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.