Spectral Energy Dynamics in Magnetohydrodynamic Turbulence

Spectral direct numerical simulations of incompressible MHD turbulence at a resolution of up to ${1024}^3$ collocation points are presented for a statistically isotropic system as well as for a setup with an imposed strong mean magnetic field. The spectra of residual energy, $E_k^{\mathrm{R}}=|E_k^{\mathrm{M}}-E_k^{\mathrm{K}}|$, and total energy, $E_k=E_k^{\mathrm{K}}+E_k^{\mathrm{M}}$, are observed to scale self-similarly in the inertial range as $E_k^{\mathrm{R}}\sim k^{-7/3}$, $E_k\sim k^{-5/3}$ (isotropic case) and $E_{k_{\perp }}^{\mathrm{R}}\sim k_{\perp }^{-2}$, $E_{k_{\perp }}\sim k_{\perp }^{-3/2}$ (anisotropic case, perpendicular to the mean field direction). A model of dynamic equilibrium between kinetic and magnetic energy, based on the corresponding evolution equations of the eddy-damped quasinormal Markovian closure approximation, explains the findings. The assumed interplay of turbulent dynamo and Alfvén effect yields $E_k^{\mathrm{R}}\sim k{E}_k^2$, which is confirmed by the simulations.

Figure 1

Total (solid line), kinetic (dashed line), and magnetic (dotted line) energy spectra in ${1024}^3$ case I simulation (normalized, time-averaged, and compensated). Dash-dotted line: $k^{-3/2}$ scaling.

Figure 2

Field-perpendicular total (solid line), kinetic (dashed line), and magnetic (dotted line) energy spectra (normalized, time-averaged, and compensated) in ${1024}^2\times 256$ case II simulation with $b_0=5$. Dash-dotted curve: high-$k$ part of field-parallel total energy spectrum. Inset: perpendicular total energy spectrum for resolutions of ${512}^2\times 256$ (dash-dotted line) to ${1024}^2\times 256$ (solid line).

Figure 3

Compensated, space-angle-integrated residual energy spectrum, $E_k^{\mathrm{R}}$, for the same system as in Fig. 1. Dash-dotted line: $k^{-2}$ scaling.

Figure 4

Compensated field-perpendicular residual energy spectrum for the same system as in Fig. 2.