Fully kinetic Biermann battery and associated generation of pressure anisotropy

The dynamical evolution of a fully kinetic, collisionless system with imposed background density and temperature gradients is investigated analytically. The temperature gradient leads to the generation of temperature anisotropy, with the temperature along the gradient becoming larger than that in the direction perpendicular to it. This causes the system to become unstable to pressure anisotropy driven instabilities, dominantly to the electron Weibel instability. When both density and temperature gradients are present and nonparallel to each other, we obtain a Biermann-like linear-in-time magnetic field growth. Accompanying particle-in-cell numerical simulations are shown to confirm our analytical results.

Figure 1

(a) Magnetic energy spectra of $B_z$ (with respect to $\left|k\right|=\sqrt{k_x^2+k_y^2\phantom{{}^2}}$) vs time from the simulation with system size $L_T/d_e=400$ ($L_T/{\lambda }_D=2000$) reported in Ref. [9]. The time when the Weibel instability begins to grow exponentially is identified with a dashed line. This estimate of the onset time ${\tau }_W$ is plotted vs system size (b) along with the predicted curve where Eq. (16) is satisfied, indicating where the Weibel growth rate exceeds that of the anisotropy predicted in Eq. (15).

Figure 2

Evolution of the averaged $B_z$ due to the perpendicular density and temperature gradients (top, green), and anisotropic temperatures $T_{xx}$ (bottom, blue) and $T_{yy}$ (middle, red), vs time along with the predicted curves at $y=0$ from Eqs. (19) and (14) (black dashed lines).