2 Theoretical Model

We consider a 1-D collisionless, four-component plasma consisting of cool inertial background electrons (at temperature ), mobile cool positrons (or electron holes; at temperature ), inertialess hot suprathermal electrons modeled by a -distribution (at temperature ), and uniformly distributed stationary ions.

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

where , and are the number density, the velocity and the pressure of the cool electrons and positrons (denoted by indices `c' and `p', respectively), is the electrostatic wave potential, the elementary charge, the electron mass, the positron mass, and denotes the specific heat ratio for degrees of freedom. For the adiabatic cool electrons and positrons in one-dimensional (), we get . Through this paper, we assume that .

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

where and are the equilibrium number density and the temperature of the hot electrons, respectively, the Boltzmann constant, and the spectral index measures the deviation from thermal equilibrium. For reality of the characteristic modified thermal velocity, , the spectral index must take . The suprathermality is measured by the spectral index , describing how it deviates from a Maxwellian distribution, i.e., low values of are associated with a significant suprathermality; on the other hand, a Maxwellian distribution is recovered in the limit .

The ions are assumed to be immobile in a uniform state, i.e., const. at all times, where is the undisturbed ion density. The plasma is quasi-neutral at equilibrium, so , that implies

where we have defined the hot-to-cool electron density ratio as , and the positron-to-cool electron density ratio as , while and are the equilibrium number densities of the cool electrons and positrons, respectively. Electrostatic waves are weakly damped in the range of [26,27,28,29], so . This region may permit the propagation of nonlinear electrostatic structures. As the positron fraction has been measured to be in low energy solar wind observations [1,2,3,5] and -0.1 in some laser-plasma experiences [14,15], we assume .

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

where 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:

The dimensionless Poisson's equation takes the following form:

where and denote the fluid density variables of the cool electrons and positrons normalized with respect to and , respectively, and the velocity variables of the cool electrons and positrons scaled by the hot electron thermal speed , and the pressure variables of the cool electrons and positrons normalized with respect to and , respectively, and the wave potential by , time and space scaled by the plasma period and the characteristic length , respectively. We have defined the cool-to-hot electron temperature ratio as , and the positron-to-hot electron temperature ratio as . Landau damping is minimized if [26,27,28], and the same for the cool positrons, i.e., (see Ref. [40]), and typically in some laser-plasma experiences [14].

2018-03-28