Parallel Wave Propagation

Let us consider wave propagation, though a warm plasma, parallel to the equilibrium magnetic field. For parallel propagation, $k_\perp\rightarrow 0$ , and, hence, from Equation (8.84), $\lambda_s\rightarrow 0$ . Making use of Equation (8.88), the matrix $T_{ij}$ simplifies to

$$T_{ij} = \left( \begin{array}{ccc} [Z(\xi_1)+Z(\xi_{-1})]/2, & {\... ...+Z(\xi_{-1})]/2, & 0 \\ [0.5ex] 0, & 0, &-Z'(\xi_0)\,\xi_0 \end{array} \right),$$ (8.92)

where, again, the only non-zero contributions are from $n=0$ and $n=\pm 1$ . The dispersion relation can be written [see Equations (5.9) and (5.10)]

$${\bf M}\cdot {\bf E} \equiv \left[\left(\frac{c}{\omega}\right)^2... ...-\left(\frac{c\,k}{\omega}\right)^2{\bf I}+{\bf K}\right]\cdot{\bf E} = {\bf0},$$ (8.93)

where

$$M_{11} = M_{22} = 1-\frac{k_\parallel^{\,2}\,c^2}{\omega^2}$$

$$\phantom{=}+\frac{1}{2}\sum_s \frac{{\mit\Pi}_s^{\,2}} {\omega\,\... ...right) + Z\left(\frac{\omega + {\mit\Omega}_s}{k_\parallel\,v_s}\right)\right],$$ (8.94)

$$M_{12} =-M_{21} = \frac{\rm i}{2}\sum_s\frac{{\mit\Pi}_s^{\,2}} {\omega\... ...right) - Z\left(\frac{\omega + {\mit\Omega}_s}{k_\parallel\,v_s}\right)\right],$$ (8.95)

$$M_{33} = 1 - \sum_s \frac{{\mit\Pi}_s^{\,2}} {(k_\parallel\,v_s)^2} \,\,Z'\left(\frac{\omega}{k_\parallel\,v_s}\right),$$ (8.96)

and $M_{13} = M_{31} = M_{23}=M_{32}=0$ .

The first root of Equation (8.93) is

$$1 +\sum_s\frac{2\,{\mit\Pi}_s^{\,2}} {(k_\parallel\,v_s)^2} \left... ...ga}{k_\parallel\,v_s}\,Z\left(\frac{\omega}{k_\parallel\,v_s}\right)\right] =0,$$ (8.97)

with the eigenvector $(0,\,0,\,E_z)$ . Here, use has been made of Equation (8.41). This root evidentially corresponds to a longitudinal, electrostatic plasma wave. In fact, it is easily demonstrated that Equation (8.97) is equivalent to the dispersion relation (8.49) that we found earlier for electrostatic plasma waves, for the special case in which the distribution functions are Maxwellians. The analysis of Section 8.4 implies that the electrostatic wave described by the previous expression is subject to significant damping whenever the argument of the plasma dispersion function becomes less than or comparable with unity: that is, whenever $\omega\stackrel {_{\normalsize <}}{_{\normalsize\sim}}k_\parallel\,v_s$ .

The second and third roots of Equation (8.93) are

$$\frac{k_\parallel^{\,2}\,c^2}{\omega^2} = 1 +\sum_s \frac{{\mit\P... ...arallel \,v_s}\,Z\left(\frac{\omega + {\mit\Omega}_s}{k_\parallel\,v_s}\right),$$ (8.98)

with the eigenvector $(E_x,\, {\rm i}\,E_x,\,0)$ , and

$$\frac{k_\parallel^{\,2}\,c^2}{\omega^2} = 1 +\sum_s \frac{{\mit\P... ...arallel\, v_s}\,Z\left(\frac{\omega - {\mit\Omega}_s}{k_\parallel\,v_s}\right),$$ (8.99)

with the eigenvector $(E_x,\, -{\rm i}\,E_x,\, 0)$ . The former root evidently corresponds to a right-handed circularly polarized wave, whereas the latter root corresponds to a left-handed circularly polarized wave. The previous two dispersion relations are essentially the same as the corresponding fluid dispersion relations, (5.90) and (5.91), except that they explicitly contain collisionless damping at the cyclotron resonances. Roughly speaking, the damping is significant whenever the arguments of the plasma dispersion functions are less than or of order unity. This corresponds to

$$\omega - \vert{\mit\Omega}_e\vert \stackrel {_{\normalsize <}}{_{\normalsize\sim}}k_\parallel\,v_e$$ (8.100)

for the right-handed wave, and

$$\omega-{\mit\Omega}_i\stackrel {_{\normalsize <}}{_{\normalsize\sim}}k_\parallel\,v_i$$ (8.101)

for the left-handed wave.

The collisionless cyclotron damping mechanism is similar to the Landau damping mechanism for longitudinal waves discussed in Section 8.3. In the former case, the resonant particles are those that gyrate about the magnetic field at approximately the same angular frequency as the wave electric field. Note that, in kinetic theory, the cyclotron resonances possess a finite width in frequency space (i.e., the incident wave does not have to oscillate at exactly the cyclotron frequency in order for there to be an absorption of wave energy by the plasma), unlike in the cold plasma model, where the resonances possess zero width.