Phase-mixing of high-frequency electrostatic oscillations in multi-component dusty plasmas

Abstract

A theoretical investigation of the phase-mixing phenomenon of Langmuir oscillations (LOs) in a multi-component dusty plasma is presented. The plasma under study is a four-component plasma consisting of electrons, positrons, singly ionised positive ions and positive (or negative) polarity dust grains. The massive dust grains are assumed to form a fixed charge neutralising background, while the electrons, positrons and ions are considered to take part in the wave dynamics. A simple perturbative analysis of the governing fluid-Maxwell’s equations leads us to an approximate relation for phase-mixing time of LOs, improving some earlier existing results. It is revealed that even though the dust grains remain motionless in the background, they still have significant impact on the phase-mixing process of LOs in such plasmas. It is found that the nature of the charge on the dust grains and the equilibrium dust density profoundly affect the phase-mixing time. The results of this investigation is expected to have relevance in space-plasma environments.

Appendix A: Solutions in original physical variables

The full first-order solutions of the original variables are as follows:

$$\begin{aligned}{} & {} n_{p}^{(1)}=\frac{1}{2}\delta \cos kx [\sigma n_{0e}-\beta (n_{0i}+n_{0d})]\nonumber \\{} & {} \qquad \quad \ \times (1-\cos {\tilde{\omega }}_{pep}t),\nonumber \\{} & {} n_{e}^{(1)}=n_{0e}\delta \cos kx - \frac{1}{2}\delta \cos kx[\sigma n_{0e}+\beta (n_{0i}+n_{0d})]\nonumber \\{} & {} \qquad \qquad \times (1-\cos {\tilde{\omega }}_{pep}t), \nonumber \\{} & {} n_{i}^{(1)}=n_{0e}\delta \cos kx(1-\sigma ) (1-\cos {\tilde{\omega }}_{pep}t),\nonumber \\{} & {} { v}_{px}^{(1)}=-\frac{\omega _{pe}}{k}\delta \sqrt{\beta } \sin kx \sin {\tilde{\omega }}_{pep}t,\nonumber \\{} & {} { v}_{ex}^{(1)}=\frac{\omega _{pe}}{k}\delta \sqrt{\beta } \sin kx \sin {\tilde{\omega }}_{pep}t,\nonumber \\{} & {} { v}_{ix}^{(1)}=-\frac{\omega _{pi}}{k}\delta \alpha ^{-1}\sqrt{1-\sigma } \sin kx \sin {\tilde{\omega }}_{pep}t. \end{aligned}$$

(A.1)

The second-order solutions of the original variables are

$$\begin{aligned} n_{p}^{(2)}\simeq & {} \frac{1}{4}\Gamma _p \delta ^2 \beta ^2 \cos 2kx({\tilde{\omega }}_{pep}t)^2,\nonumber \\ n_{e}^{(2)}\simeq & {} \frac{1}{4}\Gamma _e \delta ^2 \beta ^2 \cos 2kx({\tilde{\omega }}_{pep}t)^2,\nonumber \\ n_{i}^{(2)}\simeq & {} \frac{1}{4}n_{0i}\delta ^2\beta ^2 \left[ 1-\sigma (1-\Delta ^2)+\frac{\eta _d}{\alpha }(1-\sigma ) \right] \nonumber \\{} & {} \times \cos 2kx({\tilde{\omega }}_{pep}t)^2,\nonumber \\ { v}_{px}^{(2)}\simeq & {} -\frac{1}{4}\delta ^2\beta \frac{\omega _{pe}}{k} \left[ \alpha \beta \left( 1-\Delta ^2+\frac{\eta _d}{\alpha } \right) +1 \right] \nonumber \\{} & {} \times \sin 2kx(\omega _{pe}t), \nonumber \\ { v}_{ex}^{(2)}\simeq & {} \frac{1}{4}\delta ^2\beta \frac{\omega _{pe}}{k} \left[ \alpha \beta \left( 1-\Delta ^2+\frac{\eta _d}{\alpha } \right) -1 \right] \nonumber \\{} & {} \times \sin 2kx(\omega _{pe}t), \nonumber \\ { v}_{ix}^{(2)}\simeq & {} -\frac{1}{4}\delta ^2 \left[ \alpha \beta ^2 \frac{\omega _{pe}\Delta }{k}\left( 1-\Delta ^2+\frac{\eta _d}{\alpha }\right) +\frac{\omega _{pi}\Delta ^2\beta }{k}\right] \nonumber \\{} & {} \times \sin 2kx(\omega _{pe}t), \end{aligned}$$

(A.2)

where

$$\begin{aligned} \Gamma _p= & {} n_{0e}\left[ \left( 1-\frac{\alpha }{2} \right) -\frac{\alpha ^2 \beta }{2}\left( 1-\Delta ^2+\frac{\eta _d}{\alpha }\right) \right. \nonumber \\{} & {} \left. -\frac{\eta _d}{2}\left\{ 1+\alpha \beta \left( 1-\Delta ^2+\frac{\eta _d}{\alpha } \right) \right\} \right] \nonumber \\{} & {} -\frac{1}{2}n_{0i}\left( 1-\sigma (1-\Delta ^2)+ \frac{\eta _d}{\alpha }(1-\sigma ) \right) , \nonumber \\ \Gamma _e= & {} n_{0e}\left[ \left( 1-\frac{\alpha }{2} \right) -\frac{\alpha ^2 \beta }{2}\left( 1-\Delta ^2+\frac{\eta _d}{\alpha }\right) \right. \nonumber \\{} & {} \left. -\frac{\eta _d}{2}\left\{ 1+\alpha \beta \left( 1-\Delta ^2+\frac{\eta _d}{\alpha } \right) \right\} \right] \nonumber \\{} & {} + \frac{1}{2}n_{0i}\left( 1-\sigma (1-\Delta ^2)+ \frac{\eta _d}{\alpha }(1-\sigma ) \right) . \end{aligned}$$

(A.3)

In writing these second-order solutions, we have retained only the leading order secular terms.