Scaling and anisotropy in magnetohydrodynamic turbulence in a strong mean magnetic field

We present an analysis of the anisotropic spectral energy distribution in incompressible magnetohydrodynamic turbulence permeated by a strong mean magnetic field. The turbulent flow is generated by high-resolution pseudospectral direct numerical simulations with large-scale isotropic forcing. Examining the radial energy distribution for various angles $\theta$ with respect to ${\mathbf{B}}_0$ reveals a specific structure which remains hidden when not taking axial symmetry with respect to $B_0$ into account. For each direction, starting at the forced large scales, the spectrum first exhibits an amplitude drop around a wave number $k_0$ which marks the start of a scaling range and goes on up to a dissipative wave number $k_d\left(\theta \right)$. The three-dimensional spectrum for $k\ge k_0$ is described by a single $\theta$-independent functional form $F\left(k/k_d\right)$, with the scaling law being the same in every direction. The previous properties still hold when increasing the mean field from $B_0=5$ up to $B_0=10b_{\text{rms}}$, as well as when passing from resistive to ideal flows. We conjecture that at fixed $B_0$ the direction-independent scaling regime is reached when increasing the Reynolds number above a threshold which raises with increasing $B_0$. Below that threshold critically balanced turbulence is expected.

Figure 1

Energy contour levels of three-dimensional spectral energy density $E_3\left(k_x,k_y,k_z\right)$: (a) plane $k_z=0$, (b) plane $k_y=0$, and (c) average of energy density over all planes containing ${\mathbf{B}}_0$. (d) Anisotropic scaling law between wave numbers $k_{\perp }$ and $k_{\parallel }$ (see text), with two compensated scalings: $k_{\parallel }\left(k_{\perp }\right)k_{\perp }^{-1}$ (diamonds) and $k_{\parallel }\left(k_{\perp }\right)k_{\perp }^{-2/3}$ (crosses).

Figure 2

Plane-integrated one-dimensional perpendicular and parallel energy spectra compensated by (a) $k^{3/2}$ and (b) $k^{5/3}$, respectively. The symbol $k$ stands for the respective field-perpendicular and field-parallel wave numbers.

Figure 3

Details of spectral properties: (a) $E_3\left(k,\theta \right)$ for $\theta$ ranging from 0 to $\pi /2$, (b) $E_3\left(k/k_d,\theta \right)$, (c) $k_d^{7/2}$ (dotted curve), $k^{7/2}{E}_3\left(k,\theta \right)$ for $8\le k\le 20$ (solid curve), Eq. (7) with $m=3/2$ (dotted-dashed curve), and (d) energy contour levels of $E_3\left(k_{\parallel },k_{\perp }\right)$: simulation data (solid); Eq. (2) (dashed line, with the oblique line tracing the boundary $\chi =1$ with $k_0=5$, $B_0=5$); Eqs. (3, 7) with $\beta =3$, $\epsilon =0.158$ (dotted curve).

Figure 4

Spectra averaged in four subsets of directions. A, $14°\le \theta \le 39°$; B, $42°\le \theta \le 65°$; C, $67°\le \theta \le 76°$; D, $79°\le \theta \le 90°$; direction $\theta =90°$ as thick curve. (a) Spectra compensated by $k^{3/2}$, with the two oblique dashed lines indicating roughly the inertial range; (b) spectra compensated by $k^{5/3}$.

Figure 5

Anisotropy in ideal runs: (a) and (b) $B_0=5$ and (c) and (d) $B_0=10$. (a)–(c) $k_d^{11/3}$ (dotted curve), $k^{11/3}{E}_3\left(k,\theta \right)$ (solid curve) (a) for $6\le k\le 12$ and (b) for $5\le k\le 8$, Eq. (7) (dashed-dotted curve) with $m=5/3$, and (a) $\epsilon =0.158$ and (c) $\epsilon =0.158/2$. (b)–(d) Energy contour levels of $E_3\left(k_{\parallel },k_{\perp }\right)$: simulation data (solid curve); Eq. (2) [dashed line, with the oblique line tracing the boundary $\chi =1$ with $k_0=5$ and (b) $B_0=5$ and (d) $B_0=10$]; Eqs. (3, 7) with $\beta =3$ and (b) $\epsilon =0.158$ and (d) $\epsilon =0.158/2$ (dotted curve).