How much helicity is needed to drive large-scale dynamos?
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How much helicity is needed to drive large-scale dynamos? Simon Candelaresi Axel Brandenburg (submitted to Phys. Rev. E)
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Magnetic field generation on scales large compared with the scale of the turbulent eddies is known to be possible via the so-called alpha effect when the turbulence is helical and if the domain is large enough for the alpha effect to dominate over turbulent diffusion. Using three-dimensional turbulence simulations, we show that the energy of the resulting mean magnetic field of the saturated state increases linearly with the product of normalized helicity and the ratio of domain scale to eddy scale, provided this product exceeds a critical value around unity. This implies that large-scale dynamo action commences when the normalized helicity is larger than the inverse scale ratio. Recent findings of a smaller minimal helicity may be due to the onset of small-scale dynamo action at large magnetic Reynolds numbers. The onset of large-scale dynamo action is difficult to establish when the kinetic helicity is small.
Transcript of How much helicity is needed to drive large-scale dynamos?
How much helicity isneeded to drive
large-scale dynamos?
Simon Candelaresi
Axel Brandenburg
(submitted to Phys. Rev. E)
Closed alpha^2 Dynamo
forcing functionMomentum equation:
Helical forcing on scale
Helical motions
Helical magnetic field
normalized helicities
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(Frisch et. al. 1975, Seehafer 1996)
Predictions from the General Theory
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Kinematic phase:
(Moffatt 1978)
(Brandenburg, Subramanian 2005)
Predictions from the General Theorymean-field interpretation
Induced small-scale helical motions:
Mean-field decomposition:
(Krause, Raedler 1980)
Magnetic helicity conservation:
(Pouquet et. al. 1976)
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resistive growth for large-scale field
Triply periodic BC Magnetic helicity is conserved.
(Brandenburg, Subramanian 2005)
Predictions from the General Theory
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mean-field interpretation
Saturation magnetic field strength:
(Blackman, Brandenburg 2002)
normalized kinetic helicity
For the man magnetic field to grow:
What Pietarila Graham Finds
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(Pietarila Graham, et. al. 2012)
Fit formula:
MFT prediction
Parameters: and
Possible Issues
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● Growth rates after a fraction of the resistive time.
● Dynamo still contaminated with magnetic fields from the small-scale dynamo.
● Inaccurate fit for .
(Pietarila Graham, et. al. 2012)
Reproduction of the Predictions
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Consider the resistive phase well after the kinematic phase.
forcing function
triply periodic BC magnetic helicity is conserved
helical forcing helical motions
helical magnetic field (Beltrami field)
Parameters: and
resistive growth:
Saturation Magnetic Field
9(Candelaresi, Brandenburg 2012)
Saturation Magnetic Field
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(Blackman, Brandenburg 2002)Prediction:
(Candelaresi, Brandenburg 2012)
Saturation Magnetic Field
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Predictions:
NB:
High
Match their parameters.
No change in for high .
is underestimated for high due to viscous losses.
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(Candelaresi, Brandenburg 2012)
(Pietarila Graham, et. al. 2012)
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ABC-Flow Forcing
Forcing:
no dominant mode
Summary
● MF prediction reproduced in DNS.
● Discrepancy of (Graham) due to SSD contamination.
● ABC-flow produces oscillating modes.
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U. Frisch, A. Pouquet, J. Leorat, and A. Mazure.Possibility of an inverse cascade of magnetic helicity in magnetohydrodynamic turbulence.Journal of Fluid Mechanics, 68:769, 1975.
Frisch et al., 1975
References
H. K. Moffatt.Magnetic field generation in electrically conducting fluids.Camb. Univ. Press, 1978.
Moffatt, 1978
N. Seehafer.Nature of the α effect in magnetohydrodynamics.Phys. Rev. E, 53:1283, 1996.
Seehafer, 1996
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A. Brandenburg and K. Subramanian.Astrophysical magnetic fields and nonlinear dynamo theoryPhys. Rep. 417:1, 2005.
Brandenburg, Subramanian, 2005
F. Krause and K.-H. Raedler.Mean-field magnetohydrodynamics and dynamo theory.1980.
Krause, Raedler, 1980
References
www.nordita.org/~iomsn
Pouquet, A., Frisch, U., and Leorat, J.,Strong MHD helical turbulence and the nonlinear dynamo effect.Journal of Fluid Mechanics, 77:321, 1976.
Pouquet et al., 1976
E. G. Blackman and A. Brandenburg.Dynamic nonlinearity in large-scale dynamos with shear.Astrophys. J., 579:359, 2002.
Blackman, Brandenburg, 2002
J. Pietarila Graham, E. G. Blackman, P. D.Mininni, and A. Pouquet.Not much helicity is needed to drive large-scale dynamos.Phys. Rev. E, 85:066406, 2012.
Pietarila Graham, et. al., 2012
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