Helical MHD and aeffect

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Helical MHD and a-effect

• Axel Brandenburg (Nordita, Copenhagen)
• Kandaswamy Subramanian (IUCAA, Pune)

arXivastro-ph/0405052 Phys. Rept. (244 pages, 62
figs)
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MHD equations
Magn. Vector potential
Induction Equation
Momentum and Continuity eqns
Viscous force
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Vector potential

• BcurlA, advantage divB0
• JcurlBcurl(curlA) curl2A
• Not a disadvantage consider Alfven waves

B-formulation
A-formulation
2nd der once is better than 1st der twice!
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Pencil Code

• Started in Sept. 2001 with Wolfgang Dobler
• High order (6th order in space, 3rd order in
  time)
• Cache memory efficient
• MPI, can run PacxMPI (across countries!)
• Maintained/developed by many people (CVS!)
• Automatic validation (over night or any time)
• Max resolution so far 10243

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Helical versus nonhelical
Kida et al. (1991) helical forcing, but no
inverse cascade
Inverse cascade only when scale separation
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Allowing for scale separation
Position of the peak compatible with
No inverse cascade in kinematic regime
Decomposition in terms of Chandrasekhar-Kendall-Wa
leffe functions
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Kazantsev spectrum (kinematic)
Opposite limit, no scale separation, forcing at
kf1-2
Kazantsev spectrum confirmed (even for n/h1)
Spectrum remains highly time-dependent
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256 processor run at 10243
EM(k) not peaked at resistive scale, as
previously claimed instead kpeakRm,crit1/2 kf
6kf
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Structure function exponents
agrees with She-Leveque
third moment
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Bottleneck effect 1D vs 3D spectra
Compensated spectra (1D vs 3D)
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Relation to laboratory 1D spectra
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Bottleneck in the literature
Porter, Pouquet, Woodward (1998) using PPM,
10243 meshpoints
Kaneda et al. (2003) on the Earth simulator,
40963 meshpoints
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Helical MHD turbulence

• Helically forced turbulence (cyclonic events)
• Small large scale field grows exponentially
• Past saturation slow evolution
• ? Explained by magnetic helicity equation

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Animations
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Effects of magnetic helicity conservation
Early times h0 important
Late times steady state
By the time a steady state is reached net
magnetic helicity is generated
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Slow-down explained by magnetic helicity
conservation
molecular value!!
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Connection with a effect writhe with internal
twist as by-product
clockwise tilt (right handed)
W
? left handed internal twist
Yousef Brandenburg AA 407, 7 (2003)
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Internal twist as feedback on a (Pouquet, Frisch,
Leorat 1976)
How can this be used in practice?
Need a closure for ltj.bgt
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Rm dependence of PFL formula
St t urms kf not suppressed in Rm dependent
fashion
a is suppressed in Rm dependent fashion
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Revised nonlinear dynamo theory(originally due
to Kleeorin Ruzmaikin 1982)
Two-scale assumption
Production of large scale helicity comes at the
price of producing also small scale magnetic
helicity
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Express in terms of a
? Dynamical a-quenching (Kleeorin Ruzmaikin
1982)
no additional free parameters
Steady limit consistent with VC92
(algebraic quenching)
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Is ht quenched?can be in models with shear
Larger mean field
Slow growth but short cycles Depends
on assumption about ht-quenching!
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Additional effect of shear
Negative shear
Positive shear
Consistent with g3 and
Kitchatinov et al (1996), Kleeorin Rogachevskii
(1999)
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Effect of surface losses of current helicity

• Large scale (LS) field
• Drainage on LS dynamo
• Rm-dependent cutoff
• Shortens saturation time
• Small scale (SS) field
• Enhancement of LS dynamo

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The need for small scale losses
initial slope
Numerical experiment remove field for kgt4 every
1-3 turnover times

• large scale losses
• lower saturation level

2) higher saturation level 3) still slow time
scale
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How do magnetic helicity losses look like?
N-shaped (north) S-shaped (south)
(the whole loop corresponds to CME)
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Sigmoidal filaments
(from S. Gibson)
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Examples ofhelical structures
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Simulating solar-like differential rotation
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Results for current helicity flux
First order smoothing, and tau approximation
Vishniac Cho (2001
Expected to be finite on when there is shear
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Conclusions

• Homogeneous dynamos saturate resistively
• Entirely magnetic helicity controlled
• Inhomogeneous dynamo
• Open surface, equator
• Still many issues to be addressed
• Current helicity flux important
• Finite if there is shear
• Avoid magnetic helicity, use current helicity