We present the results of our detailed pseudospectral direct numerical simulation (DNS) studies, with up to $1024^3$ collocation points, of incompressible, magnetohydrodynamic (MHD) turbulence in three dimensions, without a mean magnetic field. Our study concentrates on the dependence of various statistical properties of both decaying and statistically steady MHD turbulence on the magnetic Prandtl number ${\rm Pr_M}$ over a large range, namely, $0.01 \leq {\rm Pr_M} \leq 10$. We obtain data for a wide variety of statistical measures such as probability distribution functions (PDFs) of moduli of the vorticity and current density, the energy dissipation rates, and velocity and magnetic-field increments, energy and other spectra, velocity and magnetic-field structure functions, which we use to characterise intermittency, isosurfaces of quantities such as the moduli of the vorticity and current, and joint PDFs such as those of fluid and magnetic dissipation rates. Our systematic study uncovers interesting results that have not been noted hitherto. In particular, we find a crossover from larger intermittency in the magnetic field than in the velocity field, at large ${\rm Pr_M}$, to smaller intermittency in the magnetic field than in the velocity field, at low ${\rm Pr_M}$. Furthermore, a comparison of our results for decaying MHD turbulence and its forced, statistically steady analogue suggests that we have strong universality in the sense that, for a fixed value of ${\rm Pr_M}$, multiscaling exponent ratios agree, at least within our errorbars, for both decaying and statistically steady homogeneous, isotropic MHD turbulence.
Depletion of Nonlinearity in Magnetohydrodynamic Turbulence: Insights from Analysis and Simulations
我们建立在流体湍流研究的最新发展[Giant \textit{et al.} Nonlinearity 27，2605（2014）]的基础上，以定义$ω^\pm= ω\pm j$的适当缩放的阶数$m$矩$D_m^{\pm}$，其中$ω$和$j$分别是三维磁流体动力学（MHD）中的涡度和电流密度。我们显示的数学分析，为单位磁普朗特数$P_M$，这些时刻可以用来确定三种可能的制度的MHD方程的解决方案，这些制度是指定的不等式$D_m^{\pm}$和$D_1^{\pm}$。然后，我们比较我们的数学结果与我们的直接数值模拟（DNSs），从而表明，3D MHD湍流就像它的流体湍流对应的所有解决方案，我们已经调查，保持在\textit{only one of these regimes}，这个制度已经耗尽非线性。我们研究的影响，我们的研究结果的指数$q^{\pm}$的能量谱$\mathcal{E}^{\pm}(k)$上的波数$k$的幂律依赖的特征，在惯性范围内的尺度。我们还评论（a）推广我们的结果的情况下$P_M \neq 1$和（b）之间的关系$D_m^{\pm}$和顺序$m$力矩的梯度的流体动力场，这是用来在湍流中的不稳定性的特征。
