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Simulations and understanding
large-scale dynamos
• Issues with global models
• Possibility of smaller scales
• Consequences of this
• Magnetic flux concentrations
• Unusual dynamo effects
Axel Brandenburg
(Nordita, Stockholm CU Boulder)
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Global models suggest
• Distributed dynamo action
– Difference to flux transport dynamos
– Would require smaller turb. diff.
ht=urms/3kf=urmsl/3
• Surface flux from upper layers
– Difference to deeply rooted tube picture
– Surface flux reamplification needed
– NEMPI: works best for large kfHp
• Mostly cylindrical W-contours
– Anti-solar differential rotation
•
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Hanasoge
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Do we need to rethink?
• In mixing length theory: l=Hp only hypothesis
– cf. Nick Featherstone’s talk
• Simulations: subgrid scale diffusion, viscosity
• Envisage reasons for (i) smaller scale flows
and/or (ii) deeper parts subadiabatic?
• But depth of convection zone still 200 Mm
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Spruit97 A changing paradigm
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Entropy rain
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Stein & Nordlund (1998) simulations
Filamentary, nonlocal shown: entropy fluctuations pos neg
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Tau approximation
upii
sjj
Ncsgu
NSus
/
supijjiiii NcsgSuususu
t
F
/2
i
su
FN
Closure
hypothesis
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Deardorff1
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Deardorff2
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Physical meaning?
lnln/ 1 pcs p
z
S pert coasting…
0 0 , 0 suus zz
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Physical meaning?
lnln/ 1 pcs p
z
S
pert
0 0 ,0 suus zz
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Why should only the top be unstable
constrad dz
dTKF
const3
16 3
TKe.g. if const
dz
dT
Power law baT 0
nTT ab
13
Polytropic index n
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Deeper parts intrinsically stable
nTT ab
13
Polytropic index n
Kramers opacity
(interior): a=1, b=-7/2
n=3.25
Entropy gradient positive (stable) for n > 3/2
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Solar opacities
n << -1 n = 3.25
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Hydrostatic
reference
solutions
Thickness only
~1Mm
111 KrH
Double Kramers-like
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Early work in the 1930s
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Original mixing length model
surface interior
unstable
stable
stable
weakly
unstable
Su rms31
conv Fassume
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New solutions
with Deardorff flux
)( adconv F
Dadconv )( F
Entropy gradient
old
new
pd
Td
ln
ln
pp HdzcSd /)(/)/( ad
arXiv:1504.03189v2
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Consequences of small scales
• Larger kf less turb. Diffusion: ht=urms/3kf
• Applications to dynamos: stronger, less turb diffusive
– Helps flux transport dynamos
• Two other important effect:
– Lambda effect differential rotation (Co smaller, Ta larger)
– Baroclinic term stronger?
– Negative effective magnetic pressure spots
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Flux emergence in global simulations
Nelson, Brown, Brun,
Miesch, Toomre (2014)
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3 scenarios
• Rising flux tubes?
• Hierachical convection?
• Self-organization as part
of the dynamo
g.B u.B g.W u.w A.B
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Sunspot decay
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Self-assembly of a magnetic spot • Minimalistic model
• 2 ingredients:
– Stratification & turbulence
• Extensions
– Coupled to dynamo
– Compete with rotation
– Radiation/ionization
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A possible mechanism
2
2122
312
212
312
21 BBUBUB
const
ijijijjiji BBUU
Breakdown of quasi-linear theory
ReM here based on forcing k
Here 15 eddies per box scale
ReM=70 means 70x15x2p=7000
based on box scale
Brandenburg et al (2011,ApJ 740, L50)
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Negative effective magnetic pressure instability
• Gas+turbulent+magnetic pressure; in pressure equil.
• B increases turbulence is suppressed
• turbulent pressure decreases
• Net effect?
Kleeorin, Rogachevskii, Ruzmaikin (1989, 1990)
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Sunspot
formation
that sucks
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Typical
downflow
speeds
Ma=0.2…0.3
Mean-field
simulation:
Neg pressure
parameterized
Brandenbur et al (2014)
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Bi-polar regions in simulations with corona
28
Warn
ecke et al. (2
013, A
pJL
777, L
37)
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Coronal loops?
Warnecke et al. (2013, ApJL 777, L37)
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First dynamo-generated bi-polar regions
30
Mitra et al. (2
01
4, arX
iv)
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Still negative effective
magnetic pressure?
Or something new?
31
Mitra et al. (2014, arXiv)
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Global models
32
Jabb
ari
et a
l. (
20
15
, ar
Xiv
)
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New aspects in mean-field concept
buBUEJ
...t JBbu
...... 2
p21
s,,t BqBBqUUuu ijjiijjiji
Ohm’s law
Theory and simulations: a effect and turbulent diffusivity
Turbulent viscosity and other effects in momentum equation
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Calculate full ij and ij tensors
• Imposed-field method
– Convection (Brandenburg et al. 1990)
• Correlation method
– MRI accretion discs (Brandenburg & Sokoloff 2002)
– Galactic turbulence (Kowal et al. 2005, 2006)
• Test field method
– Stationary geodynamo (Schrinner et al. 2005, 2007)
JBUA ε
tbuε
jijjijj JB *
turbulent emf
effect and turbulent
magnetic diffusivity
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Calculate full ij and ij 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 fields
0
sin
0
,
0
cos
0
0
0
sin
,
0
0
cos
2212
2111
kzkz
kzkz
BB
BB
pq
kjijk
pq
jij
pq
j BB ,
kzkkz
kzkkz
cossin
sincos
11311
21
1
11311
11
1
21
1
11
1
113
11
cossin
sincos
kzkz
kzkz
k
213223
113123
*
22
*
21
*
12
*
11
Example:
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Kinematic and t
independent of Rm (2…200)
1
frms31
0
rms31
0
ku
u
Sur et al. (2008, MNRAS)
1
frms
2
31
0
31
0
ku
u
uω
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Nonlocality: convolution
• Multiplication convolution
• Babcock-Leighton effect is an example
• Sharp structures in mean-field dynamos artifacts
• Convolution in x-space multiplication in k
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The 4 Roberts flows
IV flow: negative eddy diffusivity dynamo
But positive diffusion at small scales
Devlen et al. (2013)
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Time-delay dynamo for
Roberts II and III flows
xzxzxt BBB 2
2kik decay
oscillatory
With time delay
xzxzxt BtBB 2)(
xzxtxzxt BBBB 2
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Time-delay dynamo for
Roberts II and III flows
Rheinhardt et al. (2014)
Growth when
xzxtxzxt BBBB 2
])(1[
)(Re
2
22
k
kk
2/
31
frms ku
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Conclusions • Small scale deep convection
• Deep convective flux: Deardorff
• Thus marginally stable (not unstable)
• Such flows yield weaker turb diffusion
• Favor spot formation by NEMPI
• Dynamo effect from time delay