Magnetohydrodynamic turbulence mediated by reconnection

Abstract

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 $λ_c$ 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 $λ_c$. As a result, turbulent fluctuations become progressively less anisotropic at smaller scales, with the alignment angle increasing as $θ\sim (λ/λ_*)^{-4/5+β}$, where $λ_*\sim L_0 S_0^{-3/4}$ is the resistive dissipation scale. Here $L_0$ is the outer scale of the turbulence, $S_0$ is the corresponding Lundquist number, and {$0\leq β<4/5$} is a parameter. The resulting Fourier energy spectrum is $E(k_\perp)\propto k_\perp^{-11/5+2β/3}$, where $k_\perp$ is the wavenumber normal to the local mean magnetic field, and the critical scale is $λ_c\sim S_L^{-(4-5β)/(7-{20β/3})}$. The simplest model corresponds to $β=0$, 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 ${\tilde η}k^{2+2s}$ leads (in the simplest case of $β=0$) to the different transition scale $λ_c\sim L_0{\tilde S}_0^{-4/(7+9s)}$ 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.