Intermittent turbulent dynamo at very low and high magnetic Prandtl numbers

Context: Direct numerical simulations have shown that the dynamo is efficient even at low Prandtl numbers, i.e., the critical magnetic Reynolds number Rm_c necessary for the dynamo to be efficient becomes smaller than the hydrodynamic Reynolds number Re when Re -> infinity. Aims: We test the conjecture (Iskakov et al. 2007) that Rm_c actually tends to a finite value when Re -> infinity, and we study the behavior of the dynamo growth factor \gamma\ at very low and high magnetic Prandtl numbers. Methods: We use local and nonlocal shell-models of magnetohydrodynamic (MHD) turbulence with parameters covering a much wider range of Reynolds numbers than direct numerical simulations, but of astrophysical relevance. Results: We confirm that Rm_c tends to a finite value when Re -> infinity. The limit for Rm -> infinity of the dynamo growth factor \gamma\ in the kinematic regime behaves like Re^\beta, and, similarly, the limit for Re -> infinity of \gamma\ behaves like Rm^{\beta'}, with \beta=\beta'=0.4. Conclusion: Comparison with a phenomenology based on an intermittent small-scale turbulent dynamo, together with the differences between the growth rates in the different local and nonlocal models, indicate a weak contribution of nonlocal terms to the dynamo effect.