Physics of Weibel-Mediated Relativistic Collisionless Shocks

We develop a comprehensive theoretical model of relativistic collisionless pair shocks mediated by the current filamentation instability. We notably characterize the noninertial frame in which this instability is of a mostly magnetic nature, and describe at a microscopic level the deceleration and heating of the incoming background plasma through its collisionless interaction with the electromagnetic turbulence. Our model compares well to large-scale 2D3V particle-in-cell simulations, and provides an important touchstone for the phenomenology of such plasma systems.

Figure 1

Downstream-simulation frame 1D spatial profiles of the background plasma Lorentz factor ${\gamma }_{p|d}$, of its proper temperature $T_p$ (units $m_e{c}^2/k_B$), of the suprathermal beam pressure ${\xi }_b$ and of the microturbulence energy density ${\epsilon }_B$ for a 2D3V PIC simulation of initial Lorentz factor ${\gamma }_{\infty }=173$ (${\gamma }_{\infty |d}=100$ in the simulation frame). Data are light colored in regions where they cannot be measured accurately.

Figure 2

Top panel: 4 velocities $|u_{p|d}|$ and $|u_{w|d}|$ measured in the PIC simulation with ${\gamma }_{\infty |d}=100$. Bottom panel: relative 3-velocity ${\beta }_{w|p}$ between ${\mathcal{R}}_w$ and the background plasma compared to our two theoretical models, and suprathermal beam pressure ${\xi }_b$. Data are light colored in regions where they cannot be measured accurately: at $x\gtrsim 300c/{\omega }_p$, where $|{\beta }_{w|d}|\simeq 1$, the estimate of ${\beta }_w$ carries a numerical error amplified by $\sim {\gamma }_{w|d}^2$.

Figure 3

Spatial profiles of ${\gamma }_{p|d}$ extracted from PIC simulations with ${\gamma }_{\infty |d}=100$ (top) and ${\gamma }_{\infty |d}=10$ (bottom) compared to the fluid deceleration law $\sim {\xi }_b^{-1/2}$ (1.5 an ad hoc factor); the data for ${\gamma }_{\infty |d}=10$ have been offset in $x_{|d}$ by $1000c/{\omega }_p$ for clarity.

Figure 4

Trajectory of the background plasma in the temperature $T_p$ and 4-velocity $|u_{p|d}|$ plane, as measured in our reference PIC simulations (black) with ${\gamma }_{\infty |d}=100$ (top) and ${\gamma }_{\infty |d}=10$ (bottom), and as evaluated through numerical Monte Carlo integration for ${\nu }_{|w}=0.01{\omega }_p$ (red). Dashed lines indicate the expected temperatures corresponding to the fluid shock jump conditions; dotted lines show the adiabatic compression law $T_p\propto |u_p{|}^{-2/3}$.

Figure 5

Zoom on the peak of ${\epsilon }_B$ at the shock, compared to the compression law ${\epsilon }_B\propto {\beta }_w^{-2}$ (dashed green), where ${\beta }_w$ is extracted from the PIC simulation through $⟨\delta E_y^2{⟩}^{1/2}/⟨\delta B_z^2{⟩}^{1/2}$, then smoothed. The proportionality factors are ad hoc linear fits to the scaling of ${\epsilon }_B$ at large $x\gtrsim 100c/{\omega }_p$.