In the last installment, we examined a basic model of waves in cold plasmas. The two defining requirements were that the thermal fluctuations were small and ignorable (the cold piece) and that the plasma was not subjected to a magnetic field (unmagnetized). In this column, the requirement for a non-zero magnetic field ($$\vec B = 0$$) is relaxed while the requirement that the plasma is cold stays. The resulting types of wave motion are a lot richer. In contrast to the unmagnitized case, where there are two isotropic populations of electrons and ions, the presence of the magnetic field breaks the isotropy and motion of the charged particles is strongly different along the field lines as opposed to perpendicular to them.

Following again Gurnett and Bhattacharjee, the magnetic field takes on the form of

\[ \vec B = \vec B^{(0)} + \vec B^{(1)} \; ,\]

where $$\vec B^{(0)}$$ is a constant vector field and $$\vec B^{(1)}$$ carries all the spatial and temporal variations, assumed to be small compared to the constant piece. A similar form results for the density of each species. As in the unmagnetized case, the particle velocities and the electric field are zero to zeroth order.

The equations of motion for particles of the individual species, calculated from the Lorentz force law, is

\[ m_s \frac{\partial \vec v_s^{(1)}}{\partial t} = e_s \left[ \vec E^{(1)} + \vec v_s^{(1)} \times \vec B^{(0)} \right] \; .\]

Taking the Fourier transform of this equation leads to the three scalar equations
\[ – i \omega m_s v_{sx} = e_s(E_x + v_{sy}B ) \; ,\]
\[ – i \omega m_s v_{sy} = e_s(E_y – v_{sx}B ) \; ,\]
and
\[ – i \omega m_s v_{sz} = e_s E_z \; ,\]

in frequency space.

Dividing both sides by $$-i m_s$$ and using the cyclotron frequency

\[ \omega_{cs} = \frac{e_s B}{m_s} \]

on the right-hand side, allows for these three equations to be written in the matrix form as

\[ \left[ \begin{array}{ccc} -i \omega & – \omega_{cs} & 0 \\ \omega_{cs} & -i \omega & 0 \\ 0 & 0 & -i \omega \end{array} \right] \left[ \begin{array}{c} v_{sx} \\ v_{sy} \\ v_{sz} \end{array} \right] = \frac{e_s}{m_s} \left[ \begin{array}{c} E_x \\ E_y \\ E_z \end{array} \right] \; \]

or more compactly as

\[ {\mathbf M} \cdot \vec v = \vec E \; ,\]

where it is understood that all quantities are the Fourier Transforms of the corresponding state-space terms.

The goal is to express the velocties in terms of the electric field and, from them, the current density. To this end, the inverse of the matrix $${\mathbf M}$$ on the left-hand side is

\[{\mathbf M}^{-1} \equiv \left[ \begin{array}{ccc} \frac{i\omega}{\omega^2 – \omega_{cs}^2} & -\frac{\omega_{cs}}{\omega^2 – \omega_{cs}^2} & 0 \\ \frac{\omega_{cs}}{\omega^2 – \omega_{cs}^2} & \frac{i\omega}{\omega^2 – \omega_{cs}^2} & 0 \\0 & 0& \frac{i}{\omega} \end{array} \right] \; \]

from which the velocities can be written as

\[ \vec v = {\mathbf M}^{-1} \vec E \]

and the current densisty as

\[ \vec J = \sum_{s} n_s^{(0)}e_s \vec v \; .\]

Assuming the generalized Ohm’s law

\[ \vec J = \overleftrightarrow{\sigma} \vec E \; ,\]

immediately gives the form of the conductivity tensor

\[ \overleftrightarrow{\sigma} = \sum_s \frac{n_s^{(0)} e_s^2}{m_s} \left[ \begin{array}{ccc} \frac{i\omega}{\omega^2 – \omega_{cs}^2} & -\frac{\omega_{cs}}{\omega^2 – \omega_{cs}^2} & 0 \\ \frac{\omega_{cs}}{\omega^2 – \omega_{cs}^2} & \frac{i\omega}{\omega^2 – \omega_{cs}^2} & 0 \\0 & 0& \frac{i}{\omega} \end{array} \right] \; .\]

Plugging this last relation into the expression for the dielectric tensor

\[ \overleftrightarrow{K} = \overleftrightarrow{1} – \frac{\overleftrightarrow{\sigma}}{i \omega \epsilon_0} \]

gives an explicit form for the dielectric tensor for a magnetized, cold plasma

\[ \overleftrightarrow{K} = \left[ \begin{array}{ccc} 1 – \sum_s \frac{\omega_{ps}^2}{\omega^2 – \omega_{cs}^2} & -i \sum_s \frac{\omega_{ps}^2 \omega_{cs}}{\omega(\omega^2 – \omega_{cs}^2} & 0 \\ i \sum_s \frac{\omega_{ps}^2 \omega_{cs}}{\omega(\omega^2 – \omega_{cs}^2)}& 1 – \sum_s \frac{\omega_{ps}^2}{\omega^2 – \omega_{cs}^2} & 0\\0 &0 & 1 – \sum_s \frac{\omega_{ps}^2}{\omega^2} \end{array} \right] \; .\]

This expression, being unwieldy, suggests a shortcut notation be adopted. The usual convention is to define three quantities:

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

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

and

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

In terms of these expressions, the dielectric tensor is concisely written as

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

To get the dispersion relation, we need to relate the wave propagation direction to the magnetic field direction. Generally, the propagation direction will not be parallel to the magnetic field but will be off by an angle $$\theta$$. The plane containing the two vectors can be taken, without loss of generality, to be the $$x-z$$ plane. In these terms, the propagation vector $$\vec n$$ takes the form

\[\vec n \equiv \left[ \begin{array}{c}n \sin \theta \\ 0 \\ n \cos \theta \end{array} \right] \; .\]

The ancilliary tensors associated with $$\vec n$$ take the form

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

and

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

Combining the last term with the dielectric tensor gives the form for the dispersion tensor

\[ \overleftrightarrow{D}(\vec n, \omega) = \left[ \begin{array}{ccc} S-n^2 \cos^2 \theta & -i D & n^2 \cos \theta \sin \theta \\ i D & S-n^2 &0 \\ n^2 \cos \theta \sin \theta & 0 & P – n^2 \sin^2 \theta \end{array} \right] \; .\]

We obtain the dispersion relations for the various waves by setting the determinant of $$\overleftrightarrow{D}(\vec n, \omega)$$ equal to zero. For reasons that will become obvious in the next treatment, the expressions $$S$$ and $$D$$ are related to the terms $$R$$ and $$L$$ (for right- and left-handed) through the transfomations

\[ R = S + D \]

and

\[ L = S – D \; .\]

It is a simple matter to invert these relations and express $$S$$ and $$D$$ in terms of $$R$$ and $$L$$ to give

\[ S = \frac{1}{2}[ R + L ] \]

and

\[ D = \frac{1}{2}[ R – L ] \; .\]

Of particular convenience is the further expression

\[ S^2 – D^2 = RL \; .\]

Now we are in position to calculate the determinant of the dispersion tensor. It is convenient to expand along the bottom row. Doing so yields

\[ |\overleftrightarrow{D}(\vec n, \omega)| = -n^4 \sin^2 \theta \cos^2 \theta (S-n^2) \\ + (P – n^2 \sin^2 \theta) \left[ (S-n^2)(S-n^2\cos^2 \theta) – D^2 \right]\; .\]

After some careful but straightforward expasions and simplifications and using $$S^2-D^2 = RL$$, the determinant simplifies to

\[ |\overleftrightarrow{D}(\vec n, \omega)| = [S\sin^2 \theta + P \cos^2 \theta ] n^4 \\ – [RL\sin^2 \theta +PS (1+\cos^2 \theta)] n^2 + RLP \; .\]

Note that terms proportional to $$n^6$$ have canceled out. As displayed, this expression is readily solved for $$n^2$$ using the quadratic formula with

\[ A = S\sin^2 \theta + P \cos^2 \theta \; , \]

\[ B = RL\sin^2 \theta +PS (1+\cos^2 \theta) \; ,\]

and

\[ C = RLP \; .\]

Some of the wave physics can be gleaned just by examining the discriminant

\[ F^2 = B^2 – 4AC \; . \]

The first term on the right-hand side can be expanded and simplified, particularly by using $$1+\cos^2 \theta = 2 – \sin^2 \theta$$, to

\[ B^2 = (RL – PS)^2 \sin^4 \theta + 4 P^2 S^2 \cos^2 \theta + 4RLPS \sin^2 \theta \; . \]

Likewise the last term becomes

\[ 4AC = 4RLPS \sin^2 \theta – 4RLP^2 \cos^2 \theta \; .\]

Combining, grouping, and simplifying yields

\[F^2 = B^2 – 4AC = (RL-PS)^2 \sin^4 \theta + 4P^2 D^2 \cos^2 \theta \; . \]

Clearly the discriminant is always positive-definite, which in turn implies that the roots are given by

\[ n^2 = \frac{-B \pm F}{2A} \; ,\]

which are always real quantities. Thus the index of refraction is either purely real or purely imaginary and either the waves propagate without damping or they don’t propagate at all.

The last result needed before mining for specific wave types comes from a simple but important manipulation of the characteristic equation by rewriting

\[ B = RL \sin^2 \theta + PS(\sin^2 \theta + 2 \cos^2 \theta) \]

and

\[ C = RLP(\sin^2 \theta + \cos^2 \theta) \; . \]

The steps for arriving at the desired result are these. Substitute the new expressions for $$B$$ and $$C$$ into the characteristic equation. Expand and collect on $$\sin^2 \theta$$ and $$\cos^2 \theta$$ to get

\[ \sin^2 \theta [Sn^4 – (RL+PS) n^2 +RLP] \\ + \cos^2\theta [Pn^4 -2 P S n^2 + RLP ] = 0 \; .\]

Solving for $$\tan^2 \theta$$ gives

\[ \tan^2 \theta = – \frac{[Pn^4 -2 P S n^2 + RLP ]}{[Sn^4 – (RL+PS) n^2 +RLP]} \; . \]

The denominator can be factored as

\[ Sn^4 – (RL + PS)n^2 + RLP = (Sn^2 – RL)(n^2 – P) \; .\]

The numerator is a bit trickier. To factor, first substitute $$2 S = R + L $$. This step leaves the numerator in the form

\[ Pn^4 – P(R+L)n^2 + RLP = P(n^4-(R+L)n^2 + RL) \; , \]

which leads to the factoring

\[ P(n^4 – (R+L)n^2 + RL ) = P(n^2 – R)(n^2 – L) \; .\]

Substituting these new expressions in yields the final, more convenient arrangement

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

In the next column, we start with this last relationship and derive various wave modes and their behavior.