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Next: Low-Frequency Wave Propagation Up: Waves in Cold Plasmas Previous: Cutoff and Resonance

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

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


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

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

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

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

$\displaystyle {\bf v}_g = \frac{\partial\omega}{\partial {\bf k}},$ (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 8.2.)

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

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

next up previous
Next: Low-Frequency Wave Propagation Up: Waves in Cold Plasmas Previous: Cutoff and Resonance
Richard Fitzpatrick 2016-01-23