Electrostatic waves in astrophysical Druyvesteyn plasmas: I. Langmuir waves

TL;DR

This work investigates Langmuir-wave propagation in non-Maxwellian astrophysical plasmas by introducing a generalized Druyvesteyn distribution, controlled by the parameter $x$, and formulating the dispersion relation in terms of the Druyvesteyn dispersion function $Z_x(\xi)$. It presents both exact numerical solutions and analytical weak-damping approximations, validating results against the ALPS solver and showing how $x$ shapes the dispersion and damping: super-Maxwellian plasmas with $x>1$ exhibit reduced damping and flatter dispersion, while sub-Maxwellian plasmas with $x<1$ display enhanced damping and can exhibit anomalous dispersion with negative group velocity. The key contributions are the explicit Druyvesteyn dispersion framework, the demonstration of anomalous dispersion for $x<1$, and the derivation of a Druyvesteyn Bohm–Gross-type dispersion with an effective Debye length $\lambda_{De,x}$, providing a versatile tool for modeling longitudinal waves in non-Maxwellian astrophysical plasmas. These results have implications for wave generation, turbulence evolution, and radiation processes in environments like the solar wind, magnetospheres, and shocks, and motivate extensions to ion-acoustic and dusty-plasma contexts.

Abstract

Plasmas in various astrophysical systems are in non-equilibrium states as evidenced by direct in-situ measurements in the solar wind, solar corona and planetary environments as well as by indirect observations of nonthermal sources of waves and emissions. Specific to observed non-equilibrium plasmas are non-Maxwellian velocity distributions with suprathermal tails, most often described by Kappa (power-law) distributions. In this paper, we introduce an alternative modeling for linear waves in plasmas described by the generalized Druyvesteyn distribution model. This model can reproduce not only high-energy tails, but also low-energy flat-tops of velocity distributions, like those of electrons in interplanetary shocks and the solar transition region. The wave dispersion relation of longitudinal waves is derived in terms of the newly introduced Druyvesteyn dispersion function. The dispersion curves as well as damping rates of high-frequency Langmuir waves are numerically computed for the isotropic case, and their analytical approximations are provided in the limit of weak damping. We thus offer a new tool for modeling longitudinal waves, and in particular Langmuir waves under the specific conditions of Druyvesteyn distributions.

Figures (6)

• Figure 1: The generalized Druyvesteyn distribution as defined by Eq. (\ref{['druyvesteyn-dist']}), plotted as a function of the normalized particle momentum $p_{\parallel}/(m\theta)$ for $p_{\perp}=0$. The parameter $x$ controls the shape of the distribution, with $x=1$ corresponding to the Maxwellian limit. The plot highlights the gradual deviation from the Maxwellian form with varying $x$.
• Figure 2: Dispersion curves (left) and damping rates (right), both normalized to the electron plasma frequency $\omega_{p,e}$, for a generalized Druyvesteyn distributions with the Druyvesteyn parameter $x > 1$ compared to the Maxwellian ($x=1$). The wave number is normalized using the electron Debye length $\lambda_{De} = \theta d_p/(\sqrt{2} c)$. The solid lines are obtained numerically with the ALPS code directly solving Eq.(\ref{['eq:Dispersion_equation-gen']}) and the dots are the solutions obtained employing the derivative of the Druyvesteyn dispersion function Eq.(\ref{['eq:DRU1:Druyvesteyn_dispersion_function-derivative']}), with the properties discussed in Appendix \ref{['Zx-Properties']}.
• Figure 3: Dispersion curves (left) and damping rates (right), both normalized to the electron plasma frequency $\omega_{p,e}$, for a generalized Druyvesteyn distributions with the Druyvesteyn parameter $x < 1$ compared to the Maxwellian ($x=1$). The wave number is normalized using the electron Debye length $\lambda_{De} = \theta d_p/(\sqrt{2} c)$. The solid lines are obtained numerically with the ALPS code directly solving Eq.(\ref{['eq:Dispersion_equation-gen']}) and the dots are the solutions obtained employing the derivative of the Druyvesteyn dispersion function Eq.(\ref{['eq:DRU1:Druyvesteyn_dispersion_function-derivative']}), with the properties discussed in Appendix \ref{['Zx-Properties']}.
• Figure 4: Comparison of the approximate expressions (Eqs. \ref{['eq:DRU1:Druvesteyn-Bohm-Gross-dispersion_relation']} and \ref{['eq:Druyvesteyn-weak_absorption']}) (dashed lines) with the full numerical solution of Eq. (\ref{['eq:DRU1:DELD-Langmuir']}) (solid lines). The plots show dispersion relation curves (left column) and damping rates (right column) for a Druyvesteyn plasma with $x>1$ (upper row) and $x<1$ (lower row).
• Figure 5: The effective Debye length as a function of $x$.