Generals 2001 II 5
From PlasmaWiki
[edit] Part a
The electrostatic approximation assumes that 
.
This is valid for short wavelengths. The condition is that 
for all elements of the dielectric tensor 
.
[edit] Part b
The perturbed distribution function is:
With
![\beta=-\frac{k_{x}v_{\perp}}{\omega}\left[\sin\left(\phi+\Omega\tau\right)-\sin\phi\right]+\left(\omega-k_{z}v_{z}\right)\tau](http://localhost/mediawiki/images/math/7/7/b/77b4a275cc808d9d624c86b2cd785f6d.png)
The electrostatic approximation gives 
.
Then:
Rewriting the distribution function:
![\begin{array}{rcl} f_{1}\left(v_{\perp},v_{z},\phi,t\right) & = & -qe^{i\mathbf{k}\cdot\mathbf{r}-i\omega t}\int_{0}^{\infty}d\tau\, e^{-ik_{x}v_{\perp}\left[\sin\left(\phi+\Omega\tau\right)-\sin\phi\right]/\omega+i\left(\omega-k_{z}v_{z}\right)\tau}\cdot\\ & & \left[-ik_{x}\Phi U\cos\left(\phi+\Omega\tau\right)-ik_{z}\Phi\frac{\partial f_{0}}{\partial v_{z}}+ik_{z}\Phi V\cos\left(\phi+\Omega\tau\right)\right]\end{array}](http://localhost/mediawiki/images/math/7/a/0/7a039dc7a4cdc5910fdc2b7c7d2d0cd7.png)
Integrating over angles φ:
![\begin{array}{rcl} f_{1}\left(v_{\perp},v_{z},t\right) & = & 2\pi iqe^{i\mathbf{k}\cdot\mathbf{r}-i\omega t}\sum_{n=-\infty}^{\infty}\int_{0}^{\infty}d\tau\, e^{i\left(\omega-k_{z}v_{z}\right)\tau}e^{-in\Omega\tau}\cdot\\ & & J_{n}^{2}\left(\frac{k_{x}v_{\perp}}{\omega}\right)\left[k_{x}\Phi U\frac{n\omega}{k_{x}v_{\perp}}+k_{z}\Phi\frac{\partial f_{0}}{\partial v_{z}}-k_{z}\Phi V\frac{n\omega}{k_{x}v_{\perp}}\right]\end{array}](http://localhost/mediawiki/images/math/0/7/2/072812b0e38847cb87a6b8eab5c5fcae.png)
If 
, this integral will average out
to 0. Otherwise the integral is infinity. This gives:
![\begin{array}{rcl} f_{1}\left(v_{\perp},v_{z},t\right) & = & 2\pi iqe^{i\mathbf{k}\cdot\mathbf{r}-i\omega t}\sum_{n=-\infty}^{\infty}J_{n}^{2}\left(\frac{k_{x}v_{\perp}}{\omega}\right)\cdot\\ & & \left[k_{x}\Phi U\frac{n\omega}{k_{x}v_{\perp}}+k_{z}\Phi\frac{\partial f_{0}}{\partial v_{z}}-k_{z}\Phi V\frac{n\omega}{k_{x}v_{\perp}}\right]\end{array}](http://localhost/mediawiki/images/math/7/6/f/76ff9ae041a10e934fac4912442ef627.png)
Integrating over velocity space:
![\begin{array}{rcl} n_{1}\left(t\right) & = & \int v_{\perp}dv_{\perp}\int dv_{z}\,2\pi iqe^{i\mathbf{k}\cdot\mathbf{r}-i\omega t}\sum_{n=-\infty}^{\infty}\delta\left(\omega-k_{z}v_{z}-n\Omega\right)\cdot\\ & & J_{n}^{2}\left(\frac{k_{x}v_{\perp}}{\omega}\right)\left[k_{x}\Phi U\frac{n\omega}{k_{x}v_{\perp}}+k_{z}\Phi\frac{\partial f_{0}}{\partial v_{z}}-k_{z}\Phi V\frac{n\omega}{k_{x}v_{\perp}}\right]\end{array}](http://localhost/mediawiki/images/math/b/d/6/bd6dcc605a0f61cb53247cc7e7312dee.png)
One integral is over a delta function:
![\begin{array}{rcl} n_{1}\left(t\right) & = & \int dv_{\perp}\, v_{\perp}2\pi iqe^{i\mathbf{k}\cdot\mathbf{r}-i\omega t}\sum_{n=-\infty}^{\infty}J_{n}^{2}\left(\frac{k_{x}v_{\perp}}{\omega}\right)\cdot\\ & & \left[k_{x}\Phi U\frac{n\omega}{k_{x}v_{\perp}}+k_{z}\Phi\frac{\partial f_{0}}{\partial v_{z}}-k_{z}\Phi V\frac{n\omega}{k_{x}v_{\perp}}\right]_{v_{z}=\left(\omega-n\Omega\right)/k_{z}}\end{array}](http://localhost/mediawiki/images/math/2/0/2/2028e3b7617dac29483f644aa3e4c0be.png)
Using Poisson's equation:
So:
![\begin{array}{rcl} \left(k_{x}^{2}+k_{z}^{2}\right) & = & 8\pi^{2}iq^{2}e^{i\mathbf{k}\cdot\mathbf{r}-i\omega t}\int dv_{\perp}\, v_{\perp}\sum_{n=-\infty}^{\infty}J_{n}^{2}\left(\frac{k_{x}v_{\perp}}{\omega}\right)\cdot\\ & & \left[k_{x}U\frac{n\omega}{k_{x}v_{\perp}}+k_{z}\frac{\partial f_{0}}{\partial v_{z}}-k_{z}V\frac{n\omega}{k_{x}v_{\perp}}\right]_{v_{z}=\left(\omega-n\Omega\right)/k_{z}}\end{array}](http://localhost/mediawiki/images/math/2/c/6/2c613545f60dd7c6553956dcb520d767.png)
