Alpha effect versus inverse cascade

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Alpha effect versusinverse cascade

• Axel Brandenburg
• (Nordita, Copenhagen)

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Order out of disorder Hales polarity law
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Solar butterfly diagram
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Mean-field theory

• Useful if it works
• In general not just Ea B
• ht is equally important
• a and ht are tensors
• integral kernals
• nonlinearity
• memory effects

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Mean-field works in some cases

• Isotropic box simulations get Beltrami field
• Quenching compatible with 1/(1aB2/Beq2)
• Accretion discs example of anisotropy
• Shear, rotation, stratification
• Negative ayy, in spite negative helicity
• Diffusion of Bx less than diffusion of By

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Quenching of alpha

• Isotropic box simulations
• ht is catastrophically quenched
• (Cattaneo Vainshtein 1991)
• a is catastrophically quenched
• (Vainshtein Cattaneo 1992)
• Astrophysical simulations
• a small even kinematically (Rm dependence?)
• B is definitely strong

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Whats going on?

• Imposed field approach suspect?
• ltBgt forced to stay unchanced ? no a ?
• Something wrong with forcing?
• thermal/magnetic buoyancy different
• Boundary conditions? (Blackman Field 2000)
• but alpha should be determined locally?
• Need to determine a and ht simultaneously

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Helically forced MHD
Magn. Vector potential
Induction Equation
Momentum and Continuity eqns
forcing function polarized waves
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Spectra and slices of B
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Magnetic Prandtl number 100
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What causes large scale field?

• Inverse cascade of magnetic helicity
• Frisch et al. (1975), Pouquet et al. (1976)
• Intrinsically nonlinear
• Alpha-effect (nonlocal in k-space)
• Steenbeck, Krause Radler (1966)
• Exists already in linear approximation

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Large scale separation
Nonlocal inverse Transfer in k-space
consistent with

• Evidence for
• Alpha-effect.

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Convergence at large scales
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Faster growth if Rm is large
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Saturation slow-down
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Saturation behavior explained by magnetic
helicity conservation
Steady state, closed box
Small scale and large scale current helicity in
balance
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With hyperdiffusivity
Brandenburg Sarson (2002, PRL)
for ordinary hyperdiffusion
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Slow-down explained by magnetic helicity
conservation
molecular value!!
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Excellent fit!
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Comparison with quenchedmean-field models
Ought to be satisfied also for magnetically
driven instabilities!
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Other quenchings ruled out
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Taking magnetic helicity seriously
Two-scale assumption
? Dynamical a-quenching
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Dynamical quenching in a2-dynamo
? Dynamical quenching allows slightly faster
growth ? Agrees slightly better with simulations
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Universality

• With lorentzian, a needs to be adjusted
• With dynamical quenching, al/(hk1kf)

Open boundaries right amplitude
aW-dynamo shorter periods
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Conclusions

• alpha-effect, or nonlocal inverse cascade
• Resistive time scale
• Large scale field still strong
• Many quenchings can be ruled out
• Dynamical quenching is viable