Table of Contents
Dispersion function documentation for the xrays RF Ray tracing code.
Introduction
This page documents the types of dispersion functions available. As a ray moves through the plasma it must remain a solution of these functions. For a fixed wave frequency \(\omega \), the dispersion function dictates how the wave number \(\vec{k}\) and ray trajectory \(\vec{x}\) interact.
| Symbol | Unit | Description |
|---|---|---|
| \(c \) | \(\frac{m}{s}\) | Speed of light. |
| \(m_{e}\) | \(kg \) | Electron mass |
| \(t_{e}\) | \(K \) | Electron temperature |
| \(n_{e}\) | \(m^{-3}\) | Electron Denisty |
| \(q \) | \(C \) | Fundamental Charge |
| \(\vec{B}\) | \(T \) | Magnetic field |
| \(\epsilon_{0}\) | \(\frac{F}{m}\) | Vacuum permittivity |
| \(k_{B}\) | \(\frac{J}{K}\) | Boltzmann constant |
| \(\omega_{pe}\) | \(s^{-1}\) | Electron Plasma Frequency \(\omega_{pe}=\frac{n_{e}q^{2}}{\epsilon_{0}m_{e}c}\) |
| \(\omega_{ce}\) | \(s^{-1}\) | Electron Cyclotron Frequency \(\omega_{ce}=\frac{q\left|\vec{B}\right|}{m_{e}}\) |
| \(\omega_{h}\) | \(s^{-1}\) | Upper Hybrid Frequency \(\omega_{h}^{s}=\omega_{pe}^{2}+\omega_{ce}^{2}\) |
| \(\omega_{r}\) | \(s^{-1}\) | Right cutoff Frequency \(\omega_{r}=\frac{1}{2}\left(\omega_{ce}+\sqrt{\omega^{2}_{ce}+4\omega_{pe}^{2}}\right)\) |
| \(\omega_{l}\) | \(s^{-1}\) | Left cutoff Frequency \(\omega_{r}=\frac{1}{2}\left(-\omega_{ce}+\sqrt{\omega^{2}_{ce}+4\omega_{pe}^{2}}\right)\) |
| \(v_{th}\) | \(\frac{m}{s}\) | Thermal velocity \(v_{th}=\sqrt{\frac{k_{B}t_{e}}{m_{e}}}\) |
| \(\vec{n}\) | \(1 \) | \(n=\frac{\vec{k}c}{\omega}\) |
Normalization
The dispersion functions use normalized quantities for frequncy \(\omega \) and time \(t \). These are scaled to the speed of light \(c \).
| Symbol | Real Unit | Modified | Modified Unit | Description |
|---|---|---|---|---|
| \(\omega \) | \(s^{-1}\) | \(\omega'=\frac{\omega}{c}\) | \(m^{-1}\) | Wave frequency |
| \(\vec{k}\) | \(m^{-1}\) | \(\vec{k}\) | \(m^{-1}\) | Wave number |
| \(t \) | \(s \) | \(t'=tc \) | \(m \) | Time |
| \(\vec{x}\) | \(m \) | \(\vec{x}\) | \(m \) | Postion |
| \(v_{p}=\frac{\omega}{k}\) | \(\frac{m}{s}\) | \(v'_{p}=\frac{\omega'}{k}\) | \(1 \) | Phase velocity |
| \(v_{g}=\frac{\partial\omega}{\partial k}\) | \(\frac{m}{s}\) | \(v'_{g}=\frac{\partial\omega'}{\partial k}\) | \(1 \) | Group velocity |
Wave propagation
From any dispersion function, the wave propagation equations are obtained from derivatives of the dispersion relation.
\begin{equation}\frac{\partial\vec{x}}{\partial {t}}=-\frac{\frac{\partial D}{\partial\vec{k}}}{\frac{\partial D}{\partial\omega}}\end{equation}
\begin{equation}\frac{\partial\vec{k}}{\partial {t}}=\frac{\frac{\partial D}{\partial\vec{x}}}{\frac{\partial D}{\partial\omega}}\end{equation}
RF waves are traced by integrating this system of equations.
Available Dispersion Functions
The following dispersion functions are available in xrays.
Simple
This disperison function represents a wave in a vacuum.
\begin{equation}D\left(\vec{x},\vec{k},\omega\right)=\frac{\vec{k}\cdot\vec{k}}{\omega^{2}}-1\equiv 0 \end{equation}
It has no resonances or cutoffs.
Bohm Gross
This dispersion function now accounts for how occilations in the plasma propagate.
\begin{equation}D\left(\vec{x},\vec{k},\omega\right)=\omega_{pe}+\frac{3}{2}\vec{k}\cdot\vec{k}v^{2}_{th}-\omega^{2}\equiv 0 \end{equation}
It has no resonances. Waves cannot propagate below \(\omega_{pe}\) and \(v_{g}\) can never exceed \(v_{th}\).
Ordinary Wave
This disperison function represents a wave with a \(\vec{E}_{1}||\vec{B}_{0}\). That means the electric field occilates parallel to the magnetic field.
\begin{equation}D\left(\vec{x},\vec{k},\omega\right)=1-\frac{\omega^{2}_{pe}}{\omega^{2}}-\vec{n}_{\perp}\cdot\vec{n}_{\perp}\equiv 0 \end{equation}
This wave is cut off below \(\omega_{pe}\).
Extra Ordinary Wave
This disperison function represents a wave with a \(\vec{E}_{1}\perp\vec{B}_{0}\). That means the electric field occilates perpendicular to the magnetic field.
\begin{equation}D\left(\vec{x},\vec{k},\omega\right)=1-\frac{\omega_{pe}^2}{\omega^{2}}\frac{\omega^{2}-\omega_{pe}^2}{\omega^{2}-\omega_{h}^2}-\vec{n}_{\perp}\cdot\vec{n}_{\perp}\equiv 0 \end{equation}
This mode has This wave has two branches. One branch is cannot not progagate below the \(\omega_{r}\) cutoff. The other branch cannot not progagate below the \(\omega_{l}\) cutoff. As the wave approches the upper hybrid fequency, wave propagation stops as the wave number approces \(\left|\vec{k}\right|\rightarrow\infty \).
Cold Plasma
This disperison function represents a wave in a cold plasma medium.
\begin{equation}D\left(\vec{x},\vec{k},\omega\right)=det\left(\vec{\epsilon}+\vec{n}\vec{n}-\vec{n}\cdot\vec{n}\vec{I}\right)\equiv 0 \end{equation}
The quantity \(\vec{\epsilon}\) is the diaelectric tensor. Using Onsager symmetries, this tensor is defined as
\begin{equation}\vec{\epsilon}=\left(\begin{array}{ccc}\epsilon_{11}&\epsilon_{12}&0\\-\epsilon_{12}&\epsilon_{11}&0\\0&0&\epsilon_{33}\end{array}\right)\end{equation}
where
\begin{equation}\epsilon_{11}=1-\sum_{s}\frac{\frac{\omega^{2}_{p}}{\omega^{2}}}{1-\frac{\Omega^{2}_{c}}{\omega^{2}}}\end{equation}
\begin{equation}\epsilon_{12}=-i\sum_{s}\frac{\frac{\Omega_{c}}{\omega}\frac{\omega^{2}_{p}}{\omega^{2}}}{1-\frac{\Omega^{2}_{c}}{\omega^{2}}}\end{equation}
\begin{equation}\epsilon_{33}=1-\sum_{s}\frac{\omega^{2}_{p}}{\omega^{2}}\end{equation}
Note here we are including the plasma frequency for ions aswell. This dispersion function is effectively a super position of the O-Mode and X-Mode dispersion functions and as such has the same cutoffs and resonaces.
Developing new dispersion functions
This section is intended for code developers and outlines how to create new dispersion functions. New dispersion functions can be created from a subclass of dispersion::dispersion_function or any other existing dispersion_function class and overloading class methods. For convinence the dispersion::physics class contains several defined physical constants.
When a new dispersion function is subclassed from dispersion::dispersion_function an implimentation must be provided for the pure virtual method dispersion::dispersion_function::D.
