In tha last installment, the structure of plasma waves propagating along the local magnetic field direction were derived and analyzed. This column is mostly focued on the similar type of analysis for the case of waves propagating perpendicular to the local magnetic field. Along the way, a specific point that was glossed over the in the last analysis will be explored more fully. There are two reasons for this. The first is strictly the desire for these columns to be as complete as possible and a subsequent reading of the last piece as a refresher for this one showed a deficit. The second reason is that the analysis for the perpendicular case requires that point.

As before, the central relationship is the tangent form of the dispersion relation

\[ \tan^2 \theta = -\frac{P(n^2-R)(n^2 – L)}{(Sn^2 – RL)(n^2 – P)} \; , \]

where $$\theta$$ is the angle between the direction of wave propagation and the magnetic field, $$n$$ is the index of refraction, and the terms $$S$$, $$P$$, $$R$$, and $$L$$ are

\[ S = 1 – \sum_s \frac{\omega_{ps}^2}{\omega^2 – \omega_{cs}^2} \; ,\]

\[ P = 1 – \sum_s \frac{\omega_{ps}^2}{\omega^2} \; ,\]

\[ R = 1 – \sum_s \frac{\omega_{ps}^2}{\omega(\omega+\omega_{cs})} \; ,\]

and

\[ L = 1 – \sum_s \frac{\omega_{ps}^2}{\omega(\omega-\omega_{cs})} \; ,\]

respectively.

Perpendicular propagation is obtained by setting $$\theta = \pi/2$$. Since the tangent diverges at $$\pi/2$$, the appropriate dispersion relations result when the denominator goes to zero (i.e. the expression has a pole). The poles of the tangent form of the dispersion relation occur at

\[ n^2 = \frac{RL}{S} \]

and

\[ n^2 = P \; .\]

It is interesting to note that, unlike, the parallel propagation case, there are only two dispersion relations rather than three. Finding the eigenvectors leads to a subtle point that was glossed over in the last column. The intent here is to perform the computation in the proper amount of detail and then to explain why the gloss was permissible in the case of parallel propagation.

To determine the corresponding eigenvectors, it is conceptually most clean to go back to the defining equation for the dispersion tensor, which is given by:

\[ \left( \left( {\vec n}^{\times} \right)^2 + \overleftrightarrow{K} \right) \left[ \begin{array}{c} E_x \\ E_y \\ E_z \end{array} \right] = 0 \; , \]

with

\[ \overleftrightarrow{K} = \left[ \begin{array}{ccc} S & -i D & 0 \\ i D & S &0 \\ 0 & 0 & P \end{array} \right] \; ,\]

and

\[\left( {\vec n}^{\times} \right)^2 = \left[ \begin{array}{ccc} -n^2 \cos^2 \theta & 0 & n^2 \sin \theta \cos \theta \\0 & -n^2 & 0 \\n^2 \sin \theta \cos \theta & 0 & -n^2 \sin ^2 \theta \end{array} \right] \; .\]

When the direction of propagation is perpendicular, then $$\theta = \pi/2$$, and the first term becomes

\[\left( {\vec n}^{\times} \right)^2 = \left[ \begin{array}{ccc} 0 & 0 & 0 \\0 & -n^2 & 0 \\0 & 0 & -n^2 \end{array} \right] \; .\]

The resulting three equations for the electric field are:

\[S E_x – i D E_y = 0 \; ,\]

\[i D E_x + S E_y = n^2 E_y \; , \]

and

\[ P E_z = n^2 E_z \; . \]

The first equation can be solve to express $$E_x$$ in terms of $$E_y$$ as:

\[ E_x = \frac{i D}{S} E_y \, \]

independent of the actual dispersion relation. The solutions of the other two equations, however, depends intimately on the particulars of the dispersion relation.

For the case where $$n^2 = P$$, the $$y$$-equation becomes

\[ \left( S^2 – D^2 – PS \right) E_y = 0 \; , \]

for which the only solution is $$E_y = 0$$, since $$S^2 – D^2 – PS \neq 0$$. The third equation becomes

\[ P E_z = P E_z \; ,\]

which is satisfied with $$E_z$$ being assigned any value $$E_0$$. The corresponding eigenvector

\[ \left[ \begin{array}{c} 0 \\ 0 \\ P \end{array} \right] \]

is aligned along the magnetic field while its direction of propagation is in the plane perpendicular. This is known, according to Gurnett and Bhattacharjee as the ordinary mode. Note that its frequency can be arbitrary unlike the $$P$$ mode in the parallel case.

In contrast, in the case where $$n^2 = RL/S$$, the $$y$$-equation

\[ \left( S^2 – D^2 – RL \right) E_y = 0 \]

has a non-trivial solution since

\[ S^2 – D^2 = RL \]

is an identity. And so the equation is satisfied with $$E_y$$ being assigned any value $$E_0$$. The $$z$$-equation

\[ P E_z = RL/S E_z \]

can only be satisfied by $$E_z = 0$$. The corresponding eigenvector is

\[ \left[ \begin{array}{c} \frac{i D}{S} E_0 \\ E_0 \\ 0 \end{array} \right] \; . \]

This wave mode has its electric field in the $$x-y$$ plane, meaning that it has components neccessarily along and perpendicular to the direction of propagation. The parallel component yields an electrostatic piece while the perpendicular component yields an electromagnetic one. In addtion, its magnitude is no longer independent of the local magnetic field, as was the other cases examine up to this point. The dependence on magetic field strength comes from the $$S$$ and $$D$$ terms that scale $$E_x$$. This wave is termed extraordinary and leads to the hybrid resonances that are discussed at length in Section 4.4.2 of Gurnett and Bhattacharjee.

One last note before closing out this column. Last month, a seemingly simpler way was presented to find the eigenvectors for wave propagation parallel to the local magnetic field. The reason for the shortcut used was that when $$\theta = 0$$

\[\left( {\vec n}^{\times} \right)^2 = \left[ \begin{array}{ccc} -n^2 & 0 & 0 \\0 & -n^2 & 0 \\ 0 & 0 & 0 \end{array} \right] \; .\]

The basic equation

\[ \left( \left( {\vec n}^{\times} \right)^2 + \overleftrightarrow{K} \right) \left[ \begin{array}{c} E_x \\ E_y \\ E_z \end{array} \right] = 0 \; \]

becomes

\[ \left[ \begin{array}{ccc} S & -i D & 0 \\ i D & S &0 \\ 0 & 0 & P \end{array} \right] \left[ \begin{array}{c} E_x \\ E_y \\ E_z \end{array} \right] = \left[ \begin{array}{ccc} n^2 & 0 & 0 \\0 & n^2 & 0 \\ 0 & 0 & 0 \end{array} \right] \left[ \begin{array}{c} E_x \\ E_y \\ E_z \end{array} \right] \; ,\]

which is a basic eigenvalue problem for $$\overleftrightarrow{K}$$ and this is why the last column used a simple method for finding only its eigenvectors.