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:

  $\displaystyle \frac{\partial n_{c}}{\partial t}+\frac{\partial(n_{c}u_{c})}{\partial x}=0,$ (1)
  $\displaystyle \frac{\partial u_{c}}{\partial t}+u_{c}\frac{\partial u_{c}}{\par...
...artial\phi}{\partial x}-\frac{1}{m_{e}n_{c}} \frac{\partial p_{c}}{\partial x},$ (2)
  $\displaystyle \frac{\partial p_{c}}{\partial t}+u_{c}\frac{\partial p_{c}}{\partial x}+\gamma p_{c}\frac{\partial u_{c}}{\partial x}=0,$ (3)
  $\displaystyle \frac{\partial n_{p}}{\partial t}+\frac{\partial(n_{p}u_{p})}{\partial x}=0,$ (4)
  $\displaystyle \frac{\partial u_{p}}{\partial t}+u_{p}\frac{\partial u_{p}}{\par...
...artial\phi}{\partial x}-\frac{1}{m_{p}n_{p}} \frac{\partial p_{p}}{\partial x},$ (5)
  $\displaystyle \frac{\partial p_{p}}{\partial t}+u_{p}\frac{\partial p_{p}}{\partial x}+\gamma p_{p}\frac{\partial u_{p}}{\partial x}=0,$ (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:

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

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

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

  $\displaystyle \frac{\partial n}{\partial t}+\frac{\partial(nu)}{\partial x} =0,$ (10)
  $\displaystyle \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\frac {\partial\phi}{\partial x}-\frac{\sigma}{n}\frac{\partial p}{\partial x},$ (11)
  $\displaystyle \frac{\partial p}{\partial t}+u\frac{\partial p}{\partial x}+3p\frac {\partial u}{\partial x}=0,$ (12)
  $\displaystyle \frac{\partial n_{p}}{\partial t}+\frac{\partial(n_{p}u_{p})}{\partial x}=0,$ (13)
  $\displaystyle \frac{\partial u_{p}}{\partial t}+u_{p}\frac{\partial u_{p}}{\par...
...artial\phi}{\partial x}-\frac{\theta}{n_{p}}\frac{\partial p_{p} }{\partial x},$ (14)
  $\displaystyle \frac{\partial p_{p}}{\partial t}+u_{p}\frac{\partial p_{p}}{\partial x}+3p_{p}\frac{\partial u_{p}}{\partial x}=0.$ (15)

The dimensionless Poisson's equation takes the following form:

$\displaystyle \frac{\partial^{2}\phi}{\partial x^{2}} =$ $\displaystyle -\left( 1+\alpha-\beta\right) +n-\beta n_{p}$    
  $\displaystyle +\alpha\left( 1-\frac{\phi}{\kappa-\tfrac{3}{2}}\right)^{-\kappa+1/2},$ (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].

Ashkbiz Danehkar