Non-Gaussian Generalization of the Kazantsev–Kraichnan Model for a Turbulent Dynamo
We consider a natural generalization of the Kazantsev–Kraichnan model for a small-scale turbulent dynamo. This generalization takes into account the statistical time asymmetry of a turbulent flow and thus allows one to describe velocity fields with energy cascade. For three-dimensional velocity fields, a generalized Kazantsev equation is derived, and the evolution of the second-order magnetic field correlator is investigated for large but finite magnetic Prandtl numbers. It is shown that as Prm → ∞, the growth increment tends to the limit known from the T-exponential (Lagrangian deformation) method. Magnetic field generation is shown to be weaker than that in the Gaussian velocity field for any direction of the energy cascade and essentially depends on the Prandtl number.