2 Theoretical Model

We consider a 1-D collisionless, four-component plasma consisting of cool inertial background electrons (at temperature $T_{c}\neq0$), mobile cool positrons (or electron holes; at temperature $T_{p}\neq0$), inertialess hot suprathermal electrons modeled by a $\kappa$-distribution (at temperature $T_{h}\gg T_{c},T_{p}$), and uniformly distributed stationary ions.

The cool electrons and positrons are governed by the following fluid equations:

$$\frac{\partial n_{c}}{\partial t}+\frac{\partial(n_{c}u_{c})}{\partial x}=0,$$ (1)

$$\frac{\partial u_{c}}{\partial t}+u_{c}\frac{\partial u_{c}}{\par... ...artial\phi}{\partial x}-\frac{1}{m_{e}n_{c}} \frac{\partial p_{c}}{\partial x},$$ (2)

$$\frac{\partial p_{c}}{\partial t}+u_{c}\frac{\partial p_{c}}{\partial x}+\gamma p_{c}\frac{\partial u_{c}}{\partial x}=0,$$ (3)

$$\frac{\partial n_{p}}{\partial t}+\frac{\partial(n_{p}u_{p})}{\partial x}=0,$$ (4)

$$\frac{\partial u_{p}}{\partial t}+u_{p}\frac{\partial u_{p}}{\par... ...artial\phi}{\partial x}-\frac{1}{m_{p}n_{p}} \frac{\partial p_{p}}{\partial x},$$ (5)

$$\frac{\partial p_{p}}{\partial t}+u_{p}\frac{\partial p_{p}}{\partial x}+\gamma p_{p}\frac{\partial u_{p}}{\partial x}=0,$$ (6)

where $n$, $u$ and $p$ are the number density, the velocity and the pressure of the cool electrons and positrons (denoted by indices `c' and `p', respectively), $\phi$ is the electrostatic wave potential, $e$ the elementary charge, $m_{e}$ the electron mass, $m_{p}$ the positron mass, and $\gamma=(f+2)/f$ denotes the specific heat ratio for $f$ degrees of freedom. For the adiabatic cool electrons and positrons in one-dimensional ($f=1$), we get $\gamma=3$. Through this paper, we assume that $m_{e}=m_{p}$.

Following Eq. 1 in Ref. [31], the $\kappa$-distribution expression is obtained for the number density of the hot suprathermal electrons:

$$n_{h}(\phi)=n_{h,0}\left[ 1-\frac{e\phi}{k_{B}T_{h}(\kappa-\tfrac{3}{2} )}\right] ^{-\kappa+1/2}\,,$$ (7)

where $n_{h,0}$ and $T_{h}$ are the equilibrium number density and the temperature of the hot electrons, respectively, $k_{B}$ the Boltzmann constant, and the spectral index $\kappa$ measures the deviation from thermal equilibrium. For reality of the characteristic modified thermal velocity, $\left[ (2\kappa-3)k_{B}T_{h}/\kappa m_{e}\right] ^{1/2}$, the spectral index must take $\kappa>3/2$. The suprathermality is measured by the spectral index $\kappa$, describing how it deviates from a Maxwellian distribution, i.e., low values of $\kappa$ are associated with a significant suprathermality; on the other hand, a Maxwellian distribution is recovered in the limit $\kappa\rightarrow\infty$.

The ions are assumed to be immobile in a uniform state, i.e., $n_{i}=n_{i,0}=$ const. at all times, where $n_{i,0}$ is the undisturbed ion density. The plasma is quasi-neutral at equilibrium, so $Zn_{i,0}+n_{p,0}=n_{c,0}+n_{h,0}$, that implies

$$Z{n_{i,0}}/{n_{c,0}}=1+\alpha-\beta,$$ (8)

where we have defined the hot-to-cool electron density ratio as $\alpha =n_{h,0}/n_{c,0}$, and the positron-to-cool electron density ratio as $\beta=n_{p,0}/n_{c,0}$, while $n_{c,0}$ and $n_{p,0}$ are the equilibrium number densities of the cool electrons and positrons, respectively. Electrostatic waves are weakly damped in the range of $0.2\lesssim n_{c,0}/(n_{c,0}+n_{h,0})\lesssim0.8$ [26,27,28,29], so $0.25\leqslant \alpha\leqslant4$. This region may permit the propagation of nonlinear electrostatic structures. As the positron fraction has been measured to be $\phi(\mathrm{e}^{+})/(\phi(\mathrm{e}^{+})+\phi(\mathrm{e}^{-}))\lesssim 0.1$ in low energy solar wind observations [1,2,3,5] and $n(\mathrm{e}^{+})/(n(\mathrm{e}^{+})+n(\mathrm{e}^{-}))\sim0.05$-0.1 in some laser-plasma experiences [14,15], we assume $\beta\lesssim0.06$.

All four components are coupled via the Poisson's equation:

$$\frac{\partial^{2}\phi}{\partial x^{2}}=-\frac{e}{\varepsilon_{0}}\left( Zn_{i}+n_{p}-n_{c}-n_{h}\right) ,$$ (9)

where $\varepsilon_{0}$ is the permittivity constant.

Scaling by appropriate quantities, we arrive at a fluid system of our model in a dimensionless form for the cool electrons and the positrons, respectively:

$$\frac{\partial n}{\partial t}+\frac{\partial(nu)}{\partial x} =0,$$ (10)

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\frac {\partial\phi}{\partial x}-\frac{\sigma}{n}\frac{\partial p}{\partial x},$$ (11)

$$\frac{\partial p}{\partial t}+u\frac{\partial p}{\partial x}+3p\frac {\partial u}{\partial x}=0,$$ (12)

$$\frac{\partial n_{p}}{\partial t}+\frac{\partial(n_{p}u_{p})}{\partial x}=0,$$ (13)

$$\frac{\partial u_{p}}{\partial t}+u_{p}\frac{\partial u_{p}}{\par... ...artial\phi}{\partial x}-\frac{\theta}{n_{p}}\frac{\partial p_{p} }{\partial x},$$ (14)

$$\frac{\partial p_{p}}{\partial t}+u_{p}\frac{\partial p_{p}}{\partial x}+3p_{p}\frac{\partial u_{p}}{\partial x}=0.$$ (15)

The dimensionless Poisson's equation takes the following form:

$$\frac{\partial^{2}\phi}{\partial x^{2}} = -\left( 1+\alpha-\beta\right) +n-\beta n_{p}$$

$$+\alpha\left( 1-\frac{\phi}{\kappa-\tfrac{3}{2}}\right)^{-\kappa+1/2},$$ (16)

where $n$ and $n_{p}$ denote the fluid density variables of the cool electrons and positrons normalized with respect to $n_{c,0}$ and $n_{p,0}$, respectively, $u$ and $u_{p}$ the velocity variables of the cool electrons and positrons scaled by the hot electron thermal speed $c_{th}=\left( k_{B} T_{h}/m_{e}\right) ^{1/2}$, $p$ and $p_{p}$ the pressure variables of the cool electrons and positrons normalized with respect to $n_{c,0}k_{B}T_{c}$ and $n_{p,0}k_{B}T_{p}$, respectively, and the wave potential $\phi$ by $k_{B} T_{h}/e$, time and space scaled by the plasma period $\omega_{pc}^{-1}=\left( n_{c,0}e^{2}/\varepsilon_{0}m_{e}\right) ^{-1/2}$ and the characteristic length $\lambda_{0}=\left( \varepsilon_{0}k_{B}T_{h}/n_{c,0}e^{2}\right) ^{1/2}$, respectively. We have defined the cool-to-hot electron temperature ratio as $\sigma=T_{c}/T_{h}$, and the positron-to-hot electron temperature ratio as $\theta=T_{p}/T_{h}$. Landau damping is minimized if $\sigma=T_{c}/T_{h}\ll0.1$ [26,27,28], and the same for the cool positrons, i.e., $\theta=T_{p}/T_{h}\ll0.1$ (see Ref. [40]), and typically $T_{p}/T_{c} \sim0.5$ in some laser-plasma experiences [14].