Cosmic ray current-driven turbulence Cosmic ray current-driven turbulence and mean-field dynamo effectand mean-field dynamo effect

I. Rogachevskii, N. Kleeorin, A. Brandenburg, & D. Eichler(ApJ, submitted)

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MHD plasma with CRsMHD plasma with CRs

FEBJU

eic nnePt

1

ecri nnn cr4 JJB c

FBUBJBU

crcr1

41 enP

t c

To be solved with induction equationand continuity equation, isothermal EOS

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Introduces pseudoscalarIntroduces pseudoscalar

effect 0cr BJstarsin effect gΩ

/JbuBUB

t

... Bbu E

... jijjiji JB E

effect important for large-scale field in the Sun

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Comparison: alpha effect & inverse cascade in MHD (no CRs)

decaying turbulence forced turbulence

Here: magnetic helicity of opposite signs at small and large scale!

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Bi-helical fields from UlyssesBi-helical fields from Ulysses

• Taylor hypothesis– Broad k bins– Southern latitude with

opposite sign

• Small/large distances– Positive H at large k– Break point with

distance to larger k

Brandenburg, Subramanian, Balogh, & Goldstein (2011, ApJ 734, 9)

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Bell instabilityBell instability22cr42AzcB vkk

B

J

0cr

4 / kBJcJ

Bell (2004): J=2 Zirakashvili et al (2008): J=16

Continued growth in both cases! effect important?

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New simulationsNew simulations

• 5123 resolution, non-ideal (Re=Lu < 300)

• larger J parameter (80 and 800)

• most unstable k /k1= 40 and 400 (unresolved)

• measure alpha and turbulent diff. tensor

• Related to earlier work by Bykov et al. (2011)

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Bell instability Bell instability turbulence ( turbulence (JJ=80)=80)

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AnimationAnimation

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Power Power spectraspectra

• Bell mode ~25k1

• k4 spectrum

• k-5/3 at late time turbulence

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Spectral growthSpectral growth

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3 stages3 stages

• Bell instability,small scale, k/k1=40

• Accelerated large-scale growth

• Slow growth after initial saturation

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IsotropizationIsotropization

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Growth of length scaleGrowth of length scale

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Alpha tensor (test-field method)Alpha tensor (test-field method)

parallel

perp

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Normalized alpha effectNormalized alpha effect

crcr / 4/ Jc

Scales with kinetic helicity, prefactor ~0.5

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Dynamo number, turb diffDynamo number, turb diff

Critical value 1, turb diff >> microscopic value

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JJ and resolution dependence and resolution dependence

Larger resolution runs longer, can scale to higher J

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Equipartition ratio: Equipartition ratio: JJ dependent dependent

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Instantaneous Instantaneous growth rategrowth rate

• Agrees with non-ideal theory

• Amplification by many orders of magnitude

• Box Reynolds number ~ 1000

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Convergence of alphaConvergence of alpha

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Conclusions

• Growth by many orders of magnitude

• effect and turbulent diffusivity tensors

• dynamo number > 1 at late times

• k-5/3 spectrum at late times

• growth at larger length scales

• numerically converged

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Inverse cascadeInverse cascade

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Animation (poor)Animation (poor)

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Animation5Animation5

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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

argument due to Frisch et al. (1975)

k is forcedto the left