkzkz
BB ji
Tilgner & Brandenburg (2008)
cf. Cattaneo & Tobias (2009)
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Is the field in the Sun fibril?Is the field in the Sun fibril?
Käpylä et al (2008)
with rotation without rotation
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Takes many turnover timesTakes many turnover times
1
f
rms1t
1
ff
12
31
31
1t
k
k
u
U
k
UC
k
k
kkC
CCD
u
uω
Rm
=121, B
y, 512^3
LS dynamo not always excited
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Deeply rooted sunspots?Deeply rooted sunspots?
• Solar activity may not be so deeply rooted• The dynamo may be a distributed one• Near-surface shear important
Hindm
an et al. (2009, ApJ)
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Near-surface shear layerNear-surface shear layer
Benevolenskaya, Hoeksema, Kosovichev, Scherrer (1999)
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Origin of sunspotOrigin of sunspot
Theories for shallow spots:Theories for shallow spots:(i) Collapse by suppression(i) Collapse by suppression
of turbulent heat fluxof turbulent heat flux(ii) Negative pressure effects(ii) Negative pressure effects
from <from <bbiibbjj>-<>-<uuiiuujj> vs > vs BBiiBBjj
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Formation of flux concentrations
...... 221 Bijpjisji qBBquu
Recent work with Kleeorin &Recent work with Kleeorin &Rogachevskii (arXiv:0910.1835)Rogachevskii (arXiv:0910.1835)