Fully resolved array of simulations investigating the influence of the magnetic Prandtl number on magnetohydrodynamic turbulence

We explore the effect of the magnetic Prandtl number Pm on energy and dissipation in fully resolved direct numerical simulations of steady-state, mechanically forced, homogeneous magnetohydrodynamic turbulence in the range $1/32<\mathrm{Pm}<32$. We compare the spectra and show that if the simulations are not fully resolved, the steepness of the scaling of the kinetic-to-magnetic dissipation ratio with Pm is overestimated. We also present results of decaying turbulence with helical and nonhelical magnetic fields, where we find nonhelical reverse spectral transfer for $\mathrm{Pm}<1$. The results of this systematic analysis have applications including stars, planetary dynamos, and accretion disks.

Figure 1

Small circles show ${\nu }^{-1}$ and ${\eta }^{-1}$ for each of the 36 simulations appearing in Figs. 2, 3, 4. The nine large circles indicate the decaying helical and nonhelical simulations with initial spectra peaking at $k_0=40$ (see Fig. 6). The lines indicate points of constant $\text{Pm}=2^n$ for $-5\le n\le 5$. The largest and smallest values of $\eta$ and $\nu$ are 0.01 and 0.000 312 5.

Figure 2

Selected simulations: (a) kinetic energy spectra, compensated by $k^{-5/3}$, and (b) uncompensated magnetic energy spectra. The top images show data with $\text{Pm}=1$, the middle show data with $\text{Rm}\simeq 2275$, and the bottom show data with $\text{Re}\simeq 2275$. In each plot the solid line corresponds to the same simulation, with $\text{Pm}=1$ and $\text{Re}=\text{Rm}\simeq 2275$.

Figure 3

Time-averaged Alfvén ratios of simulations grouped according to resistivity $\eta$.

Figure 4

Time-averaged kinetic-to-magnetic dissipation rate ratios grouped according to resistivity $\eta$.

Figure 5

Comparison of the time-averaged kinetic-to-magnetic dissipation rate in simulations on a ${128}^3$ lattice ${({\varepsilon }_K/{\varepsilon }_M)}_{128}$ and on a ${512}^3$ lattice ${({\varepsilon }_K/{\varepsilon }_M)}_{512}$ with otherwise identical initial conditions.

Figure 6

Plot of $E_3\left(t\right)$ normalized by $E_3\left(0\right)$ for nonhelical runs (dashed lines) and helical runs (solid lines). Lines with diamond points correspond to $\text{Pm}=1$, upward-pointing small and large triangles to $\text{Pm}=4$ and 16, respectively, and downward-pointing small and large triangles to $\text{Pm}=1/4$ and $1/16$, respectively.