Generals 2004 I 4

From PlasmaWiki

The electrostatic dispersion relation is:

$k^{2}\left(1+\sum_{s}\chi_{s}\right)=0$

Plugging in for a two species plasma:

$1+\frac{2\omega_{pi}^{2}}{k^{2}v_{Ti}^{2}}\left\{ 1+\frac{\omega-kv_{D}}{kv_{Ti}}Z\left(\zeta_{i}\right)\right\} +\frac{2\omega_{pe}^{2}}{k^{2}v_{Te}^{2}}\left\{ 1+\frac{\omega-kv_{D}}{kv_{Te}}Z\left(\zeta_{e}\right)\right\} =0$

Expanding the ion term in $\zeta\gg1$ , and the electron in $\zeta\ll1$ :

$\begin{array}{rcl} 1+\frac{2\omega_{pi}^{2}}{k^{2}v_{Ti}^{2}}\left\{ 1+\frac{\omega-kv_{D}}{kv_{Ti}}\left(-\left(\frac{kv_{Ti}}{\omega-kv_{D}}\right)-\frac{1}{2}\left(\frac{kv_{Ti}}{\omega-kv_{D}}\right)^{3}\right)\right\} \\ +\frac{2\omega_{pe}^{2}}{k^{2}v_{Te}^{2}}\left\{ 1+\frac{\omega-kv_{D}}{kv_{Te}}\left(i\sqrt{\pi}e^{-\zeta^{2}}-2\frac{\omega-kv_{D}}{kv_{Te}}\right)\right\} & = & 0\end{array}$

This becomes for $\omega\ll kv_{Te}$ :