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Detection of negative effective magnetic pressure instability in turbulence simulations. Instability in stratified turbulence with B -field (here horizontal) Reproduced in mean-field magnetohydrodynamics R m dependence of turbulent transport coefficient. - PowerPoint PPT Presentation
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Detection of negative effective Detection of negative effective magnetic pressure instabilitymagnetic pressure instability
in turbulence simulationsin turbulence simulations
• Instability in stratified turbulence with Instability in stratified turbulence with BB-field -field (here horizontal)(here horizontal)
• Reproduced in Reproduced in mean-field magnetohydrodynamicsmean-field magnetohydrodynamics
• RRmm dependence of turbulent transport coefficient dependence of turbulent transport coefficient
Axel Brandenburg, Koen Kemel, Nathan Kleeorin, Dhrubaditya Mitra, Igor RogachevskiiAxel Brandenburg, Koen Kemel, Nathan Kleeorin, Dhrubaditya Mitra, Igor Rogachevskii
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SetupSetup• 3-D box, size (2)3, isothermal MHD
• Random, nonhelical forcing at kf/k1=5, 15 or 30
• Stratified in z, ~exp(-z/H), H=1, =535• Periodic in x and y• stress-free, perfect conductor in z
• Weak imposed field B0 in y
• Run for long times: what happens?
• Turnover time to=(urms kf)-1, turb diff td=(t k12)-1
• Is longer by factor 3(kf/k1)2 = 3 152 = 675
• Average By over y and t=80to
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Negative effective magnetic pressure instability in action (ReM=70, kf/k1=15)
Bra
nden
burg
et a
l. (2
011,
ApJ
L 7
40, 5
0L)
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Poorly visible in snapshots (kf/k1=15)
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Exponential growthExponential growth
• Several thousand turnover times
• Or ½ a turbulent diffusive time
• Exponential growth linear instability of an already turbulent state
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Large aspect ratio: more cells• not all cells equally strong• reasonably well seen in just y-averages (top)• Better with additional time averaging (bottom)
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Larger scale separation: 30 instead of 15
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Better visible in snapshots for kf/k1=30
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Seems to work only in certain B range
• is expected• B0/Beq~0.15• Vertical inhomo.
for stronger field
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But B structure produced deeper down
• Here more scale separation (30 instead of 15)• Field strength and growth rate the same for different B0
• Can just reposition z range of the box to get the same growth rate and field strength
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Rm dependence
Breakdown ofquasi-linear theory
ReM here based on forcing kHere 15 eddies per box scaleReM=70 means 70x15x2=7000
based on box scale
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Mean-field conceptMean-field concept
WUJBbu ...t
.........
...
(1975) Soward & Roberts (1974),Rüdiger (1974),Rädler
2p2
1s
(2010) gBrandenbur & Rheinhardt
,
,,t
BqBBqBB
UUUuu
ijjilkijklkijk
ijjikijkkijkji
effect, turbulent diffusivity, Yoshizawa effect, etceffect, turbulent diffusivity, Yoshizawa effect, etc
Turbulent viscosity and other effects in momentum equationTurbulent viscosity and other effects in momentum equation
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Earlier results for low RmEarlier results for low Rm
• Rädler (1974) computed magnetic suppression (for other reasons)
• Rüdiger (1974) works only for Pm < 8• Roberts & Soward (1975), Rüdiger et al. (1986)
Maxwell tension formally negative for Rm > 1, but invalid
• Rüdiger et al. (2012), no negative effective magnetic pressure for Rm < 1.
• Kleeorin et a. (1989, 1990, 1996), Kleeorin & Rogachevskii (1994, 2007)
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Magnetic contribution Magnetic contribution to pressure & energy to pressure & energy
different!different!
22122
31
2212
31
221
BBU
BU
B
const
ij
ij
ijjiji BBUU
Can lead to reversed mean-field Can lead to reversed mean-field buoyancy in buoyancy in stratifiedstratified system system
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Formation of flux concentrations
...... 2p2
1s BBqBBBquu ijjiji
Earlier work with Kleeorin &Earlier work with Kleeorin &Rogachevskii (2010, AN 331, 5)Rogachevskii (2010, AN 331, 5)
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Stratified runsStratified runsisothermal isothermal constant scale height, constant scale height,
density contrast, exp(2density contrast, exp(2)=535,)=535,BBeqeq varies, so B/B varies, so B/Beqeq varies, varies,
get whole curve in one runget whole curve in one run
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ConvectionConvectionnot just for forced turbulence,not just for forced turbulence,
result is robust!result is robust!
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Effective magnetic pressureEffective magnetic pressure
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Fit formula and Rm dependenceFit formula and Rm dependence
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2*
22
p0p
/1
p
p
eq/ BBβ
2peff 1 qP
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Basic mechanismBasic mechanism
Huz / 0 uu
u
y
y Bt
B g
c
BBq
t
U
s
yz2
0p1
2
2*
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2*
p
p
q
2t
rms*
2tp
A 1 kH
ukq
H
v
Anelastic: descending structure Anelastic: descending structure compression compression
BB amplifies amplifies BB amplifies amplifies
Growth rateGrowth rate
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Conclusions: new food for thoughtConclusions: new food for thought
• Essential ingredient for shallow sunspot acticity identified (ApJL 740, L50)
• Effect does not exist below Rm=1 (SOCA)• Becomes stronger with better scale
separation (e.g. 30 instead of 15)• Active regions shallow phenomena H
indman et al. (2009, A
pJ)
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Thanks to the Astrophysics Thanks to the Astrophysics group at Norditagroup at Nordita