Magnetic-Field Generation and Amplification in an Expanding Plasma

Particle-in-cell simulations are used to investigate the formation of magnetic fields $\mathbf{B}$ in plasmas with perpendicular electron density and temperature gradients. For system sizes $L$ comparable to the ion skin depth $d_i$, it is shown that $B\sim d_i/L$, consistent with the Biermann battery effect. However, for large $L/d_i$, it is found that the Weibel instability (due to electron temperature anisotropy) supersedes the Biermann battery as the main producer of $B$. The Weibel-produced fields saturate at a finite amplitude (plasma $\beta \approx 100$), independent of $L$. The magnetic energy spectra below the electron Larmor radius scale are well fitted by the power law with slope $-16/3$, as predicted by Schekochihin et al. [Astrophys. J. Suppl. Ser. 182, 310 (2009)].

Figure 1

Magnetic energy contours after saturation ($t{\omega }_{pe}=235.2$; see Fig. 3) from a 3D simulation with $L_T/d_e=50$. Lighter to darker colors represent $B^2/8\pi P_{e0}=0.0035$, 0.0071, 0.0106. Several magnetic-field lines are also displayed.

Figure 2

Out-of-plane magnetic field $B_z$ after saturation (see Fig. 3) for (a) $L_T/d_e=50$ and (b) $L_T/d_e=400$.

Figure 3

(a) Maximum $B_z$ vs time for a selection of system sizes ($L_T/d_e$). The inset shows the $L_T/d_e=400$ case (black line). The magenta line is the maximum Weibel growth rate ${\gamma }_{\max }$ at $y=0$. Dashed lines identify the times at which the spectra of Fig. 5 are calculated. (b) Maximum (asterisks) and average (diamonds) magnitudes of $B_z$ vs $L_T/d_e$. The triangle represents the maximum $B_z$ for the 3D run. The solid curve is $\max (B_z)/\sqrt{8\pi P_{e0}}=\sqrt{2}{d}_i/L_T$; the dotted line indicates $L_T/d_i=5$. The inset shows the time to maximum magnetic field $t_{\max }$ vs $L_T/d_e$. The solid line indicates $t_{\max }=L_T/v_{Te0}$.

Figure 4

Electric field in the $x$ direction $E_x$ for $L_T/d_e=25$ at $t{\omega }_{pe}=142.8$.

Figure 5

Fourier spectrum of $B_z^2$ for (a) $L_T/d_e=400$ and (b) $L_T/d_e=50$. In (a), the spectrum is shown at several different times [see Fig. 3], while in (b), the 3D (black curve) and the 2D (blue curve) simulations are shown for $t{\omega }_{pe}=235.2$. The dashed lines represent where $k{\rho }_e=1$, based on the maximum magnetic field. The solid black lines indicate a power law of $k^{-16/3}$.