# Characteristics of Nonthermal Dupree Diffusion on Space-Charge Wave in a Kappa Distribution Plasma Column with Turbulent Diffusion

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*Entropy*

**2020**,

*22*(2), 257; https://doi.org/10.3390/e22020257 (registering DOI)

## Abstract

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## 1. Introduction

## 2. Theory and Calculations

**r**and

**v**, ${v}_{L}$ is the characteristic velocity in the kappa distribution, D is the Dupree diffusion coefficient [15] owing to the Gaussian-Spatial-Diffusion (GSD) type correction term ${F}_{GSD}(k,t)\propto {e}^{-{k}^{2}D{t}^{3}/3}$ on account of the random walk caused by the plasma turbulence, k is the wave number, and t is the time. Hence, it is expected that the dispersions $\langle {(\Delta z)}^{2}\rangle $ in the spatial position increase as the cube of the elapsed time such as $\langle {(\Delta z)}^{2}\rangle =2D{t}^{3}/3$ [21]. It should be noted that Gaussian diffusion due to turbulent field fluctuations is quite similar to the spread by short-range collisions because Coulomb collisions produce random walkways in the speed space [21]. Since we are interested in investigating the nonthermal effects on the ion acoustic SCW in Lorentz turbulence (LT) plasma column, which consists of nonthermal electrons and cold ions, we consider the plasma dielectric function ${\epsilon}_{LT}(\omega ,k)$ in the frequency range where the phase velocity far exceeds the characteristic velocity of ion but is much less than that of the electron, i.e., ${v}_{Li}<<\omega /k<<{v}_{Le}$ where $\omega $ is the wave frequency, and ${v}_{Lj}$ is the characteristic velocity of the species j

## 3. Discussions

## 4. Summary

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Dispersion Relation

## Appendix B. Bessel Functions

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**Figure 1.**The scaled real part ${\overline{\omega}}_{R}$ of the wave frequency of the space-charge wave (SCW) as a function of the scaled axial wave number ${\overline{k}}_{\parallel}$ for $\overline{R}=3$ and the first-harmonic, i.e., ${\alpha}_{01}=2.4048$. The solid line is the case of $\kappa =2$. The dashed line is the case of $\kappa =3$. The dotted line is the case of $\kappa =5$. The dash-dot is the case of $\kappa \to \infty $, i.e., Maxwellian case.

**Figure 2.**The scaled real part ${\overline{\omega}}_{R}$ of the wave frequency of the SCW as a function of the scaled axial wave number ${\overline{k}}_{\parallel}$ for $\overline{R}=5$. The solid line is the case of $\kappa =2$ and the first-harmonic, i.e., ${\alpha}_{01}=2.4048$. The dashed line is the case of $\kappa =2$ and the second-harmonic, i.e., ${\alpha}_{02}=5.5201$. The dotted line is the case of $\kappa =4$ and the first-harmonic, i.e., ${\alpha}_{01}=2.4048$. The dash-dot is the case of $\kappa =4$ and the second-harmonic, i.e., ${\alpha}_{02}=5.5201$.

**Figure 3.**(

**a**) Surface plot; (

**b**) Contour plot of the scaled real part ${\overline{\omega}}_{R}$ of the wave frequency of the SCW as a function of the spectral index $\kappa $ and the scaled radius $\overline{R}$ of the cylindrical plasma column for ${\overline{k}}_{\parallel}=5$ and the first-harmonic, i.e., ${\alpha}_{01}=2.4048$.

**Figure 4.**The scaled imaginary part of the wave frequency, i.e., the scaled damping rate $\overline{\gamma}$, as a function of the scaled axial wave number ${\overline{k}}_{\parallel}$ for $\overline{D}=1$; $\overline{R}=3$; and the first-harmonic, i.e., ${\alpha}_{01}=2.4048$. The solid line is the case of $\kappa =2$. The dashed line is the case of $\kappa =3$. The dotted line is the case of $\kappa =5$. The dash-dot is the case of $\kappa \to \infty $, i.e., Maxwellian case.

**Figure 5.**The scaled damping rate $\overline{\gamma}$ of the SCW as a function of the scaled axial wave number ${\overline{k}}_{\parallel}$ for $\kappa =2$ and $\overline{R}=5$. The solid line is the case of $\overline{D}=1$ and the first-harmonic, i.e., ${\alpha}_{01}=2.4048$. The dashed line is the case of $\overline{D}=1$ and the second-harmonic, i.e., ${\alpha}_{02}=5.5201$. The dotted line is the case of $\overline{D}=3$ and the first-harmonic, i.e., ${\alpha}_{01}=2.4048$. The dash-dot is the case of $\overline{D}=3$ and the second-harmonic, i.e., ${\alpha}_{02}=5.5201$.

**Figure 6.**(

**a**) Surface plot; (

**b**) Contour plot of the scaled damping rate $\overline{\gamma}$ of the SCW as a function of the scaled radius $\overline{R}$ of the cylindrical plasma column and the spectral index $\kappa $ for ${\overline{k}}_{\parallel}=1$ and $\overline{D}=1$.

**Figure 7.**(

**a**) Surface plot; (

**b**) Contour plot of the scaled damping rate $\overline{\gamma}$ of the SCW as a function of the scaled diffusion coefficient $\overline{D}$ and the spectral index $\kappa $ for ${\overline{k}}_{\parallel}=1$ and $\overline{R}=5$.

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**MDPI and ACS Style**

Lee, M.-J.; Jung, Y.-D.
Characteristics of Nonthermal Dupree Diffusion on Space-Charge Wave in a Kappa Distribution Plasma Column with Turbulent Diffusion. *Entropy* **2020**, *22*, 257.
https://doi.org/10.3390/e22020257

**AMA Style**

Lee M-J, Jung Y-D.
Characteristics of Nonthermal Dupree Diffusion on Space-Charge Wave in a Kappa Distribution Plasma Column with Turbulent Diffusion. *Entropy*. 2020; 22(2):257.
https://doi.org/10.3390/e22020257

**Chicago/Turabian Style**

Lee, Myoung-Jae, and Young-Dae Jung.
2020. "Characteristics of Nonthermal Dupree Diffusion on Space-Charge Wave in a Kappa Distribution Plasma Column with Turbulent Diffusion" *Entropy* 22, no. 2: 257.
https://doi.org/10.3390/e22020257