Two-dimensional behavior of three-dimensional magnetohydrodynamic flow with a strong guiding field

The magnetohydrodynamic (MHD) equations in the presence of a guiding magnetic field are investigated by means of direct numerical simulations. The basis of the investigation consists of nine runs forced at the small scales. The results demonstrate that for a large enough uniform magnetic field the large scale flow behaves as a two-dimensional (2D) (non-MHD) fluid exhibiting an inverse cascade of energy in the direction perpendicular to the magnetic field, while the small scales behave like a three-dimensional (3D) MHD fluid cascading the energy forwards. The amplitude of the inverse cascade is sensitive to the magnetic field amplitude, the domain size, the forcing mechanism, and the forcing scale. All these dependences are demonstrated by the varying parameters of the simulations. Furthermore, in the case that the system is forced anisotropically in the small parallel scales an inverse cascade in the parallel direction is observed that is feeding the 2D modes $k_{\parallel }=0$.

Figure 1

Time evolution of the kinetic energy $E_{{}_K}$ (solid line) and the magnetic energy $E_{{}_M}$ (dashed line) for run R1.

Figure 2

The energy flux for run R1 in the perpendicular direction (solid line) and parallel direction (dashed line).

Figure 3

Kinetic (top panel) and magnetic (bottom panel) energy spectra of run R1. The solid lines correspond to the averaged spectra ${\overline{E}}_u\left(k_{\perp }\right),{\overline{E}}_b\left(k_{\perp }\right)$ while the dashed lines indicate the zeroth component ($k_{\parallel }=0$) of the two-dimensional energy spectra $E_u(0,k_{\perp })$ and $E_b(0,k_{\perp })$. The straight lines show for reference the power-law spectra $k^{-5/3},k^{-2},k^{-3}$.

Figure 4

Top panel: The energy flux in the perpendicular direction for R1 ($V_{{}_A}=5$, solid line), R2 ($V_{{}_A}=2$, dashed line), and R3 ($V_{{}_A}=10$, dashed-dot line). Bottom panel: The energy flux in the perpendicular direction for the same runs.

Figure 5

Top panel: The kinetic energy spectra of ${\overline{E}}_u\left(k_{\perp }\right)$ (solid line) and $E_u(0,k_{\perp })$ (dashed line) for R3, $V_{{}_A}=10$ (top lines), R1, $V_{{}_A}=5$ (middle lines), and R2, $V_{{}_A}=2$ (bottom lines). Bottom panel: The kinetic energy spectra ${\overline{E}}_u\left(k_{\perp }\right)$ (solid line) compared to the magnetic energy spectra ${\overline{E}}_b\left(k_{\perp }\right)$ (dashed line) of the same runs and with the same order. The spectra have been shifted for reasons of clarity.

Figure 6

Top panel: The energy flux in the perpendicular direction for R1 ($8<k_f\le 10$, solid line), R4 ($4<k_k\le 5$, dashed line), and R5 ($16<k_f\le 20$, dashed-dot line). Bottom panel: The energy flux in the perpendicular direction for the same runs.

Figure 7

Top panel: The kinetic energy spectra of ${\overline{E}}_u\left(k_{\perp }\right)$ (solid line) and $E_u(0,k_{\perp })$ (dashed line) for R5 (top lines), R1 (middle lines), and R4 (bottom lines). Bottom panel: The kinetic energy spectra ${\overline{E}}_u\left(k_{\perp }\right)$ (solid line) compared to the magnetic energy spectra ${\overline{E}}_b\left(k_{\perp }\right)$ (dashed line) of the same runs and with the same order. The spectra have been shifted for reasons of clarity. The arrows indicate the location of the forcing.

Figure 8

Top panel: The energy flux in the perpendicular direction for R1 ($M=0$, solid line), R6 ($M=0.4$, dashed line), and R7 ($M=0.6$, dashed-dot line). Bottom panel: The energy flux in the perpendicular direction for the same runs.

Figure 9

Top panel: The kinetic energy spectra of ${\overline{E}}_u\left(k_{\perp }\right)$ (solid line) and $E_u(0,k_{\perp })$ (dashed line) for R1, $M=0$ (top lines), R6, $M=0.4$ (middle lines), and R7 $M=0.6$ (bottom lines). Bottom panel: The kinetic energy spectra ${\overline{E}}_u\left(k_{\perp }\right)$ (solid line) compared to the magnetic energy spectra ${\overline{E}}_b\left(k_{\perp }\right)$ (dashed line) of the same runs and with the same order. The spectra have been shifted for reasons of clarity.

Figure 10

The vector potential spectrum ${\overline{E}}_{{}_A}$ for three different times.

Figure 11

Top panel: The energy flux in the perpendicular direction for R1 (solid line), R8 (dashed line), and R9 (dashed-dot line). Bottom panel: The energy flux in the perpendicular direction for the same runs.

Figure 12

Top panel: The kinetic energy spectra of ${\overline{E}}_u\left(k_{\perp }\right)$ (solid line) and $E_u(0,k_{\perp })$ (dashed line) for R8 (top lines), R1 (middle lines), and R9 (bottom lines). Bottom panel: The kinetic energy spectra ${\overline{E}}_u\left(k_{\perp }\right)$ (solid line) compared to the magnetic energy spectra ${\overline{E}}_b\left(k_{\perp }\right)$ (dashed line) of the same runs and with the same order. The spectra have been shifted for reasons of clarity.