Generals 2003 I 5
From PlasmaWiki
Bernstein waves are hot plasma electrostatic waves. The hot plasma electrostatic dispersion function is:
![k_{\perp}^{2}+k_{\|}^{2}+\sum_{s}\frac{4\pi n_{s}q_{s}^{2}}{T_{\|}}\left[1+\sum_{n}\frac{\left(\omega-k_{\|}V-n\Omega\right)T_{\perp}+n\Omega T_{\|}}{k_{\|}w_{\|}T_{\perp}}e^{-\lambda}I_{n}\left(\lambda\right)Z\left(\zeta_{n}\right)\right]](http://localhost/mediawiki/images/math/6/2/8/6288cdfdd7a95d1fccca4af1d18ce9d8.png)
Bernstein waves occur at V = 0, 
. Taking just the first
term of Z as 
(corresponding to 
:
![k_{\perp}^{2}+k_{\|}^{2}+\sum_{s}\frac{4\pi n_{s}q_{s}^{2}}{T_{\|}}\left[1-\sum_{n}\frac{\left(\omega-n\Omega\right)T_{\perp}+n\Omega T_{\|}}{k_{\|}w_{\|}T_{\perp}}e^{-\lambda}I_{n}\left(\lambda\right)\frac{k_{\|}w_{\|}}{\left(\omega-n\Omega\right)}\right]=0](http://localhost/mediawiki/images/math/e/8/8/e88cc69844aa6286c55d0d3c70abd2e4.png)
Taking the limit:
![k_{\perp}^{2}+\sum_{s}\frac{4\pi n_{s}q_{s}^{2}}{T_{\|}}\left[1-\sum_{n}\frac{\left(\omega-n\Omega\right)T_{\perp}+n\Omega T_{\|}}{T_{\perp}\left(\omega-n\Omega\right)}e^{-\lambda}I_{n}\left(\lambda\right)\right]=0](http://localhost/mediawiki/images/math/1/1/8/118e5f9bf9167f124ef0e4a4068b952c.png)
We only have positrons and electrons, so assuming quasineutrality:
![k_{\perp}^{2}+\frac{8\pi n_{e}e^{2}}{T_{\|}}\left[1-\sum_{n}\frac{\left(\omega-n\Omega\right)T_{\perp}+n\Omega T_{\|}}{T_{\perp}\left(\omega-n\Omega\right)}e^{-\lambda}I_{n}\left(\lambda\right)\right]=0](http://localhost/mediawiki/images/math/d/8/9/d89f1c5f6a56dea252db2e16e6bf702a.png)
