Self-Organization of a Plasma due to 3D Evolution of the Weibel Instability

The nonlinear evolution of the thermal Weibel instability is studied by using three-dimensional particle-in-cell simulations. After a fast saturation due to a reduction in the temperature anisotropy, the instability evolves to a quasistationary state which includes a single mode long wavelength helical magnetic field and a finite degree of temperature anisotropy. The nonlinear stability of this state is explained by periodic variations of the temperature anisotropy axis. At long time scales the magnetic field, wave number, and temperature anisotropy slowly evolve to the decreasing magnitudes.

Figure 1

Time evolution of the spatially averaged thermal energies $⟨T_{\perp }⟩/T_{\perp 0}$ (1), $⟨T_{\parallel }⟩/T_{\perp 0}$ (2), and magnetic field energy $W_B=⟨B^2⟩/8\pi n{T}_{\perp 0}$ (3). The inset shows the initial exponential growth of the magnetic field energy (3) and analytical prediction from the linear kinetic theory (4). The total energy (5) demonstrates the accuracy of energy conservation in 3D simulation. The time unit is the plasma wave period $2\pi /{\omega }_{pe}$.

Figure 2

Spatial variations of the magnetic field and the $k$ spectrum of magnetic field energy density ($m_i=k{L}_i/2\pi$, $L_i=15c/{\omega }_{pe}$, $i=x,y,z$) at different time moments, $t=3$ [(a),(d)], $t=10$ [(b),(e)], and $t=121$ [(c),(f)]. Magnetic field [in units of $m_{e}c{\omega }_{pe}/(2\pi e)$] and magnetic field energy spectral density are normalized to their maximum absolute values: 0.063 (a), 0.19 (b), 0.34 (c), $3.5\times {10}^{-6}$ (d), $5.2\times {10}^{-3}$ (e), and 0.052 (f). The top panels (a)–(c) show magnetic field lines crossing the $z$ axis at the center of the computational box.

Figure 3

The electron phase space for $z\simeq 2.2c/{\omega }_{pe}$ at different times.

Figure 4

Spectral density of the magnetic field energy as a function of $k_z$ at different times: $t=15$ (1), $t=50$ (2), $t=102$ (3), and $t=210$ (4) ($k_{\mathrm{m}\mathrm{i}\mathrm{n}}=2\pi {\omega }_{pe}/60c$).