Numerical Demonstration of Fluctuation Dynamo at Low Magnetic Prandtl Numbers

Direct numerical simulations of incompressible nonhelical randomly forced MHD turbulence are used to demonstrate for the first time that the fluctuation dynamo exists in the limit of large magnetic Reynolds number $\mathrm{Rm}\gg 1$ and small magnetic Prandtl number $\mathrm{Pm}\ll 1$. The dependence of the critical ${\mathrm{Rm}}_c$ for dynamo on the hydrodynamic Reynolds number Re is obtained for $1\lesssim \mathrm{Re}\lesssim 6700$. In the limit $\mathrm{Pm}\ll 1$, ${\mathrm{Rm}}_c$ is about 3 times larger than for the previously well-established dynamo at large and moderate Prandtl numbers: ${\mathrm{Rm}}_c\lesssim 200$ for $\mathrm{Re}\gtrsim 6000$ compared to ${\mathrm{Rm}}_c\sim 60$ for $\mathrm{Pm}\ge 1$. It is not yet possible to determine numerically whether the growth rate of the magnetic energy is $\propto {\mathrm{Rm}}^{1/2}$ in the limit $\mathrm{Rm}\to \infty$, as it should be if the dynamo is driven by the inertial-range motions at the resistive scale.

Figure 1

$\mathrm{\text{Growth}}/\mathrm{\text{decay}}$ rate of $⟨B^2⟩$ vs Pm for $\eta =4\times {10}^{-3}$ ($\mathrm{Rm}\sim 60$), $\eta =2\times {10}^{-3}$ ($\mathrm{Rm}\sim 110$), $\eta ={10}^{-3}$ ($\mathrm{Rm}\sim 230$), $\eta =5\times {10}^{-4}$ ($\mathrm{Rm}\sim 450$), and $\eta =2.5\times {10}^{-4}$ ($\mathrm{Rm}\sim 830$).

Figure 2

The stability curve ${\mathrm{Rm}}_c$ vs Re. “Error bars” connect (Re, Rm) for decaying and growing runs used to obtain points on the stability curve. Stability curves based on the Laplacian and hyperviscous runs are shown separately. For comparison, we also plot the ${\mathrm{Rm}}_c(\mathrm{Re})$ curve obtained in simulations employing TG1 [19] and TG2 forcing [13] (the three highest-Re points in the latter case were obtained by large-eddy simulations). The values of Re and Rm are recalculated according to our definitions using the forcing wave number $k_0$ rather than the dynamical integral scale as in 13, 19.

Figure 3

Normalized spectra of kinetic energy (compensated by $k^{5/3}$) and (growing) magnetic energy for fixed $\eta =5\times {10}^{-4}$ (the $\mathrm{Rm}\sim 450$ sequence from Fig. 1) and increasing Re: (a) Laplacian runs, (b) hyperviscous runs.