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 \mathrm { Rm } _ { c } that is necessary for a dynamo to be efficient becomes smaller than the hydrodynamic Reynolds number \mathrm { Re } when \mathrm { Re } \rightarrow \infty .

Aims : We test the conjecture that \mathrm { Rm } _ { c } tends to a finite value when \mathrm { Re } \rightarrow \infty , 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 , that is of astrophysical relevance .

Results : We confirm that \mathrm { Rm } _ { c } tends to a finite value when \mathrm { Re } \rightarrow \infty .
As \mathrm { Rm } \rightarrow \infty , the limit to the dynamo growth factor \gamma in the kinematic regime follows \mathrm { Re } ^ { \beta } , and , similarly , the limit for \mathrm { Re } \rightarrow \infty of \gamma behaves like \mathrm { Rm } ^ { \beta ^ { \prime } } , with \beta \approx \beta ^ { \prime } \approx 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 .