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.

1. How much helicity is
needed to drive
large-scale dynamos?
(submitted to Phys. Rev. E)
Simon Candelaresi
Axel Brandenburg

2. Closed alpha^2 Dynamo
Momentum equation: forcing function
Helical forcing on scale
Helical motions
Helical magnetic field
(Frisch et. al. 1975, Seehafer 1996)
normalized helicities
2

3. Predictions from the General Theory
Kinematic phase:
(Moffatt 1978)
(Brandenburg, Subramanian 2005)
3

4. Predictions from the General Theory
mean-field interpretation
Mean-field decomposition:
Induced small-scale helical motions:
(Krause, Raedler 1980)
Magnetic helicity conservation:
(Pouquet et. al. 1976)
Triply periodic BC Magnetic helicity is conserved.
resistive growth for large-scale field (Brandenburg,
Subramanian 2005)
4

5. Predictions from the General Theory
mean-field interpretation
Saturation magnetic field strength:
(Blackman, Brandenburg 2002)
For the man magnetic field to grow:
normalized kinetic helicity
5

6. What Pietarila Graham Finds
Parameters: and
MFT prediction
Fit formula:
(Pietarila Graham, et. al. 2012)
6

7. Possible Issues
● 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)
7

8. Reproduction of the Predictions
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
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9. Saturation Magnetic Field
resistive growth:
(Candelaresi, Brandenburg 2012)
9

10. Saturation Magnetic Field
Prediction: (Blackman, Brandenburg 2002)
(Candelaresi, Brandenburg 2012)
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11. Saturation Magnetic Field
Predictions:
NB:
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12. High
(Pietarila Graham, et. al. 2012)
Match their parameters.
No change in for high . (Candelaresi, Brandenburg 2012)
is underestimated for high due to viscous losses.
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13. ABC-Flow Forcing
Forcing:
no dominant mode
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14. Summary
● MF prediction reproduced in DNS.
● Discrepancy of (Graham) due to SSD contamination.
● ABC-flow produces oscillating modes.
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15. References
Frisch et al., 1975
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.
Seehafer, 1996
N. Seehafer.
Nature of the α effect in magnetohydrodynamics.
Phys. Rev. E, 53:1283, 1996.
Brandenburg, Subramanian, 2005
A. Brandenburg and K. Subramanian.
Astrophysical magnetic fields and nonlinear dynamo theory
Phys. Rep. 417:1, 2005.
Moffatt, 1978
H. K. Moffatt.
Magnetic field generation in electrically conducting fluids.
Camb. Univ. Press, 1978.
Krause, Raedler, 1980
F. Krause and K.-H. Raedler.
Mean-field magnetohydrodynamics and dynamo theory.
1980.
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16. References
Pouquet et al., 1976
Pouquet, A., Frisch, U., and Leorat, J.,
Strong MHD helical turbulence and the nonlinear dynamo effect.
Journal of Fluid Mechanics, 77:321, 1976.
Blackman, Brandenburg, 2002
E. G. Blackman and A. Brandenburg.
Dynamic nonlinearity in large-scale dynamos with shear.
Astrophys. J., 579:359, 2002.
Pietarila Graham, et. al., 2012
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.
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