Magnetic field spreading from stellar and galactic dynamos into the exterior

TL;DR

This paper investigates how magnetic fields generated by stellar and galactic dynamos diffuse into their exterior when the surrounding medium is poorly conducting, rather than assuming a current-free vacuum. It develops a mean-field dynamo framework with an α effect, turbulent diffusivity, and viscosity, including the displacement current as an optional ingredient, and solves the axisymmetric equations in a finite dynamo region embedded in a diffusive exterior. The key findings are that dipolar fields decay as $r^{-3}$ while quadrupolar fields acquire a slow toroidal component that decays as $r^{-2}$, with the magnetosphere expanding ballistically during exponential growth and diffusively during saturation; displacement current is shown to be negligible for the cases considered. The work yields observable predictions for Faraday rotation measures and synchrotron emission that could distinguish dipole from quadrupole configurations, while arguing that the outskirts of galaxies cannot easily magnetize the intergalactic voids by simple superposition, reinforcing primordial-origin scenarios. These results provide a physically motivated boundary condition for dynamos and offer testable signatures for current and future radio telescopes.

Abstract

The exteriors of stellar and galactic dynamos are usually modeled as a current-free potential field. A more realistic description might be that of a force-free magnetic field. Here, we suggest that, in the absence of outflows, neither of those reflect the actual behavior when the magnetic field spreads diffusively into a more poorly conducting turbulent exterior outside dynamo. In particular, we show that the usual ordering of the dipole magnetic field being the most slowly decaying one is altered, and that the quadrupole can develop a toroidal component that decays even more slowly with radial distance. This behavior is best seen for spherical dynamo volumes and becomes more complicated for oblate ones. In either case, however, those fields are confined within a magnetosphere beyond which the field drops exponentially. The magnetosphere expands ballistically (i.e., linearly in time $t$) during the exponential growth phase of the dynamo, but diffusively proportional to $t^{1/2}$ during the saturated phase. We demonstrate that the Faraday displacement current, which plays a role in a vacuum, can safely be neglected in all cases. For quadrupolar configurations, the synchrotron emission from the magnetosphere is found to be constant along concentric rings. The total and the polarized radio emissions from the dipolar or the quadrupolar configurations display large scale radial trends that are potentially distinguishable with existing radio telescopes. The superposition of magnetic fields from galaxies in the outskirts of the voids between galaxy clusters can therefore not explain the void magnetization of the intergalactic medium, reinforcing the conventional expectation that those fields are of primordial origin.

Figures (8)

• Figure 1: Colorscale representation of $\ln|\bm{B}|$ vs $r$ and $t$ for Run B. Yellow (blue) shades denote large (small) fields. The dynamo operates in $0\leq r\leq 1$, as can be seen by the elevated field strength close to $r=0$.
• Figure 2: Radial dependence of $\langle \bm{B}^2\rangle^{1/2}$ at times $t/\tau_\mathrm{diff}=30$, 35, and 50 (dotted lines), as well as $t/\tau_\mathrm{diff}=100$, 300, and 1000 (solid lines), for (a) a dipolar field with $C_\eta=50$ (Run A), (b) a quadrupolar field with $C_\eta=50$ (Run B), and (c) a quadrupolar field with $C_\eta=10$ (Run C). The asymptotic fall-offs $\propto r^{-3}$ for the dipolar field and $\propto r^{-2}$ for the quadrupolar fields are marked with dashed-dotted lines. The red lines denote the fit given by Eq. (\ref{['eq:Bfit']}) with the $q_{\rm diff}$ values listed in Table \ref{['TSummary']}.
• Figure 3: Radial dependence of (a) $a_{11}(r)$ and $b_{21}(r)$ for the dipole (Run A) and (b) $a_{21}(r)$ and $b_{11}(r)$ for the quadrupole (Run B). The asymptotic slopes are $a_{11}/(B_{\rm eq} R)\approx0.052\,(r/R)^{-2}$ and $b_{21}/B_{\rm eq}\approx0.3\,(r/R)^{-3}$ for the dipole and $b_{11}/B_{\rm eq}\approx0.132\,(r/R)^{-2}$ and $a_{21}/(B_{\rm eq} R)\approx0.037\,(r/R)^{-3}$ for the quadrupole, and are marked with dashed-dotted lines. For a vacuum field, $b_{11}$ would be zero. The red and blue lines give the scalings of $\overline{A}_\phi$ and $\overline{B}_\phi$, respectively, which are opposite for the dipolar and quadrupolar cases.
• Figure 4: Radial magnetic field profiles compensated (a) by $r^3$ for the dipolar case (Runs D--F), and (b) by $r^2$ for the quadrupolar case (Run G--I). For the solid lines, the displacement current is neglected (Runs D and G), while for the dashed and dotted lines it is included with $c R/\eta_{\rm eff}=1$ (Runs E and H) and 0.5 (Runs F and I), respectively. The red, blue, and black lines correspond to the times $t/\tau_\mathrm{diff}=20$, 200, and 800.
• Figure 5: Radial magnetic field profiles for Runs J, K, and L. For Run J, we have $r_\mathrm{in}/R=0.2$ instead of 0.1, while for Runs K and L, we have $h/R=0.5$ and 0.2, respectively. The times are (a) $t/\tau_\mathrm{diff}=0.1$, 0.3, and 1.5 for the dotted lines and 10, 100, and 942 for the solid lines, (b) 0.1, 0.2, and 0.8 for the dotted lines and 10, 50, and 530 for the solid lines, (c) 0.2, 0.4, and 0.6 for the dotted lines and 500, 1200, 2700, and 5990 for the solid lines.