Electrostatic solitons and Alfvén waves generated by streaming instability in electron-positron plasmas

Electron-positron plasmas may be present in laboratory experiments and certain astrophysical environments. Due to the inertia symmetry there have been some debates about whether the nonlinear electrostatic solitons generated easily in electron-proton plasmas may develop in electron-positron plasmas. In this study we show the formation of electrostatic solitons and electromagnetic Alfvén waves in extensive magnetized electron-positron plasma system generated by streaming instabilities based on electromagnetic particle simulations along with fluid theory analyses. In particular, the four-component beam-plasma system may lead to the formation of interlacing electron and positron solitons in the early evolution stage (say, $t<20{\omega }_p^{-1}$, where ${\omega }_p^{}$ is the plasma frequency) and large-amplitude Alfvén waves in the later phase (say, $t>150{\omega }_p^{-1}$). The magnetic-field perturbations are generated by the beam and firehose-type instabilities associated with temperature anisotropy of ${\mathrm{T}}_{\parallel }>{\mathrm{T}}_{\perp }$ resulting from the parallel plasma heating by the electrostatic instability. It is shown that the growth rates and the dominant wavelengths of both electrostatic and electromagnetic instabilities are consistent with the fluid theory. The coexistence of electrostatic solitons and electromagnetic Alfvén waves with significant magnetic field fluctuations in the same system is a unique feature of electron-positron plasmas and the unified theory behind the formation mechanisms is well addressed in the paper.

Figure 1

The imaginary part of the wave frequency ${\omega }_i$ derived from the linear fluid theory as functions of wave number $k$ and $u_0$ for (a) the longitudinal mode and (b) the transverse mode.

Figure 2

The imaginary part of the wave frequency ${\omega }_i$ derived from the linear fluid theory as functions of wave number $k$ and ${\omega }_c$ for the transverse mode.

Figure 3

(a), (b) Time evolution of the logarithm of the longitudinal mode electric field $E_x$ and (c) the transverse mode electric field $E_y$ with ${\omega }_c/{\omega }_p=0.0$ (circle symbols), $0.1$ (cross symbols), $0.2$ (plus symbols), $0.3$ (star symbols), $0.4$ (square symbols), $0.5$ (diamond symbols), and $1.0$ (triangle symbols) based on the simulation results, respectively. The slope of the corresponding straight lines in each panels are the maximum growth rates of the unstable mode derived from the linear theory.

Figure 4

Temporal and spatial evolution of (a) $E_x(x,t)$, (b) $E_y(x,t)$, and (c) $B_y(x,t)$ and the corresponding time evolution of the wave number for (d) $E_x(k,t)$, (e) $E_y(k,t)$, and (f) $B_y(k,t)$ for the case of ${\omega }_c/{\omega }_p=0.4$.

Figure 5

The magnetic field hodograms at (a) $t=168{\omega }_p^{-1}$ and (b) $t=256{\omega }_p^{-1}$ for the case of ${\omega }_c/{\omega }_p=0.4$. The circle symbol in each panels denotes the value of $(B_y,B_z)$ at $x=0$.

Figure 6

Phase space of the particles for the cases of ${\omega }_c/{\omega }_p=0.0$ at (a) $t=64{\omega }_p^{-1}$ and (d) $t=168{\omega }_p^{-1}$, ${\omega }_c/{\omega }_p=0.4$ at (b) $t=64{\omega }_p^{-1}$ and (e) $t=168{\omega }_p^{-1}$, and ${\omega }_c/{\omega }_p=1.0$ at (c) $t=64{\omega }_p^{-1}$ and (f) $t=168{\omega }_p^{-1}$. The blue, black, red, and green dots denote background positrons, beam positrons, background electrons, and beam electrons, respectively.

Figure 7

Time evolution of the logarithm of the electric potential drop $\mathrm{\Delta }\varphi$ for the simulation cases with ${\omega }_c/{\omega }_p=0.0$ (red curve), $0.4$ (cyan curve), and $1.0$ (black curve).

Figure 8

Velocity distributions of positrons at $t=0$ (solid curve), $t=64{\omega }_p^{-1}$ (dashed curve), and $t=256{\omega }_p^{-1}$ (dot-dashed curve) for the cases of (a) ${\omega }_c/{\omega }_p=0.0$, (b) ${\omega }_c/{\omega }_p=0.4$, and (c) ${\omega }_c/{\omega }_p=1.0$. The panels from top to bottom denote the velocity distributions $f\left(v_x\right)$, $f\left(v_y\right)$, and $f\left(v_z\right)$, respectively.

Figure 9

Time evolution of plasma beta $\beta$ for background positrons (top panels) and beam positrons (middle panels) and the quantity $\delta B^2/B_0^2$ (bottom panels) for the cases of (a) ${\omega }_c/{\omega }_p=0.2$, (b) ${\omega }_c/{\omega }_p=0.4$, and (c) ${\omega }_c/{\omega }_p=1.0$. The solid, dashed, and dot-dashed curves in top and middle panels denote $({\beta }_{\parallel }-{\beta }_{\perp })$, ${\beta }_{\parallel }$, and ${\beta }_{\perp }$, respectively.