On the double-adiabatic equations in the relativistic regime

Abstract

We revisit the double adiabatic evolution equations and extend them to the relativistic and ultrarelativistic regimes. We analytically solve the relativistic, time-dependent drift kinetic equation for a homogeneous, magnetized, collisionless plasma and obtain a solution explicitly dependent on the magnetic field and density variations. In the case of an initial relativistic Maxwellian distribution, a natural extension to an anisotropic Maxwell-Jüttner is obtained. We calculate the moments of this time-dependent solution and obtain analytical expressions for the evolution of the perpendicular and parallel pressures in the ultrarelativistic case. We numerically solve the moment equations in the relativistic case and obtain general expressions for the double-adiabatic equations in this regime. We confirm our results using fully kinetic particle-in-cell simulations of shearing and compressing boxes. Our findings can be readily applied to relativistic species including cosmic-rays and electron-positron pairs, present in astrophysical plasmas like pulsar wind nebulae, astrophysical jets, black hole accretion flows, and Van Allen radiation belts.

Figures (8)

• Figure 1: Panel $a$: The evolution of the ion perpendicular pressure for run Shb0.5m8d0.2wcis800 (open blue circles), for initial $k_BT_i^{\text{init}}/m_ic^2=0.2$. The solid blue line shows the evolution of $P_{\perp}$ according to eqn. \ref{['eq:Pperp_MJ']}, numerically integrated. Panel $b$: The evolution of the ion parallel pressure for run Shb0.5m8d0.2wcis800 (open orange circles). The solid orange line shows the evolution of $P_{\parallel}$ according to eqn. \ref{['eq:Ppar_MJ']}, numerically integrated. Panel $c$: The evolution of the electron perpendicular pressure for run Shb0.5m8d0.2wcis800 (open blue circles), for initial $k_BT_e^{\text{init}}/m_ec^2=1.6$ (given a mass ratio $m_i/m_e = 8$). The solid blue line shows the evolution of $P_{\perp}$ according to eqn. \ref{['eq:Pperp_MJ']}, numerically integrated. Panel $d$: The evolution of the electron parallel pressure for run Shb0.5m8d0.2wcis800 (open orange circles). The solid orange line shows the evolution of $P_{\parallel}$ according to eqn. \ref{['eq:Ppar_MJ']}, numerically integrated. In panels $a$ and $c$, the nonrelativistic CGL evolution for $P_{\perp}$ is shown in dashed gray line. In panels $b$ and $d$, the nonrelativistic CGL evolution for $P_{\parallel}$ is shown in dashed gray line.
• Figure 2: Panel $a$: The evolution of the ion perpendicular pressure for run Shb0.5m1836d30wcis3200 (open black circles), for initial $k_BT_i^{\text{init}}/m_ic^2=30$. The dashed blue line shows the evolution of $P_{\perp}$ according to eqn. \ref{['eq:Pperp_UR']}. The solid orange line shows the evolution of $P_{\perp}$ according to eqn. \ref{['eq:Pperp_MJ']}, numerically integrated. Panel $b$: The evolution of the ion parallel pressure for run Shb0.5m1836d30wcis3200 (open black circles). The dashed magenta line shows the evolution of $P_{\parallel}$ according to eqn. \ref{['eq:Ppar_UR']}. The solid green line shows the evolution of $P_{\parallel}$ according to eqn. \ref{['eq:Ppar_MJ']}, numerically integrated. Panel $c$: The evolution of the electron perpendicular pressure for run Shb0.5m1836d30wcis3200 (open black circles), for initial $k_BT_e^{\text{init}}/m_ec^2=55080$. The dashed blue line shows the evolution of $P_{\perp}$ according to eqn. \ref{['eq:Pperp_UR']}. The solid orange line shows the evolution of $P_{\perp}$ according to eqn. \ref{['eq:Pperp_MJ']}, numerically integrated. Panel $d$: The evolution of the electron parallel pressure for run Shb0.5m1836d30wcis3200 (open black circles). The dashed magenta line shows the evolution of $P_{\parallel}$ according to eqn. \ref{['eq:Ppar_UR']}. The solid green line shows the evolution of $P_{\parallel}$ according to eqn. \ref{['eq:Ppar_MJ']}, numerically integrated. In panels $a$ and $c$, the nonrelativistic CGL evolution for $P_{\perp}$ is shown in dashed gray line. In panels $b$ and $d$, the nonrelativistic CGL evolution for $P_{\parallel}$ is shown in dashed gray line.
• Figure 3: Panel $a$: The evolution of the ion perpendicular temperature for run Compb0.5m8d0.2wcis800 (open blue circles), for initial $k_BT_i^{\text{init}}/m_ic^2=0.2$. The solid blue line shows the evolution of $T_{\perp}$ according to eqn. \ref{['eq:Pperp_Compressing_MJ']}, numerically integrated. Panel $b$: The evolution of the ion parallel temperature for run Compb0.5m8d0.2wcis800 (open orange circles). The solid orange line shows the evolution of $T_{\parallel}$ according to eqn. \ref{['eq:Ppar_Compressing_MJ']}, numerically integrated. Panel $c$: The evolution of the electron perpendicular temperature for run Compb0.5m8d0.2wcis800 (open blue circles), for initial $k_BT_e^{\text{init}}/m_ec^2=1.6$. The solid blue line shows the evolution of $T_{\perp}$ according to eqn. \ref{['eq:Pperp_Compressing_MJ']}, numerically integrated. Panel $d$: The evolution of the electron parallel temperature for run Compb0.5m8d0.2wcis800 (open orange circles). The solid orange line shows the evolution of $T_{\parallel}$ according to eqn. \ref{['eq:Ppar_Compressing_MJ']}, numerically integrated. In panels $a$ and $c$ the nonrelativistic CGL evolution for $P_{\perp}$ is shown in dashed gray line for a compressing motion. In panels $b$ and $d$, the nonrelativistic CGL evolution for $P_{\parallel}$ is shown in dashed gray line for a compressing motion.
• Figure 4: Panel $a$: The evolution of the ion perpendicular temperature for run Compb0.5m1836d30wcis3200 (open blue circles), for initial $k_BT_i^{\text{init}}/m_ic^2=30$. The solid blue line shows the evolution of $T_{\perp}$ according to eqn. \ref{['eq:Pperp_compressing_UR']}. Panel $b$: The evolution of the ion parallel temperature for run Compb0.5m1836d30wcis3200 (open orange circles). The solid orange line shows the evolution of $T_{\parallel}$ according to eqn. \ref{['eq:Ppar_compressing_UR']}. In panels $a$ and $c$ the nonrelativistic CGL evolution for $P_{\perp}$ is shown in dashed gray line for a compressing motion. In panels $b$ and $d$, the nonrelativistic CGL evolution for $P_{\parallel}$ is shown in dashed gray line for a compressing motion.
• Figure 5: Panel $a$: The histograms of the ion perpendicular momentum for run Shb0.5m8d0.2wcis800 at $t\cdot s = 0$ (light blue bars) and $t\cdot s = 1$ (yellow bars). The marginal distribution $f_{\text{aMJ}}(p_{\perp},t)$, eqn. \ref{['eq:fperp_marginal']}, is shown at $t\cdot s = 0$ (solid blue line) and $t\cdot s = 1$ (solid red line), for $\theta=0.2$. Panel $b$: The histograms of the ion parallel momentum for run Shb0.5m8d0.2wcis800 at $t\cdot s = 0$ (light blue bars) and $t\cdot s = 1$ (yellow bars). The marginal distribution $f_{\text{aMJ}}(p_{\parallel},t)$, eqn. \ref{['eq:fpar_marginal']}, is shown at $t\cdot s = 0$ (solid blue line) and $t\cdot s = 1$ (solid red line), for $\theta=0.2$. Panels $c$ and $d$ show the same quantities as panels $a$ and $b$, respectively, but for run Shb0.5m8d30wcis3200 (i.e. $\theta=30$).