Dynamo effect in the Kraichnan magnetohydrodynamic turbulence

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-Kasantzev model velocity field augmented with a viscous scale cutoff. 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 Prandtl number case the dynamo effect is always present. Our analytical estimates are in good agreement with previous numerical studies of the Kraichnan-Kasantzev dynamo by D. Vincenzi.

30 pages, 7 postscript figures

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• Growth rate of small-scale dynamo at low magnetic Prandtl numbers
• Steady state existence of passive vector fields under the Kraichnan model
• Fluctuation Dynamo and Turbulent Induction at Small Prandtl Number
• Kazantsev dynamo in turbulent compressible flows
• Dissipative magnetic structures and scales in small-scale dynamos
• Suppression of small-scale dynamo in time irreversible turbulence
• Small-Scale Kinematic Dynamo and Non-Dynamo in Inertial-Range Turbulence
• Evolution of localized magnetic field perturbations and the nature of turbulent dynamo
• Reynolds number dependence of Lyapunov exponents of turbulence and fluid particles
• Role of the sonic scale in the growth of magnetic field in compressible turbulence
• Virtual states and exponential decay in small-scale dynamo
• Small-scale turbulent dynamo for low-Prandtl number fluid: comparison of the theory with results of numerical simulations
• Small-scale dynamo with finite correlation times
• An upper critical dimension for dynamo action: A $d$-dimensional closure model study