Turbulent dynamo in the shell model and the Kazantsev-Kraichnan approach

We investigate small-scale magnetic energy generation in a turbulent conducting flow using two different approaches. One is based on the Kazantsev-Kraichnan (KK) model, developed for a random velocity field with short time correlations, and the other uses the shell magnetohydrodynamic (MHD) method, which describes a turbulent energy cascade on a finite number of spectral shells. We consider only the kinematic (linear) regime and show an exponential growth of magnetic energy for sufficiently large magnetic Reynolds numbers (Rm) in both approaches. By comparing the dependencies of the growth rates at different Rm, we try to answer the question how the model assumptions-of short time correlations or of finite spectral shells-affect the turbulent dynamo possibility and the position of the Rm threshold at which this generation starts. We analyze the generation behavior at different Reynolds and magnetic Prandtl numbers, which are not explicitly included in the KK model but taken into account in the shell approach. Based on the obtained results, we try to discuss why the small-scale generation has not been observed in real MHD dynamo experiments so far.