Magnetohydrodynamic Turbulence Mediated by Reconnection

Magnetic field fluctuations in MHD turbulence can be viewed as current sheets that are progressively more anisotropic at smaller scales. As suggested by Loureiro & Boldyrev 2017 and Mallet et al. 2017, below a certain critical thickness $\lambda_c$ such current sheets become tearing-unstable. We propose that the tearing instability changes the effective alignment of the magnetic field lines in such a way as to balance the eddy turnover rate at all scales smaller than $\lambda_c$. As a result, turbulent fluctuations become progressively less anisotropic at smaller scales, with the alignment angle increasing as $\theta \sim (\lambda/\lambda_*)^{-4/5+\beta}$, where $\lambda_*\sim L_0 S_0^{-3/4}$ is the resistive dissipation scale. Here $L_0$ is the outer scale of the turbulence, $S_0$ is the corresponding Lundquist number, and {$0\leq \beta <4/5$} is a parameter. The resulting Fourier energy spectrum is $E(k_\perp)\propto k_\perp^{-11/5+2\beta/3}$, where $k_\perp$ is the wavenumber normal to the local mean magnetic field, and the critical scale is $\lambda_c\sim S_L^{-(4-5\beta)/(7-{20\beta/3})}$. The simplest model corresponds to $\beta=0$, in which case the predicted scaling formally agrees with one of the solutions obtained in (Mallet et al. 2017) from a discrete hierarchical model of abruptly collapsing current sheets, an approach different and complementary to ours. We also show that the reconnection-mediated interval is non-universal with respect to the dissipation mechanism. Hyper-resistivity of the form ${\tilde \eta}k^{2+2s}$ leads (in the simplest case of $\beta=0$) to the different transition scale $\lambda_c\sim L_0{\tilde S}_0^{-4/(7+9s)}$ and the energy spectrum $E(k_\perp)\propto k_\perp^{-(11+9s)/(5+3s)}$, where ${\tilde S}_0$ is the corresponding hyper-resistive Lundquist number.

磁流体动力学（Magnetohydrodynamics, MHD）湍流中的磁场涨落可被视为在更小尺度上各向异性逐渐增强的电流片。正如Loureiro与Boldyrev（2017）以及Mallet等人（2017）所提出的，当电流片的厚度低于某一临界厚度$lambda_c$时，其会变得具有撕裂不稳定性。我们提出，撕裂不稳定性会改变磁场线的有效取向，使得所有小于$lambda_c$的尺度上的涡旋周转速率达到平衡。其结果是，湍流涨落在更小尺度上的各向异性逐渐减弱，取向角满足$ heta sim (lambda/lambda_*)^{-4/5+eta}$，其中$lambda_*sim L_0 S_0^{-3/4}$为电阻耗散尺度。此处$L_0$为湍流的外尺度，$S_0$为对应的隆德奎斯特数，且参数满足$0leq eta <4/5$。由此得到的傅里叶能谱为$E(k_perp)propto k_perp^{-11/5+2eta/3}$，其中$k_perp$为垂直于局域平均磁场的波数，临界尺度为$lambda_csim S_L^{-(4-5eta)/(7-20eta/3)}$。最简单的模型对应$eta=0$，此时预测的标度律与Mallet等人（2017）中通过离散层级式突变坍缩电流片模型得到的其中一组解形式一致——该方法与我们的研究路径既有所区别又互为补充。我们还证明了，重连介导的区间对耗散机制不具有普适性。形如$ ilde{eta}k^{2+2s}$的超电阻率会（在$eta=0$的最简单情形下）带来不同的过渡尺度$lambda_csim L_0 ilde{S}_0^{-4/(7+9s)}$与傅里叶能谱$E(k_perp)propto k_perp^{-(11+9s)/(5+3s)}$，其中$ ilde{S}_0$为对应的超电阻率隆德奎斯特数。