Wave propagation parallel to the magnetic field

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 Eq. (976), $\lambda_s\rightarrow 0$. Making use of the asymptotic expansion (980), the matrix $T_{ij}$ simplifies to

$$T_{ij} = \left( \begin{array}{ccc} [Z(\xi_1)+Z(\xi_{-1})]/2 ... .../2 & 0 \\ [0.5ex] 0 & 0 &-Z'(\xi_0)\,\xi_0 \end{array}\right),$$ (984)

where, again, the only non-zero contributions are from $n=0$ and $n=\pm 1$. The dispersion relation can be written [see Eq. (435)]

$${\bf M}\!\cdot\!{\bf E} = {\bf0},$$ (985)

where

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

$$+\frac{1}{2}\sum_s \frac{\omega_{p\,s}^{~2}} {\omega\,\,k_z v_s} ... ...k_z\,v_s}\right) + Z\!\left(\frac{\omega + {\Omega}_s}{k_z\,v_s}\right)\right],$$ (986)

$$M_{12} = -M_{21} = \frac{\rm i}{2}\sum_s\frac{\omega_{p\,s}^{~2}} {\omega\... ...}{k_z\,v_s}\right) - Z\left(\frac{\omega + {\Omega}_s}{k_z\,v_s}\right)\right],$$ (987)

$$M_{33} = 1 - \sum_s \frac{\omega_{p\,s}^{~2}} {(k_z\,v_s)^2} \,\,Z'\!\left(\frac{\omega}{k_z\,v_s}\right),$$ (988)

and $M_{13} = M_{31} = M_{23}=M_{32}=0$. Here, $v_s= \sqrt{2\,T_s/m_s}$ is the species-$s$ thermal velocity.

The first root of Eq. (985) is

$$1 +\sum_s\frac{2\,\omega_{p\,s}^{~2}} {(k_z\,v_s)^2} \left[1... ...{k_z\,v_s}\,Z\!\left(\frac{\omega}{k_z\,v_s}\right)\right] =0,$$ (989)

with the eigenvector $(0,0,E_z)$. Here, use has been made of Eq. (935). This root evidentially corresponds to a longitudinal, electrostatic plasma wave. In fact, it is easily demonstrated that Eq. (989) is equivalent to the dispersion relation (945) that we found earlier for electrostatic plasma waves, for the special case in which the distribution functions are Maxwellians. Recall, from Sects. 6.3-6.5, that the electrostatic wave described by the above expression is subject to significant damping whenever the argument of the plasma dispersion function becomes less than or comparable with unity: i.e., whenever $\omega\stackrel {_{\normalsize <}}{_{\normalsize\sim}}k_z\,v_s$.

The second and third roots of Eq. (985) are

$$\frac{k_z^{~2}\,c^2}{\omega^2} = 1 +\sum_s \frac{\omega_{p\,... ..._z v_s}\,Z\!\left(\frac{\omega + {\Omega}_s}{k_z\,v_s}\right),$$ (990)

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

$$\frac{k_z^{~2}\,c^2}{\omega^2} = 1 +\sum_s \frac{\omega_{p\,... ..._z v_s}\,Z\!\left(\frac{\omega - {\Omega}_s}{k_z\,v_s}\right),$$ (991)

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 above two dispersion relations are essentially the same as the corresponding fluid dispersion relations, (514) and (515), except that they explicitly contain collisionless damping at the cyclotron resonances. As before, the damping is significant whenever the arguments of the plasma dispersion functions are less than or of order unity. This corresponds to

$$\omega - \vert{\Omega}_e\vert \stackrel {_{\normalsize <}}{_{\normalsize\sim}}k_z\,v_e$$ (992)

for the right-handed wave, and

$$\omega-{\Omega}_i\stackrel {_{\normalsize <}}{_{\normalsize\sim}}k_z\,v_i$$ (993)

for the left-handed wave.

The collisionless cyclotron damping mechanism is very similar to the Landau damping mechanism for longitudinal waves discussed in Sect. 6.3. In this case, the resonant particles are those which gyrate about the magnetic field with approximately the same angular frequency as the wave electric field. On average, particles which gyrate slightly faster than the wave lose energy, whereas those which gyrate slightly slower than the wave gain energy. In a Maxwellian distribution there are less particles in the former class than the latter, so there is a net transfer of energy from the wave to the resonant particles. 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.