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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)

4

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

qq

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