Numerical exploration of electromagnetic electron whistler-cyclotron instability in Vasyliunas–Cairns distributed non-thermal plasmas: A kinetic theory approach

Abstract

A natural compression/expansion in the extended plasma systems generates the temperature anisotropy in the particle distribution. Such a kinetic anisotropy acts as source of free energies to instigate the generation of a variety of instabilities. These microinstabilities in turn are extensively involved in the enhancement of electromagnetic fluctuations and scatter the particles to reach the quasistable states, with relatively lower anisotropies. One of them is the instability associated with right hand circularly polarized electromagnetic electron whistler-cyclotron mode which significantly contributes to checking/defining the perpendicular electron temperature in large scale extended space plasmas. Its transverse dielectric response function in hybrid non-thermal Vasyliunas–Cairns distributed plasmas (which simultaneously incorporates the characteristics of both type of non-thermalities, i.e., $\alpha$ and $\kappa$) is calculated by using the well-known dispersion relation presented by Gary, 1993 [1]. The obtained dielectric response function is solved numerically to procure the real and imaginary frequencies of whistler instability. The impact of important physical parameters, i.e., non-thermality parameters $\alpha$ and $\kappa$ for different temperature anisotropy A and plasma beta $\beta$ is studied on the numerically calculated real frequency and growth rate of whistler instability. Both the real and imaginary frequencies are found sensitive to the respective instability thresholds. It is also investigated that the real frequency and growth rate are remarkably supported by the hybrid non-thermality of $\alpha$ and $\kappa$. Contemporary analysis is highly pertinent to comprehend the various magnetized space plasma environments where mixed non-thermal distributions are in frequently existent.

Appendix: A

This appendix section include the integrals ($I_{1},I_{2},I_{3},I_{4},I_{5}$ and $I_{6}$) that appear in the Eq. (4) with their respective solutions.

$$\begin{aligned} I_{1}& =\frac{2}{\kappa v_{th_{\perp e}(VC)}^{2}}A_{VC}(-\kappa -1)\int _{-\infty }^{+\infty }dv_{\parallel }\frac{\left\{ 1+\frac{ kv_{\parallel }}{\omega }A\right\} }{\omega -kv_{\parallel }\pm \Omega _{c}}\int _{0}^{\infty }v_{\perp }^{3}dv_{\perp } \nonumber \\ & \qquad \times \left[ 1+\frac{v_{\parallel }^{2}}{\kappa v_{th_{\parallel e}(VC)}^{2}}+\frac{v_{_{\perp }}^{2}}{\kappa v_{th_{\perp e}(VC)}^{2}}\right] ^{-\kappa -2} \end{aligned}$$

(22)

$$\begin{aligned} =\frac{1}{\pi \alpha _{0}}\left\{ \frac{A}{\omega }+\left( \frac{A}{\omega } \xi +\frac{1}{kv_{th_{\parallel e}(VC)}}\right) Z_{\kappa 1}(\xi )\right\} \end{aligned}$$

(23)

$$\begin{aligned} I_{2}&=\frac{2\alpha }{\kappa v_{th_{\perp e}(VC)}^{6}}A_{VC}(-\kappa -1)\int _{-\infty }^{+\infty }dv_{\parallel }\frac{\left\{ 1+\frac{ kv_{\parallel }}{\omega }A\right\} }{\omega -kv_{\parallel }\pm \Omega _{c}}\nonumber \\ & \quad \times \int _{0}^{\infty }v_{\perp }^{7}dv_{\perp }\left[ 1+\frac{ v_{\parallel }^{2}}{\kappa v_{th_{\parallel e}(VC)}^{2}}+\frac{ v_{_{\perp }}^{2}}{\kappa v_{th_{\perp e}(VC)}^{2}}\right] ^{-\kappa -2} \end{aligned}$$

(24)

$$\begin{aligned} & =2\alpha \frac{3\kappa ^{2}}{(\kappa -2)(\kappa -1)}\frac{1}{\pi \left[ 1+ \frac{15}{4}\alpha \frac{\kappa ^{2}}{(\kappa -\frac{5}{2})(\kappa -\frac{3}{ 2})}\right] } \nonumber \\ & \quad \quad \times \left\{ \frac{A}{\omega }\frac{(\kappa -2)(\kappa -1)}{(\kappa -\frac{ 5}{2})(\kappa -\frac{3}{2})}+\left( \frac{A}{\omega }\xi +\frac{1}{ kv_{th_{\parallel e}(VC)}}\right) Z_{\kappa 2}(\xi )\right\} \end{aligned}$$

(25)

$$\begin{aligned} I_{3}&=\frac{2\alpha }{\kappa v_{th_{\perp e}(VC)}^{2}v_{th_{\parallel e}(VC)}^{4}}A_{VC}(-\kappa -1)\int _{-\infty }^{+\infty }v_{\parallel }^{4}dv_{\parallel }\frac{\left\{ 1+\frac{kv_{\parallel }}{\omega } A\right\} }{\omega -kv_{\parallel }\pm \Omega _{c}}\nonumber \\ & \quad \times \int _{0}^{\infty }v_{\perp }^{3}dv_{\perp }\left[ 1+\frac{ v_{\parallel }^{2}}{\kappa v_{th_{\parallel e}(VC)}^{2}}+\frac{ v_{_{\perp }}^{2}}{\kappa v_{th_{\perp e}(VC)}^{2}}\right] ^{-\kappa -2} \end{aligned}$$

(26)

$$\begin{aligned} & =\alpha \frac{1}{(\pi )[1+\frac{15}{4}\alpha \frac{\kappa ^{2}}{(\kappa - \frac{5}{2})(\kappa -\frac{3}{2})}]}\frac{1}{kv_{th_{\parallel e}(VC)}} \nonumber \\ & \qquad \left\{ \frac{\xi }{2}\frac{\kappa }{(\kappa -\frac{3}{2})}+\xi ^{3}+kv_{th_{\parallel e}(VC)}\frac{A}{\omega }\right. \nonumber \\ & \qquad \times \left( \frac{3}{4}\frac{\kappa ^{2}}{(\kappa -\frac{5}{2})(\kappa - \frac{3}{2})}+\xi ^{2}\frac{\kappa }{2}\frac{1}{(\kappa -\frac{3}{2})}+\xi ^{4}\right) \nonumber \\ & \qquad \left. +\left( 1+kv_{th_{\parallel e}(VC)}\frac{A}{\omega }\xi \right) \xi ^{4}Z_{\kappa 1}(\xi )\right\} \end{aligned}$$

(27)

$$\begin{aligned} I_{4}& =\frac{4\alpha }{\kappa v_{th_{\perp e}(VC)}^{4}v_{th_{\parallel e}(VC)}^{2}}A_{VC}(-\kappa -1)\int _{-\infty }^{+\infty }v_{\parallel }^{2}dv_{\parallel }\frac{\left\{ 1+\frac{kv_{\parallel }}{\omega } A\right\} }{\omega -kv_{\parallel }\pm \Omega _{c}}\nonumber \\ & \quad \times \int _{0}^{\infty }v_{\perp }^{5}dv_{\perp }\left[ 1+\frac{ v_{\parallel }^{2}}{\kappa v_{th_{\parallel e}(VC)}^{2}}+\frac{ v_{_{\perp }}^{2}}{\kappa v_{th_{\perp e}(VC)}^{2}}\right] ^{-\kappa -2} \end{aligned}$$

(28)

$$\begin{aligned} & =4\alpha \frac{1}{(\pi )[1+\frac{15}{4}\alpha \frac{\kappa ^{2}}{(\kappa - \frac{5}{2})(\kappa -\frac{3}{2})}]}\frac{\kappa }{(\kappa -1)}\frac{1}{ kv_{th_{\parallel e}(VC)}} \nonumber \\ & \qquad \times \left\{ \xi \frac{(\kappa -1)}{(\kappa -\frac{3}{2})} +kv_{th_{\parallel e}(VC)}\frac{A}{\omega }\right. \nonumber \\ &\quad \quad \times \left( \frac{\kappa (\kappa -1)}{2(\kappa -\frac{5}{2})(\kappa -\frac{ 3}{2})}+\xi ^{2}\frac{(\kappa -1)}{(\kappa -\frac{3}{2})}\right) \nonumber \\ & \qquad \left. + \,(kv_{th_{\parallel e}(VC)}\frac{A}{\omega }\xi +1)\xi ^{2}Z_{\kappa 3}(\xi )\right\} \end{aligned}$$

(29)

$$\begin{aligned} I_{5}&=\frac{4\alpha }{v_{th_{\perp e}(VC)}^{4}}A_{VC}\int _{-\infty }^{+\infty }dv_{\parallel }\frac{\{1+\frac{kv_{\parallel }}{\omega }A\}}{ \omega -kv_{\parallel }\pm \Omega _{c}}\nonumber \\ & \quad \times \int _{0}^{\infty }v_{\perp }^{5}dv_{\perp }\left[ 1+\frac{ v_{\parallel }^{2}}{\kappa v_{th_{\parallel e}(VC)}^{2}}+\frac{ v_{_{\perp }}^{2}}{\kappa v_{th_{\perp e}(VC)}^{2}}\right] ^{-\kappa -1} \end{aligned}$$

(30)

$$\begin{aligned} I_{5} &=-4\alpha \frac{\kappa ^{2}}{(\kappa -1)(\kappa -2)}\frac{1}{(\pi ) \left[ 1+\frac{15}{4}\alpha \frac{\kappa ^{2}}{(\kappa -\frac{5}{2})(\kappa - \frac{3}{2})}\right] }\frac{1}{k} \nonumber \\ & \quad \times \left\{ \frac{kA}{\omega }\frac{(\kappa -2)(\kappa -1)}{(\kappa - \frac{5}{2})(\kappa -\frac{3}{2})}+\left( \frac{kA}{\omega }\xi +\frac{1}{ v_{th_{\parallel e}(VC)}}\right) Z_{\kappa 2}(\xi )\right\} \end{aligned}$$

(31)

$$\begin{aligned} I_{6}&=\frac{4\alpha }{v_{th_{\perp e}(VC)}^{2}v_{th_{\parallel e}(VC)}^{2}} A_{VC}\int _{-\infty }^{+\infty }v_{\parallel }^{2}dv_{\parallel }\frac{ \left\{ 1+\frac{kv_{\parallel }}{\omega }A\right\} }{\omega -kv_{\parallel }\pm \Omega _{c}}\nonumber \\ & \quad \times \int _{0}^{\infty }v_{\perp }^{3}dv_{\perp }\left[ 1+\frac{ v_{\parallel }^{2}}{\kappa v_{th_{\parallel e}(VC)}^{2}}+\frac{ v_{_{\perp }}^{2}}{\kappa v_{th_{\perp e}(VC)}^{2}}\right] ^{-\kappa -1} \end{aligned}$$

(32)

$$\begin{aligned} & =-2\alpha \frac{\kappa }{(\kappa -1)}\frac{1}{(\pi )\left[ 1+\frac{15}{4} \alpha \frac{\kappa ^{2}}{(\kappa -\frac{5}{2})(\kappa -\frac{3}{2})}\right] }\frac{1}{kv_{th_{\parallel e}(VC)}}\left\{ \xi \frac{(\kappa -1)}{(\kappa - \frac{3}{2})}\right. \nonumber \\ & \quad +kv_{th_{\parallel e}(VC)}\frac{A}{\omega }\left( \frac{\kappa (\kappa -1)}{ 2(\kappa -\frac{5}{2})(\kappa -\frac{3}{2})}+\xi ^{2}\frac{(\kappa -1)}{ (\kappa -\frac{3}{2})}\right) \nonumber \\ & \quad \left. +(kv_{th_{\parallel e}(VC)}\frac{A}{\omega }\xi +1)\xi ^{2}Z_{\kappa 3}(\xi )\right\} \end{aligned}$$

(33)