Physics of relativistic collisionless shocks: The scattering-center frame

In this first paper of a series dedicated to the microphysics of unmagnetized, relativistic collisionless pair shocks, we discuss the physics of the Weibel-type transverse current filamentation instability that develops in the shock precursor, through the interaction of an ultrarelativistic suprathermal particle beam with the background plasma. We introduce in particular the notion of the “Weibel frame,” or scattering center frame, in which the microturbulence is of mostly magnetic nature. We calculate the properties of this frame, using first a kinetic formulation of the linear phase of the instability, relying on Maxwell-Jüttner distribution functions, then using a quasistatic model of the nonlinear stage of the instability. Both methods show that (i) the Weibel frame moves at subrelativistic velocities relative to the background plasma, therefore at relativistic velocities relative to the shock front; (ii) the velocity of the Weibel frame relative to the background plasma scales with ${\xi }_{\mathrm{b}}$, i.e., the pressure of the suprathermal particle beam in units of the momentum flux density incoming into the shock; and (iii) the Weibel frame moves slightly less fast than the background plasma relative to the shock front. Our theoretical results are found to be in satisfactory agreement with the measurements carried out in dedicated large-scale 2D3V particle-in-cell simulations.

Figure 1

Spatial profiles of the main hydrodynamic quantities characterizing the background plasma and the beam of suprathermal particles in the shock precursor, as a function of distance to the shock front $x_{|\mathrm{d}}$ in units of $c/{\omega }_{\mathrm{p}}$ (simulation frame). (a–c) Profiles extracted from a PIC simulation of shock Lorentz factor ${\gamma }_{\infty }=17$ (corresponding to a relative upstream-downstream Lorentz factor ${\gamma }_{\infty |\mathrm{d}}=10$) at time $t=3600{\omega }_{\mathrm{p}}^{-1}$. (d–f) From a PIC simulation of shock Lorentz factor ${\gamma }_{\infty }=173$ (${\gamma }_{\infty |\mathrm{d}}=100$) at time $t=6900{\omega }_{\mathrm{p}}^{-1}$. (a, d) Lorentz factor of the background plasma in the simulation (downstream) rest frame (dark gray) of the beam in the simulation frame (light red) and the relative Lorentz factor between the beam and the background plasma (light blue). (b, e) Proper temperature of the background plasma (dark gray) and of the beam (light red). (c, f) Proper density of the background plasma (dark gray) and of the beam (light red). The asymmetry of the beam-plasma configuration is manifest in those figures.

Figure 2

Spatial profiles of normalized electromagnetic field energy densities in transverse magnetic field $\delta B_z$ (black), transverse electric field $\delta E_y$ (blue), and longitudinal electric field $\delta E_x$ (red), as a function of distance to the shock (units of $c/{\omega }_{\mathrm{p}}$). (a, b) Extracted from simulation ${\gamma }_{\infty }=17$ (simulation frame ${\gamma }_{\infty |\mathrm{d}}=10$). (c, d) Simulation ${\gamma }_{\infty }=173$ (simulation frame ${\gamma }_{\infty |\mathrm{d}}=100$). Panels (a) and (c) show the energy densities measured in the simulation frame; panels (b) and (d) show the corresponding energy densities deboosted to the Weibel frame ${\mathcal{R}}_{\mathrm{w}}$ (where $\delta E_{y|\mathrm{w}}$ vanishes by definition). The velocity of ${\mathcal{R}}_{\mathrm{w}}$ is measured in the simulation as ${\beta }_{\mathrm{w}|\mathrm{d}}={\langle \delta E_y^2\rangle }^{1/2}/{\langle \delta B_z^2\rangle }^{1/2}$, where the average is taken over the transverse dimension of the simulation box. See text for details.

Figure 3

Parameters ${\chi }_{\mathrm{b}}$ and ${\tilde{\chi }}_{\mathrm{p}}$ of the beam and the background plasma as a function of distance to the shock front (units of $c/{\omega }_{\mathrm{p}}$), extracted from PIC simulations with ${\gamma }_{\infty }=17$ (a) and ${\gamma }_{\infty }=173$ (b). We recall that the hydrodynamic (kinetic) regime for the beam and/or the plasma corresponds to ${\chi }_{\mathrm{b}}\gg 1$ (${\chi }_{\mathrm{b}}\ll 1$) and/or ${\tilde{\chi }}_{\mathrm{p}}\gg 1$ (${\tilde{\chi }}_{\mathrm{p}}\ll 1$), respectively. This figure thus indicates that the plasma should be described in the kinetic regime, and that the beam regime is intermediate.

Figure 4

Theoretical estimates of the relative velocity between the Weibel frame ${\mathcal{R}}_{\mathrm{w}}$ and the background plasma, ${\beta }_{\mathrm{w}|\mathrm{p}}$, as a function of distance to the shock front (units of $c/{\omega }_{\mathrm{p}}$), compared with the velocity extracted from our reference PIC simulations through the ratio ${〈\delta E_y^2〉}^{1/2}/{〈\delta B_z^2〉}^{1/2}$ (solid green); where this value cannot be measured accurately from the simulation, the data are colored in light green (see text for details). The red circle symbol/dashed curve uses the numerical solution to the general kinetic dispersion relation (28) to derive ${\zeta }_{\max }$, while the purple square/dashed curve plots the analytic approximations derived in Sec. 3c. (a) ${\gamma }_{\infty }=17$ (${\gamma }_{\infty |\mathrm{d}}=10$). (b) ${\gamma }_{\infty }=173$ (${\gamma }_{\infty |\mathrm{d}}=100$).

Figure 5

Evolution of the nonlinearity parameter of the background plasma ${\mathrm{\Xi }}_{\mathrm{p}|\mathrm{w}}$ as a function of distance to the shock (units of $c/{\omega }_{\mathrm{p}}$). (a) ${\gamma }_{\infty |\mathrm{d}}=10$ at time $t=3600{\omega }_{\mathrm{p}}^{-1}$. (b) ${\gamma }_{\infty |\mathrm{d}}=100$ at time $t=6900{\omega }_{\mathrm{p}}^{-1}$.

Figure 6

Similar to Fig. 4, for our theoretical estimate of ${\beta }_{\mathrm{w}|\mathrm{p}}$ obtained assuming nonlinear equilibrium filaments, as developed in Sec. 4. The purple diamond symbol/dashed curve uses (86) to compute the ${\beta }_{\mathrm{w}|\mathrm{p}}$. (a) ${\gamma }_{\infty }=17$ (${\gamma }_{\infty |\mathrm{d}}=10$). (b) ${\gamma }_{\infty }=173$ (${\gamma }_{\infty |\mathrm{d}}=100$).