Waves in Unmagnetized Plasmas

Let us now investigate the cold-plasma dispersion relation in detail. It is instructive to first consider the limit in which the equilibrium magnetic field is zero. In the absence of a magnetic field, there is no preferred direction, so we can, without loss of generality, assume that ${\bf k}$ is directed along the $z$-axis (i.e., $\theta=0$). In the zero magnetic field limit (i.e., ${{\mit\Omega}}_e, {{\mit\Omega}}_i \rightarrow 0$), the eigenmode equation (5.42) reduces to

$$\left(\!\begin{array}{ccc} P-n^2, & 0, & 0\\ 0, & P-n^2, & 0 \\ ... ...! \right)\left(\!\begin{array}{c} E_x\\ E_y\\ E_z\end{array}\!\right) = {\bf0},$$ (5.68)

where

$$P \simeq 1 - \frac{{{\mit\Pi}}_e^{2}}{\omega^2}.$$ (5.69)

Here, we have neglected ${{\mit\Pi}}_i$ with respect to ${{\mit\Pi}}_e$.

It is clear from Equation (5.68) that there are two types of waves. The first possesses the eigenvector $(0,\,0,\,E_z)$, and has the dispersion relation

$$1- \frac{{{\mit\Pi}}_e^{2}}{\omega^2} = 0.$$ (5.70)

The second possesses the eigenvector $(E_x,\, E_y,\, 0)$, and has the dispersion relation

$$1 - \frac{ {{\mit\Pi}}_e^{2}}{\omega^2} - \frac{k^2\,c^2}{\omega^2} = 0.$$ (5.71)

Here, $E_x$, $E_y$, and $E_z$ are arbitrary non-zero quantities.

The former wave has ${\bf k}$ parallel to ${\bf E}$, and is, thus, a longitudinal (with respect to the electric field) wave. This wave is known as the plasma wave, and possesses the fixed frequency $\omega = {{\mit\Pi}}_e$. Now, if ${\bf E}$ is parallel to ${\bf k}$ then it follows from Equation (5.5) that ${\bf B} = {\bf0}$. In other words, the plasma wave is purely electrostatic in nature. In fact, the plasma wave is an electrostatic oscillation of the type discussed in Section 1.4. Because $\omega$ is independent of ${\bf k}$, the so-called group-velocity (Fitzpatrick 2013),

$${\bf v}_g = \frac{\partial\omega}{\partial {\bf k}},$$ (5.72)

associated with a plasma wave, is zero. As is demonstrated in Section 6.7, the group-velocity is the propagation velocity of localized wave packets. It is clear that the plasma wave is not a propagating wave, but instead has the property than an oscillation set up in one region of the plasma remains localized in that region. It should be noted, however, that in a “warm” plasma (i.e., a plasma with a finite thermal velocity) the plasma wave acquires a non-zero, albeit very small, group velocity. (See Section 7.2.)

The latter wave is a transverse wave, with ${\bf k}$ perpendicular to ${\bf E}$. There are two independent linear polarizations of this wave, which propagate at identical velocities, just like a vacuum electromagnetic wave. The dispersion relation (5.71) can be rearranged to give

$$\omega^2 = {{\mit\Pi}}_e^{2} + k^2 c^2,$$ (5.73)

showing that this wave is just the conventional electromagnetic wave, whose vacuum dispersion relation is $\omega^2=k^2 c^2$, modified by the presence of the plasma. An important conclusion, which follows immediately from the previous expression, is that this wave can only propagate if $\omega\geq {{\mit\Pi}}_e$. Because ${{\mit\Pi}}_e$ is proportional to the square root of the electron number density, it follows that electromagnetic radiation of a given frequency can only propagate through an unmagnetized plasma when the electron number density falls below some critical value.