Large-eddy simulation for decaying magnetohydrodynamic turbulence at low magnetic Reynolds number

Abstract

Large-eddy simulations have been conducted to investigate the decay law of homogeneous turbulence influenced by a magnetic field within a cubic domain, employing periodic boundary conditions. The initial integral Reynolds number is approximately 1000, while the initial interaction number $N$ ranges from 0.1–100. The results reveal that the Joule cone angle $\theta$, half of the Joule cone, decays as $\cos \theta \sim t^{-1/2}$ when $N \gg 1$. In the nonlinear stage, small-scale vortices gradually recover and restore three-dimensionality. Moreover, the corresponding critical state at small scales, marking the transition from quasi-two-dimensional structure to the onset of three-dimensionality, has been quantitatively defined. During the linear stage, based on the true magnetic damping number ($\tau _t = \rho / (\sigma {\boldsymbol{B}}^2 \cos ^2 \psi )$, where $\sigma$, $\boldsymbol{B}$ and $\psi$ denote the electrical conductivity, magnetic field and the angle between the wavevector and $\boldsymbol{B}$ in Fourier space, respectively), Moffatt’s decay law, $K \sim t^{-1/2}$, manifests at distinct times and zones in the Fourier space, with $K$ signifying turbulent kinetic energy. In the nonlinear stage, for $N \gg 1$, a $-3$ slope in the energy power spectrum is prominently observed over an extended period. The near-equivalence of the characteristic time scales of inertial and Lorentz forces in the inertial subrange suggests a quasiequilibrium state between energy transfer and Joule dissipation in Fourier space, thereby corroborating the hypothesis proposed by Alemany et al. 1979 Journal de Mecanique 18(2): 277–313. Additionally, it is observed that pressure mediates energy transfer from horizontal kinetic energy ($K_{\parallel }$) to vertical kinetic energy ($K_{\bot }$), accelerating the decay of $K_{\parallel }$. Notably, concurrent inverse and direct energy transfers emerge during the decay process. Our analysis reveals that the ratio $R$ of the maximum inverse to maximum direct energy flux correlates with the dimensionality of the turbulence, following the scaling law $R\sim (\cos \theta )^{-2.2}$.