Numerical viscosity and resistivity in MHD turbulence simulations

ABSTRACT For magnetohydrodynamical (MHD) turbulence simulations to accurately capture the underlying physics, we must understand numerical dissipation. Here, we quantify numerical viscosity and resistivity in the subsonic and supersonic turbulence regimes, with Mach numbers $\mathcal {M}= 0.1$ and 10, respectively. We find that the hydrodynamic (${\mathrm{Re}}$) and magnetic Reynolds numbers (${\mathrm{Rm}}$) on the turbulence driving scale $\ell _{\mathrm{turb}}$ in a cubic domain of side length L with a total of $N^3$ resolution elements are well described by ${\mathrm{Re}}=[2(N/N_{\mathrm{Re}})(\ell _{\mathrm{turb}}/L)]^{p_{\mathrm{Re}}}$ and ${\mathrm{Rm}}=[2(N/N_{\mathrm{Rm}})(\ell _{\mathrm{turb}}/L)]^{p_{\mathrm{Rm}}}$. We provide two sets of fit values of $(N_{\mathrm{Re}}, p_{\mathrm{Re}}, N_{\mathrm{Rm}}, p_{\mathrm{Rm}})$: one with $p_{\mathrm{Re}}$ and $p_{\mathrm{Rm}}$ fixed at their theoretical values, and the other one allowing all four parameters to vary. The sets for $\mathcal {M}=0.1$ are $(1.57_{-0.12}^{+0.10},4/3,1.55_{-0.14}^{+0.45},4/3)$ and $(0.83_{-0.08}^{+0.09},1.20_{-0.02}^{+0.02},4.19_{-4.05}^{+2.95},1.60_{-0.33}^{+0.18})$, respectively. For $\mathcal {M}=10$, they are $(3.55_{-0.56}^{+0.78},3/2,1.03_{-0.11}^{+0.12},3/2)$ and $(10.46_{-0.85}^{+0.96},1.90_{-0.04}^{+0.04},0.44_{-0.23}^{+0.61},1.32_{-0.09}^{+0.17})$. The resulting magnetic Prandtl numbers (${\mathrm{Pm}}={\mathrm{Rm}}/{\mathrm{Re}}$) are consistent with constant values of $1.0_{-0.2}^{+0.3}$ for $\mathcal {M}= 0.1$, and $6.2_{-4.8}^{+5.6}$ for $\mathcal {M}= 10$. These results apply when the magnetic energy ($E_{\mathrm{mag}}$) is $\lesssim 10{{\ \rm per\ cent}}$ of the turbulent kinetic energy ($E_{\mathrm{kin}}$). When $E_{\mathrm{mag}}/E_{\mathrm{kin}}\sim 0.1-1$, ${\mathrm{Rm}}$ is reduced by a factor $\sim 3$ (implying an increase in $N_{\mathrm{Rm}}$ by a factor $\sim 2$) for $\mathcal {M}=0.1$, while ${\mathrm{Rm}}$ for $\mathcal {M}=10$ and ${\mathrm{Re}}$ (for any $\mathcal {M}$) remain largely unaffected. We compare our ${\mathrm{Re}}- N$ relation with 14 other simulations from the literature, which use a large range of different numerical methods (with and without Riemann solvers, different reconstruction schemes and orders, and smoothed particle hydrodynamics), and find that they all agree with the ${\mathrm{Re}}- N$ relations above to within a factor of three. We further compare these results to target ${\mathrm{Re}}$ and ${\mathrm{Rm}}$ values in simulations using explicit dissipation from the literature. These literature comparisons and our relations allow users to assess what value of ${\mathrm{Re}}$ and ${\mathrm{Rm}}$ can be reached at a given N, ensuring that physical dissipation dominates over numerical dissipation.