Electromagnetic Field Generation in the Downstream of Electrostatic Shocks Due to Electron Trapping

A new magnetic field generation mechanism in electrostatic shocks is found, which can produce fields with magnetic energy density as high as 0.01 of the kinetic energy density of the flows on time scales $\sim 10^4{\omega }_{pe}^{-1}$. Electron trapping during the shock formation process creates a strong temperature anisotropy in the distribution function, giving rise to the pure Weibel instability. The generated magnetic field is well confined to the downstream region of the electrostatic shock. The shock formation process is not modified, and the features of the shock front responsible for ion acceleration, which are currently probed in laser-plasma laboratory experiments, are maintained. However, such a strong magnetic field determines the particle trajectories downstream and has the potential to modify the signatures of the collisionless shock.

Figure 1

Shock formation in the 3D simulation for mass ratio $m_p/m_e=100$, $u_0=\pm 0.015$, and $\mathrm{\Delta }\gamma =0.015$: (a) Perpendicular electromagnetic field, (b) box-averaged electrostatic field $⟨E_1⟩$ (black), and 2D slice of magnetic field in (a) at $z=30c/{\omega }_{pe}$ showing the extension of the filaments. Time is $t{\omega }_{pe}=460$.

Figure 2

Temporal evolution of (a) normalized magnetic energy density and (b) thermal velocities $v_{\text{th},\parallel }$ (dash dotted), $v_{\text{th},\perp }$ (dotted), and anisotropy $A$ (solid) in a 2D simulation with $m_p/m_e=1836$, $u_0=\pm 0.1$, and $\mathrm{\Delta }\gamma =10$ measured over $\mathrm{\Delta }{x}_1=0.7c/{\omega }_{pe}$ at the center of the simulation box. The black dashed line in (a) is $\text{exp}(2{\sigma }_{m}t)$.

Figure 3

Electron phase spaces $(u_x,x)$ of left beam with $u_0=0.1$ (blue) and right beam with $u_0=-0.1$ (gray) (a),(b) and electron distribution functions measured at $x=500c/{\omega }_{pe}$ for momentum $u_x$ (solid lines) for left and right beams in blue and gray (c),(d), respectively, for the 2D simulations shown in Fig. 2 and for $t{\omega }_{pe}=100$ (a),(c) and 1000 (b),(d). The dashed lines indicate the distribution of the transverse momentum $u_y$.

Figure 4

Parameter scan of the (a) maximum energy of the electrostatic potential normalized to the upstream proton kinetic energy obtained from the simulations (the solid line represents a linear fit with $c_s/u_0$), and (b) associated maximum anisotropy $A$ (with linear fit $\sqrt{u_0}/\mathrm{\Delta }\gamma$) for 2D simulations with $m_p/m_e=1836$.

Figure 5

Proton density (blue) and normalized magnetic field energy density (orange) obtained from 2D simulations for an electrostatic shock (ES) and a Weibel-mediated shock (EM). The dashed lines indicate the shock front with the transition from the upstream (u) to the downstream region (d).