On the accuracy of simulations of turbulence

Abstract

The widely recognized issue of adequate spatial resolution in numerical simulations of turbulence is studied in the context of two-dimensional magnetohydrodynamics. The familiar criterion that the dissipation scale should be resolved enables accurate computation of the spectrum, but fails for precise determination of higher-order statistical quantities. Examination of two straightforward diagnostics, the maximum of the kurtosis and the scale-dependent kurtosis, enables the development of simple tests for assessing adequacy of spatial resolution. The efficacy of the tests is confirmed by examining a sample problem, the distribution of magnetic reconnection rates in turbulence. Oversampling the Kolmogorov dissipation scale by a factor of 3 allows accurate computation of the kurtosis, the scale-dependent kurtosis, and the reconnection rates. These tests may provide useful guidance for resolution requirements in many plasma computations involving turbulence and reconnection.

Small Scale Processes in the Solar Wind

“…The latter are chosen to achieve both high Reynolds numbers and to ensure adequate spatial resolution. A detailed discussion of these issues has been given by Wan et al (2010). We report on runs with resolution from 4096 2 up to 16384 2 grid points, reaching Reynolds numbers R ν = R µ ∼ 10000.…”

Magnetic reconnection as an element of turbulence

“…The latter are chosen to achieve both high Reynolds numbers and to ensure adequate spatial resolution. A detailed discussion of these issues has been given by Wan et al (2010), and will be reviewed here in Sect. 6.…”

“…In this section we will briefly review the widely recognized issue of adequate spatial resolution in numerical simulations of turbulence, in the context of 2D MHD (Donzis et al, 2008;Wan et al, 2010). Generally, it is widely thought that a good criterion for the quality of numerics is given by the spectra: a nice power spectrum is synonym of a good simulation.…”

“…Here we argue that this is not sufficient, since a slightly inadequate resolution fails for precise determination of higher-order statistical quantities, and therefore fails to accurately describe reconnection. In particular, in Wan et al (2010) it was suggested that oversampling the Kolmogorov dissipation scale η by a factor of 3 allows accurate computation of the kurtosis, the scale-dependent kurtosis, and the reconnection rates [see Eq. (14) for a definition of η].…”

“…Recently, Wan et al (2010) demonstrated that an insufficient numerical (grid) resolution may strongly damage the small scale properties of both turbulence and reconnection. Here we summarize these main results, providing suggestions and numerical recipes in order to correctly simulate the problem of reconnection in turbulence.…”