Magnetic Field Amplification by a Nonlinear Electron Streaming Instability

Magnetic field amplification by relativistic streaming plasma instabilities is central to a wide variety of high-energy astrophysical environments as well as to laboratory scenarios associated with intense lasers and electron beams. We report on a new secondary nonlinear instability that arises for relativistic dilute electron beams after the saturation of the linear Weibel instability. This instability grows due to the transverse magnetic pressure associated with the beam current filaments, which cannot be quickly neutralized due to the inertia of background ions. We show that it can amplify the magnetic field strength and spatial scale by orders of magnitude, leading to large-scale plasma cavities with strong magnetic field and to very efficient conversion of the beam kinetic energy into magnetic energy. The instability growth rate, saturation level, and scale length are derived analytically and shown to be in good agreement with fully kinetic simulations.

Figure 1

2D simulation results of a dilute, ultrarelativistic electron beam propagating through an electron-ion plasma. Magnetic field profiles for cases with (a),(c) stationary ions and (b),(d) mobile ions, taken at (a),(b) saturation of the Weibel instability ($t=1750{\omega }_p^{-1}$) and (c),(d) saturation of a nonlinear electron streaming instability ($t=5250{\omega }_p^{-1}$). The energy distribution of these simulations is shown in (e), where the solid lines correspond to the mobile ion simulations in (b),(d) and dashed lines correspond to stationary ion simulations in (a),(c). The beam electron (${\epsilon }_b$), background electron (${\epsilon }_e$), and ion (${\epsilon }_i$) kinetic energy densities as well as the magnetic field energy density (${\epsilon }_B$) are all normalized to the initial beam electron kinetic energy density.

Figure 2

(a) Cross-sectional profile of the magnetic field and density structure of Fig. 1 taken by averaging the magnetic field and densities from $x=390c/{\omega }_p$ to $x=410c/{\omega }_p$. The formation of these cavities is sketched in (b),(c). (b) Fields at saturation of the Weibel instability before ions have moved. (c) Subsequent formation of background plasma cavities expanding due to magnetic pressure.

Figure 3

Comparison of analytic theory (solid lines) with 1D (circles) and 2D $y\text{−}z$ (squares) simulations for a dilute, ultrarelativistic electron beam propagating through an electron-ion background. We use slab geometry theory given that the theoretical cylindrical growth rate differs only by 15%. (a) Nonlinear instability growth rate. (b) Temporal evolution of the dominant magnetic field length scale ${\lambda }_B(t)$ for density ratio $\alpha =0.1$, beam Lorentz factor ${\gamma }_{b0}=1000$, and mass ratio $m_i/Z{m}_e=1836$, with the growth rate taken from the magnetic energy overlaid as dotted (1D) and dashed (2D) lines. (c) Saturation magnetic field length scale ${\lambda }_{B,\mathrm{sat}}$ compared to Eq. (3). (d) Saturation magnetization ${\epsilon }_B$ compared to Eq. (4). All parameter scans in (a),(c),(d) use as fixed parameters $\alpha =0.01$, ${\gamma }_{b0}=1000$, and $m_i/Z{m}_e=1836$.

Figure 4

3D simulation results of a dilute, ultrarelativistic electron beam propagating (in the $x$ direction) through an electron-ion background. The magnetic field (a) and ion density (b) are reported at $t=13000{{\omega }_p}^{-1}$ near saturation of the nonlinear instability, showing the amplification of large-scale magnetic fields and the generation of density cavities. The opacity scales linearly with the value.