Intermittent turbulent dynamo at very low and high magnetic Prandtl numbers

Context. Direct numerical simulations of plasmas have shown that the dynamo effect is efficient even at low Prandtl numbers, i.e., the critical magnetic Reynolds number Rm(c) that is necessary for a dynamo to be efficient becomes smaller than the hydrodynamic Reynolds number Re when Re ->infinity. Aims. We test the conjecture that Rm(c) tends to a finite value when Re ->infinity, and we study the behavior of the dynamo growth factor. 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, that is of astrophysical relevance. Results. We confirm that Rm(c) tends to a finite value when Re ->infinity. As Rm ->infinity, the limit to the dynamo growth factor. in the kinematic regime follows Re(beta), and, similarly, the limit for Re ->infinity of gamma behaves like Rm(beta'), with beta approximate to beta' approximate to 0.4. Conclusions. Our 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 that nonlocal terms contribute weakly to the dynamo effect.