Hall-magnetohydrodynamic small-scale dynamos

Magnetic field generation by dynamo action is often studied within the theoretical framework of magnetohydrodynamics (MHD). However, for sufficiently diffuse media, the Hall effect may become non-negligible. We present results from three-dimensional simulations of the Hall-MHD equations subjected to random nonhelical forcing. We study the role of the Hall effect in the dynamo efficiency for different values of the Hall parameter. For small values of the Hall parameter, the small-scale dynamo is more efficient, displaying faster growth and saturating at larger amplitudes of the magnetic field. For larger values of the Hall parameter, saturation of the magnetic field is reached at smaller amplitudes than in the MHD case. We also study energy transfer rates among spatial scales and show that the Hall effect produces a reduction of the direct energy cascade at scales larger than the Hall scale, therefore leading to smaller energy dissipation rates. Finally, we present results stemming from simulations at large magnetic Prandtl numbers, which is the relevant regime in the hot and diffuse interstellar medium. In the range of magnetic Prandtl numbers considered, the Hall effect moves the peak of the magnetic energy spectrum as well as other relevant magnetic length scales toward the Hall scale.

Figure 1

Kinetic (thin lines) and magnetic (thick lines) energies vs time for $ϵ=0$, 0.05, and 0.10 (from top to bottom).

Figure 2

Kinetic (thin lines) and magnetic (thick lines) dissipation rates vs time for $ϵ=0$, 0.05, and 0.10 (from top to bottom).

Figure 3

Magnetic energy vs time, showing the exponential growth rate during the early linear dynamo regime.

Figure 4

Ratio between $ϵ\left|J_k\right|$ and $\left|U_k\right|$ at different times (labeled). The top frame corresponds to the run with $ϵ=0.05$ and the bottom frame to $ϵ=0.10$. The vertical gray line in each frame corresponds to the Hall wave number $k_{ϵ}=1/ϵ$.

Figure 5

Total-energy spectrum (thick trace) at $t=72$ for $ϵ=0.00$ (top), $ϵ=0.05$ (center), and $ϵ=0.10$ (bottom). In each frame magnetic energy spectra at $t=18,36,72$ are also shown (corresponding to the thin lines from bottom to top). The Kolmogorov and Kazantsev spectra are overlaid (dotted traces) for reference.

Figure 6

Spectral distribution of current density, i.e., $k^2{E}_b\left(k\right)$ vs $k$ for three different values of the Hall parameter (labeled). The dotted trace corresponds to the Kazantsev slope $k^{7/2}$. The arrows indicate the average wave number $k_J$ [see Eq. (6)] for each distribution.

Figure 7

Energy transfer rate contour plots on the $\left(\kappa ,Q\right)$ plane for the run with $ϵ=0.05$ at the stationary regime. Both $\kappa$ and $Q$ run from zero to $k_{max}=85$. Light-gray filled contours correspond to levels at [0.001,0.010,0.100] of the maximum positive value, while dark-gray contours display the same levels at negative energy transfer rates. The top frame shows the total transfer rate, with a peak value of 0.36, and the bottom frame shows the Hall transfer rate, with a peak value of $4\times {10}^{-4}$.

Figure 8

Energy fluxes vs $k$, normalized by the total dissipation rate $D$. The black line corresponds to a time average at the stationary regime of the run with $ϵ=0$, while the gray line is for $ϵ=0.05$.

Figure 9

Total-energy spectrum (thick trace) at $t=72$ for different runs (as labeled). Magnetic energy spectra at $t=18,36,72$ (thin lines from bottom to top in each panel) are also shown. The Kolmogorov and Kazantsev slopes are overlaid (dotted trace) for reference.

Figure 10

Magnetic energy vs time for the three runs as labeled.

Figure 11

Spectral distribution of current density, i.e., $k^2{E}_b\left(k\right)$ vs $k$ for three runs with different values of $ϵ$ and Pm (labeled). The dotted trace corresponds to the Kazantsev slope $k^{7/2}$. The arrows indicate the average wave number $k_J$ [see Eq. (6)] for each of the current density distributions.