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 M = 0.1 and 10, respectively. We find that the hydrodynamic (Re) and magnetic Reynolds numbers (Rm) on the turbulence driving scale turb in a cubic domain of side length L with a total of N3 resolution elements are well described by Re = [2(N/NRe)(turb/L)]pRe and Rm = [2(N/NRm)(turb/L)]pRm . We provide two sets of fit values of (NRe, pRe, NRm, pRm): one with pRe and pRm fixed at their theoretical values, and the other one allowing all four parameters to vary. The sets for M = 0.1 are (1.57+0.10 −0.12, 4/3, 1.55+0.45 −0.14, 4/3) and (0.83+0.09 −0.08, 1.20+0.02 −0.02, 4.19+2.95 −4.05, 1.60+0.18 −0.33), respectively. For M = 10, they are (3.55+0.78 −0.56, 3/2, 1.03+0.12 −0.11, 3/2) and (10.46+0.96 −0.85, 1.90+0.04 −0.04, 0.44+0.61 −0.23, 1.32+0.17 −0.09). The resulting magnetic Prandtl numbers (Pm = Rm/Re) are consistent with constant values of 1.0+0.3 −0.2 for M = 0.1, and 6.2+5.6 −4.8 for M = 10. These results apply when the magnetic energy (Emag) is 10 per cent of the turbulent kinetic energy (Ekin). When Emag/Ekin ∼ 0.1 − 1, Rm is reduced by a factor ∼ 3 (implying an increase in NRm by a factor ∼ 2) for M = 0.1, while Rm for M = 10 and Re (for any M) remain largely unaffected. We compare our 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 Re − N relations above to within a factor of three. We further compare these results to target Re and Rm values in simulations using explicit dissipation from the literature. These literature comparisons and our relations allow users to assess what value of Re and Rm can be reached at a given N, ensuring that physical dissipation dominates over numerical dissipation. Show more