Radiation reaction effects in relativistic plasmas: The electrostatic limit

We study the evolution of electrostatic plasma waves, using the relativistic Vlasov equation extended by the Landau-Lifshitz radiation reaction, accounting for the back-reaction due to the emission of single particle Larmor radiation. In particular, the Langmuir wave damping is calculated as a function of wave number, initial temperature, and initial electric field amplitude. Moreover, the background distribution function loses energy in the process, and we calculate the cooling rate as a function of initial temperature and initial wave amplitude. Finally, we investigate how the relative magnitude of wave damping and background cooling varies with the initial parameters. In particular, it is found that the relative contribution to the energy loss associated with background cooling decreases slowly with the initial wave amplitude.

Figure 1

The electric field $E$ over time for $\delta =0.005$ using the low-temperature limit.

Figure 2

The evolution of the energy loss rate in the cold limit [given by spatial integration over the right-hand side of Eq. (15)] as a function of time, for $E_0=1$, $k=0.95$, and $\delta =0.02$.

Figure 3

The spatial dependence of the electric field just before wave breaking occurs at $t=1.2$. The used parameters were $E_0=1$, $k=0.95$, and $\delta =0.02$.

Figure 4

The energy loss rate $\langle {\mathrm{\Gamma }}_c\rangle$ of a Langmuir wave for $\delta =0.02$ as a function of wave number $k$ for two values of the initial electric field amplitude, for $E_0=10$ (upper panel) and for $E_0=3$ (lower panel).

Figure 5

The variation of peak momentum before wave breaking, $P_{\mathrm{wb}}$, with wave number $k$ for $E_0=3$ and $\delta =0.02$.

Figure 6

$T_{\mathrm{rel}}$ as a function of time $t$. In the first panel we have $E_0=3$, $\delta =0.01$, and $p_{\mathrm{th}}=\left(0.1,0.3,0.5\right)$. In the second panel we have $p_{\mathrm{th}}=0.3$, $\delta =0.01$, and $E_0=\left(1,2,3\right)$.

Figure 7

$T_{\mathrm{rel}}$ for $\delta ={10}^{-5}$, $p_{\mathrm{th}}=0.1$, and $E_0=20$.

Figure 8

The dots show the damping ${\mathrm{\Gamma }}_H$ plotted as a function of initial amplitude $E_0$ using $p_{\mathrm{th}}=0.3$ and $\delta =0.005$ (upper panel). For the lower panel, the damping ${\mathrm{\Gamma }}_H$ is plotted as a function of $p_{\mathrm{th}}$ for $E_0=3$ and $\delta =0.005$. The triangles show the period-time $T_p$ for the same parameters as used for the damping as read off on the right-hand side axis.

Figure 9

${\mathrm{\Gamma }}_c$ as a function of the initial amplitude $E_0$ for $\delta =0.005$ and $p_{\mathrm{th}}=(0.1,0.3,0.6)$ and the low-temperature solution “ODE.”

Figure 10

The ratio R (cooling energy loss over total energy loss) as a function of initial field amplitude $E_0$ using $\delta =0.0005$ and $p_{\mathrm{th}}=0.3,0.5$.