Dynamo Effect in the Kraichnan Magnetohydrodynamic Turbulence |
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Authors: | Heikki Arponen Peter Horvai |
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Institution: | (1) Department of Mathematics and Statistics, Helsinki University, P.O. Box 68, 00014 Helsinki, Finland;(2) Science & Finance, Capital Fund Management, 6-8 Bd Haussmann, 75009 Paris, France |
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Abstract: | The existence of a dynamo effect in a simplified magnetohydrodynamic model of turbulence is considered when the magnetic Prandtl
number approaches zero or infinity. The magnetic field is interacting with an incompressible Kraichnan-Kazantsev model velocity
field which incorporates also a viscous cutoff scale. An approximate system of equations in the different scaling ranges can
be formulated and solved, so that the solution tends to the exact one when the viscous and magnetic-diffusive cutoffs approach
zero. In this approximation we are able to determine analytically the conditions for the existence of a dynamo effect and
give an estimate of the dynamo growth rate. Among other things we show that in the large magnetic Prandtl number case the
dynamo effect is always present. Our analytical estimates are in good agreement with previous numerical studies of the Kraichnan-Kazantsev
dynamo by Vincenzi (J. Stat. Phys. 106:1073–1091, 2002). |
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Keywords: | Dynamo Magnetohydrodynamic Turbulence Kraichnan-Kazantsev |
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