Hall magnetohydrodynamic turbulence with a magnetic Prandtl number larger than unity

Turbulence structures with the magnetic Prandtl number larger than unity are studied by means of direct numerical simulations of homogeneous, isotropic, and incompressible Hall magnetohydrodynamic (MHD) turbulence driven by a random force. Spectral and spatial structures on the scales smaller than the ion skin depth are focused upon in this numerical paper. The numerical simulations reveal the emergence of a new power law in the velocity field whereas it is not observed in MHD turbulence simulation without the Hall term. An order estimate of the energy budget in the spectral space shows that this new power law appears in association with the Hall effect and that a balance between the Lorentz force and the viscous dissipation is crucial for formation of the power law. A resemblance to an elastic turbulence is found in the power-law-formation mechanism. Frequent eruptions of strong current ribbons accompanying strong palinstrophy density are observed, showing generation of the palinstrophy density by the Lorentz force at the scales below the ion skin depth. These properties in spectral and spatial structures characterize a high magnetic Prandtl number Hall MHD turbulence at the scales smaller than the ion skin depth.

Figure 1

Shell-averaged and time-averaged energy spectra $E_K\left(k\right)$ and $E_M\left(k\right)$ in (a) runs $A$ and $B$ and in (b) run $C$.

Figure 2

(a) A comparison of $E_M\left(k\right)$ and $E_K\left(k\right)$ of run $B$ among the computations of $N^3={256}^3,{512}^3$, and ${1024}^3$. (b) A comparison of the compensated energy spectra $k^{8/3}{E}_M\left(k\right),k^{5/2}{E}_M\left(k\right)$, and $k^{7/3}{E}_M\left(k\right)$ of run $B$ among the computations of $N^3={256}^3,{512}^3$, and ${1024}^3$.

Figure 3

Energy transfer functions $T_{u_j{\partial }_j{u}_i}\left(k\right)$ and $T_{JB}$ in run $B$. Negative values are not plotted.

Figure 4

Isosurfaces of the enstrophy density $Q$ (green, or gray in the printed version), the palinstrophy $P$ (red, or dark gray in the printed version), and current density $I$ (light gray) in runs $A\text{–}C$ in (a)–(c), respectively. Isosurfaces of the palinstrophy density $P$ (red) are drawn in (b) and (c), whereas they are not drawn in (a) for clarity in the figure.

Figure 5

Contours of the viscous term $-\nu u_i{\partial }_k{\partial }_k{u}_i$ (color map) and Lorentz-force term $u_i\left({\epsilon }_{ijk}{J}_j{B}_k\right)$ (contour lines) in Eq. (3) of run $B$ on a plane of $128\times 128$ grid points.