Physics of relativistic collisionless shocks. II. Dynamics of the background plasma

In this second paper of a series, we discuss the dynamics of a plasma entering the precursor of an unmagnetized, relativistic collisionless pair shock. We discuss how this background plasma is decelerated and heated through its interaction with a microturbulence that results from the growth of a current filamentation instability in the shock precursor. We make use, in particular, of the reference frame ${\mathcal{R}}_{\mathrm{w}}$ in which the turbulence is mostly magnetic. This frame moves at relativistic velocities towards the shock front at rest, decelerating gradually from the far to the near precursor. In a first part, we construct a fluid model to derive the deceleration law of the background plasma expected from the scattering of suprathermal particles off the microturbulence. This law leads to the relationship ${\gamma }_{\mathrm{p}}\sim {\xi }_{\mathrm{b}}^{-1/2}$ between the background plasma Lorentz factor ${\gamma }_{\mathrm{p}}$ and the normalized pressure of the beam ${\xi }_{\mathrm{b}}$; it is found to match nicely the spatial profiles observed in large-scale 2D3V particle-in-cell simulations. In a second part, we model the dynamics of the background plasma at the kinetic level, incorporating the inertial effects associated with the deceleration of ${\mathcal{R}}_{\mathrm{w}}$ into a Vlasov-Fokker-Planck equation for pitch-angle diffusion. We show how the effective gravity in ${\mathcal{R}}_{\mathrm{w}}$ drives the background plasma particles through friction on the microturbulence, leading to efficient plasma heating. Finally, we compare a Monte Carlo simulation of our model with dedicated PIC simulations and conclude that it can satisfactorily reproduce both the heating and the deceleration of the background plasma in the shock precursor, thereby providing a successful one-dimensional description of the shock transition at the microscopic level.

Figure 1

In black, the Lorentz factor of the background plasma, ${\gamma }_{\mathrm{p}|\mathrm{d}}$, vs distance to the shock $x_{|d}$ (in units of $c/{\omega }_{\mathrm{p}}$), extracted from PIC simulations of a pair shock of Lorentz factor ${\gamma }_{\infty |d}=10$ [(a)] and ${\gamma }_{\infty |d}=100$ [(b)]. The law ${\gamma }_{\mathrm{p}|\mathrm{d}}=1.2{\xi }_{\mathrm{b}}^{-1/2}$ is overplotted in blue, using as input the profile ${\xi }_{\mathrm{b}}\left(x\right)$ extracted from the same PIC simulations and the same ad hoc numerical prefactor 1.2 in both panels. The law ${\gamma }_{\mathrm{p}|\mathrm{d}}\sim {\xi }_{\mathrm{b}}^{-1/2}$ is that predicted by Eq. (10) in the relativistic decelerating regime ($1\ll {\gamma }_{\mathrm{p}}\ll {\gamma }_{\infty }$). The numerical data is light colored in regions where it cannot be measured accurately.

Figure 2

In red, the normalized pressure of suprathermal particles ${\xi }_{\mathrm{b}}$ vs the distance to the shock $x_{|d}$ (in units of $c/{\omega }_{\mathrm{p}}$), as extracted from PIC simulations of a pair shock of Lorentz factor ${\gamma }_{\infty |d}=10$ [(a)] and ${\gamma }_{\infty |d}=100$ [(b)]. The law ${\left(0.15\right)}^{-1}{p}_{\mathrm{p}}/F_{\infty }$ is overplotted in gray, using the same ad hoc numerical prefactor ${\left(0.15\right)}^{-1}$ in both panels. This scaling $p_{\mathrm{p}}\propto F_{\infty }{\xi }_{\mathrm{b}}$ is that predicted by Eq. (12) in the relativistic decelerating regime ($1\ll {\gamma }_{\mathrm{p}}\ll {\gamma }_{\infty }$). As before, data is light colored in regions where it cannot be measured accurately.

Figure 3

Evolution of the microturbulence strength parameter ${\epsilon }_B$ in the near precursor, as a function of distance to the shock $x_{|d}$ (in units of $c/{\omega }_{\mathrm{p}}$), extracted from PIC simulations of a pair shock of Lorentz factor ${\gamma }_{\infty |d}=10$ [(a)] and ${\gamma }_{\infty |d}=100$ [(b)], in black. The MHD-like compression law ${\epsilon }_B\propto 1/{\beta }_{\mathrm{w}}^2$, with ${\beta }_{\mathrm{w}}$ extracted from the same PIC simulations, is overplotted in dashed green, for comparison. The proportionality factor is chosen to match the ${\epsilon }_B\left(x_{|d}\right)$ curve in the precursor at distances $x_{|d}\gtrsim 100c/{\omega }_{\mathrm{p}}$ in order to remove its possible spatial dependence. This choice corresponds to ${\epsilon }_B=\left(1.2\times {10}^{-2}-1.6\times {10}^{-5}{x}_{|\mathrm{d}}\right){\beta }_{\mathrm{w}}^{-2}$ in (a) and ${\epsilon }_B=0.026{\beta }_{\mathrm{w}}^{-2}$ in (b).

Figure 4

Profiles of the background plasma four-velocity $|u_{\mathrm{p}|\mathrm{d}}|$ in the simulation frame ${\mathcal{R}}_d$ as a function of distance to shock $x_{|d}$ (in units of $c/{\omega }_{\mathrm{p}}$). In red, the results of the Monte Carlo computation, for different values of the scattering frequency (light red: ${\nu }_{|\mathrm{w}}={\omega }_{\mathrm{p}}$; medium red: ${\nu }_{|\mathrm{w}}=0.1{\omega }_{\mathrm{p}}$; and dark red: ${\nu }_{|\mathrm{w}}=0.01{\omega }_{\mathrm{p}}$); in black, the value measured in the PIC simulation. We stress that the values extracted from the PIC simulation at $x\lesssim 10c/{\omega }_{\mathrm{p}}$ are inaccurate because the distinction between background plasma particles and shock-heated particles becomes difficult. The dotted gray line shows the four-velocity $|u_{\mathrm{w}}|$, inferred from the PIC simulation, and used as input in the Monte Carlo computation. (a): ${\gamma }_{\infty |d}=10$ corresponding to ${\gamma }_{\infty }=17$; (b): ${\gamma }_{\infty |d}=100$ corresponding to ${\gamma }_{\infty }=173$.

Figure 5

Same as Fig. 4 for the profiles of the temperature $T_{\mathrm{p}}$ as a function of distance to shock $x_{|d}$ (in units of $c/{\omega }_{\mathrm{p}}$), in the simulation frame. We stress that the values extracted from the PIC simulation at $x\lesssim 10c/{\omega }_{\mathrm{p}}$ are inaccurate because the distinction between background plasma particles and shock-heated particles becomes difficult. The dashed gray line indicates the expected final values of $T_{\mathrm{p}}$ corresponding to the fluid shock-crossing conditions for a relativistic unmagnetized shock in two dimensions.

Figure 6

Same as Fig. 4, but now representing the trajectory of the background plasma in the plane $(T_{\mathrm{p}},|u_{\mathrm{p}}\left|\right)$. We stress that the values extracted from the PIC simulation at large $T_{\mathrm{p}}$ are inaccurate because the distinction between background plasma particles and shock-heated particles becomes difficult. The dashed gray line indicates the expected final values of $T_{\mathrm{p}}$ corresponding to the fluid shock-crossing conditions for a relativistic unmagnetized shock in two dimensions (the corresponding value of $u_{\mathrm{p}}^x$ is zero in the simulation frame). The dotted line shows the adiabatic law of compression for a nonrelativistic gas, $T_{\mathrm{p}}\propto {\left|u_{\mathrm{p}}^x\right|}^{-2/3}$.

Figure 7

Spatial profile of ${\gamma }_{\mathrm{p}|\mathrm{d}}{\beta }_{\mathrm{p}|\mathrm{d}}{n}_{\mathrm{p}}/\left({\gamma }_{\infty |d}{\beta }_{\infty |d}{n}_{\infty }\right)$ extracted from our two reference PIC simulations, vs the distance to the shock $x_{|d}$ (in units of $c/{\omega }_{\mathrm{p}}$). This quantity decreases towards the shock as a result of a finite scattering frequency, because once the momentum of a particle has changed sign, this particle is no longer characterized in this postprocessing as a background plasma particle. This figure shows that about half of the particles are scattered at least once over the crossing of the precursor.