1 Introduction

Electron-acoustic waves may occur in plasmas characterized by a co-existence of two distinct electron populations, here referred to as `` cool'' and ``hot'' electrons. These are electrostatic waves of high frequency (in comparison with the ion plasma frequency), propagating at a phase speed which lies between the hot and cool electron thermal velocities. On such a fast (high frequency) dynamical scale, the positive ions may safely be assumed to form a uniform stationary charge background simply providing charge neutrality, yet playing no essential role in the dynamics. The cool electrons provide the inertia necessary to maintain the electrostatic oscillations, while the restoring force comes from the hot electron pressure.

A matter of importance in electrostatic wave propagation (although inevitably overlooked in fluid plasma models) is Landau damping, which becomes stronger when the phase velocity approaches the thermal velocity of either electron component, thus the wave can propagate in the plasma only within a restricted range of parameter values. It turns out that electron-acoustic waves are weakly damped for a temperature ratio $ T_{c}/T_{h} \lesssim0.1$ and provided that the cool electrons represent an intermediate fraction of the total electron density: $ 0.2\lesssim n_{c}/(n_{c}+n_{h})\lesssim0.8$ . [1,2,3,4] The wavenumber $ k$ to minimize damping lies roughly between $ 0.2\lambda_{Dc}^{-1}$ and $ 0.6\lambda_{Dc}^{-1}$ (where $ \lambda_{Dc}$ is the cool electron Debye length). These results on bi-Maxwellian plasmas were later extended to include the effect of the excess suprathermality of the hot electrons [5] (a physical feature to be discussed below). It was found that excess suprathermal electrons do cause a modification of the damping curves, but the overall qualitative conclusion remains unchanged: electron-acoustic waves survive Landau damping over a wide range of parameter values. [5] However, care must be taken in the choice of plasma configuration when studying nonlinear electron-acoustic structures, so as to ensure that one is considering a region of parameter space in which Landau damping is minimized.

Electron-acoustic waves occur in laboratory experiments [6,7] and space plasmas, e.g., in the Earth's bow shock [8,9,10] and in the auroral magnetosphere [1,11]. They are associated with Broadband Electrostatic Noise (BEN), a common high-frequency background activity, regularly observed by satellite missions in the plasma sheet boundary layer (PSBL) [12,13,14]. BEN emission includes a series of isolated bipolar pulses, within a frequency range from $ \sim10$ Hz up to the local electron plasma frequency ($ \sim10$ kHz) [12]. This clearly suggests that BEN is related to electron dynamics rather than to the ions [12,14].

In the standard bi-Maxwellian picture, the two electron species would each be assumed to be in a (different) thermal Maxwellian distribution, parameterized via two distinct temperature values, $ T_{c}$ and $ T_{h}$, respectively [15,16,17]. Contrary to this picture, space and laboratory plasmas often possess an excess population of suprathermal electrons, a fact which is reflected in a power law distribution at high velocity (above the electron thermal speed). This excess suprathermality phenomenon is well modeled by a generalized Lorentzian or $ \kappa $-distribution [18,19,20,22,21]. The common form of the isotropic (three-dimensional) generalized Lorentzian or $ \kappa $-distribution function is given by [18,20,22]

$\displaystyle f_{\kappa}(v)=n_{0}(\pi\kappa\theta^{2})^{-3/2}\frac{\Gamma(\kapp...
...kappa-\frac{1}{2})}\left( 1+\frac{v^{2}}{\kappa\theta^{2}}\right) ^{-\kappa-1},$ (1)

where $ n_{0}$ is the equilibrium number density of the electrons, $ v$ the velocity variable, and $ \theta$ the most probable speed, which acts as a characteristic “modified thermal speed", and is related to the usual thermal speed $ v_{th,e}=(2k_{B}T_{e}/m_{e})^{1/2}$ by $ \theta=v_{th,e}\left[ (\kappa
-\tfrac{3}{2})/\kappa\right] ^{1/2}$. Here $ k_{B}$ is the Boltzmann constant, $ m_{e}$ the electron mass and $ T_{e}$ the temperature of an equivalent Maxwellian having the same energy content. [22] The term involving the Gamma function ($ \Gamma$) arises from the normalization of $ f_{\kappa}(v)$, viz., $ \int f_{\kappa}(v)d^{3}v=n_{0}$. Here, suprathermality is denoted by the spectral index $ \kappa $, with $ \kappa> \frac{3}{2}$, for reality of the most probable speed, $ \theta$. [22] Low values of $ \kappa $ are associated with a significant number of suprathermal particles; on the other hand, for $ \kappa\rightarrow\infty$ a Maxwellian distribution is recovered.

The $ \kappa $-distribution was first applied to model velocity distributions observed in space plasmas that were Maxwellian-like at lower velocities, but had a power-law form at higher speeds [23], and was later applied in a variety of studies, successfully fitting many real space observations, e.g., [9,24,25,21]. Typical $ \kappa $ values usually lie in the range $ 2<\kappa<6$. For example, observations in the earth's foreshock satisfy $ 3<\kappa_{e}<6$, [9] measurements of plasma sheet electron and ion distributions yield $ \kappa_{i}=4.7$ and $ \kappa_{e}=$ $ 5.5$ (here, $ e$ denotes electrons and $ i$ ions), [24] and coronal electrons in the solar wind are modeled with $ 2<\kappa_{e}<6$ [25]. Recent observations of the radial distribution of the electron population in Saturn's magnetosphere also point towards a kappa distribution ( $ \kappa_{e}\simeq2.9$ - $ 4.2)$  [26]. Therefore, we focus our interest in the following on the range $ 2<\kappa<6$; in fact, the Maxwellian limit is already practically attained for values above $ \kappa\simeq10$.

A linear analysis of electron-acoustic waves was first carried out by assuming an unmagnetized Maxwellian homogeneous plasma, which exhibited a heavily damped acoustic-like solution in addition to Langmuir waves and ion-acoustic waves. [27] Those early results were later extended to take into account the effect of excess suprathermal particles [5,28], whose presence in fact results in an increase in the Landau damping at small wavenumbers, in particular when the hot electron component is dominant [5,29]. Studies of linear and nonlinear electron-acoustic waves in plasmas with nonthermal electrons have received a great deal of interest in recent years [30,31,34,32,33]. Negative potential solitary structures were shown to exist in a two-electron plasma, either for Maxwellian [30] or for nonthermal [31,35] hot electrons. Interestingly, either incorporation of finite inertia [32,33] or the addition of a beam component [36,37] may lead to the existence of positive and negative potential solitons. A recent investigation has established the properties of modulated electron-acoustic wavepackets in kappa-distributed plasmas, and has studied the effect of suprathermality on the amplitude (modulational) stability. [38]

In this paper, we study the linear and nonlinear dynamics of electron-acoustic waves in a plasma consisting of cool adiabatic electron and hot $ \kappa $-distributed electrons, in addition to stationary ions. The paper is organised as follows. An electron-plasma-fluid model is presented in Section 2. In Section 3, a linear dispersion relation is derived and discussed. In Section 4, the Sagdeev pseudopotential method is employed to investigate the occurrence of stationary profile electrostatic solitary waves. In Section 5, we depict the existence domain of the electron-acoustic solitary waves. Section 6 is devoted to a parametric investigation of the form of the Sagdeev pseudopotential and of the characteristics of electron-acoustic solitary waves. Our results are summarized in Section 7.

Ashkbiz Danehkar