Generals 2004 I 3

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[edit] Part a

The collision operator is:

$C\left(f_{e}\right)=\nu_{ei}\left(\frac{v_{Te}}{v}\right)^{3}\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial f_{e}}{\partial\theta}\right)$

The steady state Vlasov equation is:

$-\frac{e}{m}\mathbf{E}\cdot\frac{\partial f_{e}}{\partial\mathbf{v}}=C\left(f_{e}\right)$

If we linearize this:

$-\frac{e}{m}\mathbf{E}\cdot\frac{\partial f_{0e}}{\partial\mathbf{v}}=C\left(f_{1e}\right)$

Using $f 0 e = f M$ , this becomes:

$-\frac{e}{m}E_{\|}\cos\theta\frac{-2v}{v_{Te}^{2}}f_{M}=\nu_{ei}\left(\frac{v_{Te}}{v}\right)^{3}\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial f_{1e}}{\partial\theta}\right)$

If we guess $f_{1e}=g\left(v\right)\cos\theta$ :

$-\frac{e}{m}E_{\|}\cos\theta\frac{-2v}{v_{Te}^{2}}f_{M}=\nu_{ei}\left(\frac{v_{Te}}{v}\right)^{3}2g\left(v\right)\cos\theta$

So that:

$g\left(v\right)=\frac{eE_{\|}}{m\nu_{ei}v_{Te}}\left(\frac{v}{v_{Te}}\right)^{4}f_{M}$

Then:

$j_{\|}=e\int v_{\|}f_{1}d\mathbf{v}=2\pi\int v^{2}\sin\theta dvd\theta\, v\cos\theta\, f_{1}$

Plugging in (and using $p = cosθ$ )

$j_{\|}=2\pi e\int_{-1}^{1}p^{2}dp\int v^{3}dv\frac{eE_{\|}}{m\nu_{ei}v_{Te}}\left(\frac{v}{v_{Te}}\right)^{4}f_{M}$

Plugging in the maxwellian and doing the first integral:

$j_{\|}=\frac{4\pi e}{3}\int v^{3}dv\frac{eE_{\|}}{m\nu_{ei}v_{Te}}\left(\frac{v}{v_{Te}}\right)^{4}\frac{2n_{0}}{\pi^{3/2}v_{Te}^{3}}e^{-v^{2}/v_{t}^{2}}$

Transforming to $x = v / v T e$ :

$j_{\|}=\frac{4\pi e}{3}\frac{2eE_{\|}n_{0}}{m\nu_{ei}\pi^{3/2}}\int dx\, x^{7}e^{-x^{2}}=\frac{8e^{2}E_{\|}n_{0}}{m\nu_{ei}\pi^{1/2}}=\sigma_{\|}E_{\|}$

Then the resistivity is:

$\eta=\frac{E_{\|}}{j_{\|}}=\frac{m\nu_{ei}\pi^{1/2}}{8e^{2}n_{0}}$

[edit] Part b

The parallel resistivity is changed in the banana regime by conductivity reduction, which occurs because trapped particles cannot carry current. The new resistivity will thus be:

$\eta_{banana}=\frac{1}{\sigma_{\|}\left(1-\sqrt{\epsilon}\right)}=\frac{m\nu_{ei}\pi^{1/2}}{8e^{2}n_{0}\left(1-\epsilon^{1/2}\right)}$

Where $ε = r / R 0$ , and $ε 1 / 2$ is the fraction of trapped particles. The regime is valid as long as the collision frequency is less than the bounce frequency. The bounce frequency is given by: