Bump-on-Tail Instability in Cairns Distributed Plasma

Abstract

The growing rate of bump-on-tail instability has been investigated in unmagnetized plasma with Cairns distribution. The growing rate depends significantly on the number density ratio n₁/n₂, temperature ratio T₁/T₂, and the nonthermal parameter α. The influence of these parameters on the maximum growing rate is analyzed numerically by using solar data. The higher α can enhance the growing rate of bump-on-tail instability. The maximum growth rate also will increase with the decrease of number density ratio n₁/n₂ and the increase of temperature ratio T₁/T₂. These interesting results will be helpful for understanding some phenomena in space plasma.

Appendix

The real part of the dispersion relation is expressed as

$$D_{r} (k,\omega_{r} ) = 1 + \frac{{\omega_{{p{\text{e}}}}^{2} }}{{k^{2} }}\int_{ - \infty }^{\infty } {\frac{{\partial f_{e0} /\partial v_{z} }}{{\omega_{r} /k - v_{z} }}} dv_{z} = 0$$

(A1)

For electrons, in the case $\omega /kv_{T} \gg 1$, the integral term in the dispersion relation can be approximated as

$$\int^{\infty}_{-\infty} {\frac{{\partial f_{e0} /\partial v_{z} }}{{\omega_{r} /k - v_{z} }}} dv_{z} = \int_{ - \infty }^{\infty } {\frac{{\partial f_{e0} }}{{\partial v_{z} }}\left( {\frac{k}{{\omega_{r} }} + \frac{{k^{2} v_{z} }}{{\omega {}_{r}^{2} }} + \frac{{k^{3} v_{z}^{2} }}{{\omega {}_{r}^{3} }} + \cdots } \right)} dv_{z}$$

(A2)

$$\begin{array}{l} \frac{{\partial f_{e0} }}{{\partial v_{z} }} = \frac{1}{{n_{e} }}\frac{1}{{(1 + 3\alpha )\left( {2\pi } \right)^{1/2} v_{T1} }}\left( {n_{1} \left\{ {4\frac{{\alpha v_{z}^{3} }}{{v_{T1}^{4} }}\exp \left( { - \frac{{v_{z}^{2} }}{{2v_{T1}^{2} }}} \right) - \frac{{v_{z} }}{{v_{T1}^{2} }}\left( {1 + \alpha \frac{{v_{z}^{4} }}{{v_{T1}^{4} }}} \right)\exp \left( { - \frac{{v_{z}^{2} }}{{2v_{T1}^{2} }}} \right)} \right\}} \right. + \hfill \\ \frac{{n_{2} }}{2}\frac{{v_{T1} }}{{v_{T2} }}\left\{ {4\frac{{\alpha \left( {v_{z} - v_{0} } \right)^{3} }}{{v_{T2}^{4} }}\exp \left( { - \frac{{\left( {v_{z} - v_{0} } \right)^{2} }}{{2v_{T2}^{2} }}} \right) - \frac{{\left( {v_{z} - v_{0} } \right)}}{{v_{T2}^{2} }}\left( {1 + \alpha \frac{{\left( {v_{z} - v_{0} } \right)^{4} }}{{v_{T2}^{4} }}} \right)\exp \left( { - \frac{{\left( {v_{z} - v_{0} } \right)^{2} }}{{2v_{T2}^{2} }}} \right) + } \right. \hfill \\ \left. {\left. {4\frac{{\alpha \left( {v_{z} - v_{0} } \right)^{3} }}{{v_{T2}^{4} }}\exp \left( { - \frac{{\left( {v_{z} - v_{0} } \right)^{2} }}{{2v_{T2}^{2} }}} \right) - \frac{{\left( {v_{z} - v_{0} } \right)}}{{v_{T2}^{2} }}\left( {1 + \alpha \frac{{\left( {v_{z} - v_{0} } \right)^{4} }}{{v_{T2}^{4} }}} \right)\exp \left( { - \frac{{\left( {v_{z} - v_{0} } \right)^{2} }}{{2v_{T2}^{2} }}} \right)} \right\}} \right) \hfill \\ \end{array}$$

(A3)

Combining Eq. (A2) and (A3), we can get

$$\int^{\infty}_{-\infty} {\frac{{\partial f_{e0} /\partial v_{z} }}{{\omega_{r} /k - v_{z} }}} dv_{z} = \int^{\infty}_{-\infty} {\frac{{\partial f_{e0} }}{{\partial v_{z} }}\left( {\frac{k}{{\omega_{r} }} + \frac{{k^{2} v_{z} }}{{\omega {}_{r}^{2} }} + \frac{{k^{3} v_{z}^{2} }}{{\omega {}_{r}^{3} }} + \cdots } \right)} dv_{z}$$

$$\begin{array}{l} = - \frac{{n_{1} }}{{n_{e} }}\left( {\frac{{k^{2} }}{{\omega_{r}^{2} }}{ + }\frac{{3\left( {1 + 15\alpha } \right)}}{1 + 3\alpha }\frac{{k^{4} v_{T1}^{2} }}{{\omega_{r}^{4} }}} \right) - \frac{{n_{2} }}{{2n_{e} }}\left( {\frac{{2k^{2} }}{{\omega_{r}^{2} }} + \frac{{6\left( {1 + 15\alpha } \right)}}{1 + 3\alpha }\frac{{k^{4} v_{T2}^{2} }}{{\omega_{r}^{4} }}} \right) \hfill \\ = - \frac{{n_{1} }}{{n_{e} }}\frac{{k^{2} }}{{\omega_{r}^{2} }} - \frac{{n_{2} }}{{n_{e} }}\frac{{k^{2} }}{{\omega_{r}^{2} }} - \frac{{n_{1} }}{{n_{e} }}\frac{{3\left( {1 + 15\alpha } \right)}}{1 + 3\alpha }\frac{{k^{4} v_{T1}^{2} }}{{\omega_{r}^{4} }} - \frac{{n_{2} }}{{n_{e} }}\frac{{3\left( {1 + 15\alpha } \right)}}{1 + 3\alpha }\frac{{k^{4} v_{T2}^{2} }}{{\omega_{r}^{4} }} \hfill \\ \end{array}$$

(A4)

Assuming the cases, i.e., $n_{1} \gg n_{2}$, $n_{1} T_{1} \gg n_{2} m_{e} v_{0}^{2}$, then we can get

$$D_{r} (k,\omega_{r} ) = 1 - \frac{{n_{1} }}{{n_{e} }}\frac{{\omega_{{p{\text{e}}}}^{2} }}{{\omega_{r}^{2} }} - \frac{{n_{1} }}{{n_{e} }}\frac{{3\left( {1 + 15\alpha } \right)}}{(1 + 3\alpha )}k^{2} v_{{T{1}}}^{2} \frac{{\omega_{{p{\text{e}}}}^{{2}} }}{{\omega_{r}^{4} }} = 0$$

(A5)

$$D_{r} (k,\omega_{r} ) = 1 - \frac{{\omega_{{p{1}}}^{2} }}{{\omega_{r}^{2} }} - \frac{{3\left( {1 + 15\alpha } \right)}}{(1 + 3\alpha )}k^{2} \lambda_{{D{1}}}^{2} \frac{{\omega_{{p{1}}}^{4} }}{{\omega_{r}^{4} }} = 0$$

(A6)

where we have used $\lambda_{{D{1}}}^{2} = v_{{T{1}}}^{2} /\omega_{{p{1}}}^{{2}}$.