About the effects of solar wind suprathermal electrons on electrostatic waves

Abstract

Electrostatic (ES) waves generated in space plasmas, e.g., Langmuir and ion-acoustic waves, are subject to multiple applications, such as plasma diagnosis, generation of radio emissions, and the acceleration and heating of resonant populations. The dispersion properties of these waves are well known for idealized plasmas, i.e., with Maxwellian distributions, but in the solar wind and terrestrial magnetosphere plasma particles exhibit Kappa distributions with high energy tails enhanced by suprathermal populations. This paper proposes a realistic analysis of these populations and their influence on ES waves, which often is hindered by a misinterpretation of Kappa distributions. Of particular importance in the analysis of ES waves is the Debye wavelength, the correct derivation of which shows, as expected, an increase (and not a decrease) in the presence of suprathermal electrons. Based on these new evaluations, we show how the suprathermal electrons self-consistently modify the properties of ES waves. For Langmuir waves, the positive slope of the frequency increase with the wave-number is markedly enhanced, involving more resonant particles from the high-energy tails, and thus leading to enhanced damping rates. In contrast, ion-acoustic waves are supported by suprathermal electrons, which increase the kinetic energy contrast between electrons and protons, and thus reduce the damping rate of ion-acoustic waves. Obviously, suprathermal protons do not affect Langmuir waves, but inhibit ion-acoustic modes. The present results provide both the methodology and the theoretical tools necessary to understand the physical processes involving these waves in non-ideal plasmas from space.

Appendix: Kappa model with \(\kappa \)-independent temperature

By contrast to the original Kappa distribution from (1), in the literature we can also find another Kappa approach with temperature (also defined by the second order moment) independent of $\kappa$ parameter (Bryant 1996; Podesta 2005)

$$\begin{aligned} F_{\kappa }(v) = & {\frac{n_{\kappa }}{(2 \pi k_{B} T / m_{e})^{3}}} { \frac{\Gamma (\kappa )}{\kappa ^{1/2} \Gamma (\kappa - 1/2)}} \\ & \times \left [1 + {\frac{m_{e} v^{2}}{2 (\kappa -3/2) k_{B} T}} \right ]^{-\kappa -1}, \end{aligned}$$

(22)

whose relevance is still disputable (Lazar et al. 2015, 2016). In this case the Maxwellian limit takes the same form as in (2)

$$\begin{aligned} F_{\kappa \to \infty} (v) = f_{M} (v) & = { \frac{n_{\kappa }}{(2 \pi k_{B}T/m_{e})^{3/2}}} \exp \left (-{ \frac{m_{e} v^{2}}{2 k_{B} T}} \right ) \\ & = {\frac{n_{\kappa }}{(\pi \theta )^{3/2}}} \exp \left (-{ \frac{v^{2}}{\theta ^{2}}} \right ). \end{aligned}$$

(23)

However, the Debye length for Kappa electrons changes as follows

$$\begin{aligned} \lambda _{D,\kappa}= \left ({\frac{\kappa -3/2}{\kappa -1/2}}\right )^{1/2} \lambda _{D,{\mathrm{M}}}, \end{aligned}$$

(24)

where $\lambda _{D,{\mathrm{M}}} = [m_{e} \theta ^{2} /(8 \pi n e^{2})]^{1/2}$ is obviously the same.

A graphical comparison of the Debye lengths ratios $\lambda _{D,\kappa}/\lambda _{D,{\mathrm{M}}}$, as given by Eqs. (4) (red) and (24) (black-dotted), is provided in Fig. 5. The relevant one (red) shows a natural lowering of $\lambda _{D,\kappa}$ with increasing the power-exponent $\kappa$, due to the decrease of the mean kinetic energy or temperature of electrons. For low values of $\kappa \to 3/2$, the Debye length is, however, not much superior and remains comparable to the Maxwellian limit of the core, i.e., that obtained in the absence of suprathermals.

Fig. 5

Debye length ratios as a function of $\kappa (> 3/2)$, from Eq. (4) (red), and Eq. (24) (black-dotted)

For the other Kappa model with $\kappa$-independent temperature, the variation of the Debye length reduces to very small values with decreasing $\kappa \to 3/2$. This results is not only contradictory, but even unphysical, because the mean kinetic energy or temperature of electrons does not change.