Waves in Hot Magnetized Plasma #
Just like in the cold plasma dispersion relation, finding the dielectric tensor helps with the dispersion relation. The process involves multiple steps. We will only look at the overview here.
The process utilizes the method of characteristics, where the perturbation of \( f(\vec v) \) due to the wave is found by integrating along unperturbed orbits. Note: the orbits will be spirals, \( \vec R(t) = \vec R(\vec x(t), \vec v(t), t) \). The steps of the process are:
- Calculate \( \left. \dv{f}{t} \right|_{\vec R} \) which is the change in the distribution function along this trajectory.
- Linearize the evolution equation with \( \vec E = \vec E_1 \) and \( \vec B = \vec B_0 + \vec B_1 \)
- Express the perturbations as Fourier expansions
- Integrate the equation of motion for the particles, and get expressions for the particl velocity
- Calculate \( \vec J = q \int \vec v f_1 \dd \vec v \)
- Use \( \vec J = \vec \sigma \cdot \vec E \) and \( \vec \epsilon = \vec \epsilon_b + i \vec \sigma / \omega \) to construct the effective dielectric tensor \( \vec K = \vec \epsilon / \epsilon_0 \)
We will, as usual, use a coordinate system such that \( \vec B_0 = B_0 \vu z \). We define \( \psi \) to be the angle made by \( \vec k \) in the x-y plane:
\[\vec B_0 = B_0 \vu z\] \[k_\perp ^2 = k_x ^2 + k_y ^2\] \[k_x = k_\perp \cos \psi \qquad k_y = k _\perp \sin \psi\]
The form of the effective dielectric tensor (from Swanson) is:
\[\vec K = \begin{bmatrix} K_1 + \sin^2 \psi K_0 & K_2 - \cos \psi \sin \psi K_0 & \cos \psi K_4 + \sin \psi K_5 \\ -K_2 - \cos \psi \sin \psi K_0 & K_1 + \cos ^2 \psi K_0 & \sin \psi K_4 - \cos \psi K_5 \\ \cos \psi K_6 - \sin \psi K_7 & \sin \psi K_6 + \cos \psi K_7 & K_3 \end{bmatrix}\]where the dielectric coefficients \( K_j \) can be found in the text.
Recall that the wave equation
\[\vec n \cross (\vec n \cross \vec E) + \frac{\vec \epsilon}{\epsilon_0} \cdot \vec E = 0\]is still valid. For \( \vec k = k_x \vu x + k_y \vu y \) this is:
\[\begin{bmatrix} \kappa_{xx} - k_z ^2 - k_y ^2 & \kappa_{xy} + k_x k_y & \kappa_{xz} + k_x k_z \\ \kappa_{yx} + k_y k_x & \kappa_{yy} + k_z ^2 - k_x ^2 & \kappa_{yz} + k_x k_z \\ \kappa_{zx} + k_z k_x & \kappa_{zy} + k_z k_y & \kappa_{zz} - k_{\perp}^2 \end{bmatrix} \begin{bmatrix} E_x \\E_y \\E_z \end{bmatrix} = 0\]where \( \kappa_{ij} \equiv (\omega^2 / c^2)K_{ij} \). The dispersion relation is obtained by setting the determinant of the matrix to zero. The dispersion relation may be written either in terms of dimensionless quantities \( K_{ij}, n_x, n_y, n_z \) or in terms of \( \kappa_{ij}, k_x, k_y, k_z \) as is convenient.
If we define \( \gamma \equiv k_z ^2 - \kappa_1 \) then the dispersion relation can be written:
\[\begin{aligned} 0 & = \kappa_3 \left[ \gamma (\gamma - \kappa_0 + k_\perp ^2) + \kappa_2 ^2 \right] \\ & \quad + k_\perp ^2 \left[(\gamma - \kappa_0 + k_\perp ^2)\kappa _1 - \kappa_2 ^2 \right] \\ & \quad + \kappa_4 (\gamma - \kappa_0 + k_\perp ^2)(2 k_\perp k_z + \kappa_4) \\ & \quad - \kappa_5 \left[\gamma \kappa_5 + 2 \kappa_2(k_\perp k_z + \kappa_4)\right] \end{aligned}\]Kinetic Effects on R-wave #
For parallel propagation, \( k_\perp = \lambda = 0 \) and the tensor elements simplify since the infinite sums reduce to either one or two terms. Additionally, we find that \( K_0 = K_4 = K_5 = 0 \), so the dispersion relation is
\[(\gamma ^2 + \kappa _2 ^2) \kappa_3 = 0\]The solutions are \( \kappa_3 = 0 \) and \( k_z ^2 = \kappa _1 \pm \kappa_2 \). The first solution is unaffected by the magnetic field, and is identical to the ion-acoustic wave dispersion relation. This is the equivalent of the O-wave in the cold plasma dispersion relation.
The other two roots are the R- and L-waves, whose dispersion relation is:
\[\begin{aligned} n_{R,L}^2 & = K_1 \pm i K_2 \\ & = 1 + \sum_j \frac{\omega_{pj}^2}{\omega k_z v_{lj}} \left[Z_{1j}\left( \frac{1 \pm \epsilon_j}{2} \right) + Z_{-1j} \left(\frac{1 \mp \epsilon_j}{2}\right)\right] \end{aligned}\]where
\[Z_{\pm1} = \left( 1 - \frac{k_z - v_0}{\omega}\right) Z(\zeta_{\pm 1}) + \frac{k_z v_l}{2 \omega} \left( 1 - \frac{T_\perp}{T_\parallel} \right) Z^\prime (\zeta_{\pm 1})\] \[v_0 \equiv \text{drift velocity}\] \[v_l ^2 = \frac{2 k_B T_\parallel}{m} \equiv \text{longitudinal thermal speed}\] \[Z(\zeta) = \frac{1}{\sqrt{\pi}} \int _{-\infty} ^{\infty} \frac{e^{-x^2}}{x - \zeta} \dd x \quad \text{(Plasma dispersion function)}\] \[\epsilon_j \equiv q_j / |q_j|\] \[\zeta_{\pm j} = \frac{\omega \pm \omega_{cj} - k_z v_0}{k_z v_{lj}}\]
With no drift (\( v_0 = 0 \)) and isotropic temperature (\( T_\perp = T_\parallel \)):
\[n_R ^2 = 1 + \frac{\omega_{pe}^2}{\omega k_z v_{le}}Z_{-1 e} + \frac{\omega_{pi}^2}{\omega k_z v_{li}} Z_{+1i}\]For large values of \( \omega /k_z v_{li} \),
\[Z_{+1i} = Z(\zeta_{+1i}) = Z\left( \frac{\omega + \omega_{ci}}{k_z v_{li}} \right) \approx 0\]And at the electron cyclotron resonance, \( \omega = \omega_{ce} \rightarrow \zeta_{-1e} = 0 \) so \( Z_{-1e} = i \sqrt{\pi} \) \[\rightarrow n_R ^2 = 1 + i \sqrt{\pi} \frac{\omega_{pe}^2}{\omega k_z v_{le}}\]
Slightly away from resonance, for weak damping \( |\omega_i| \ll |\omega_r| \):
\[\frac{\omega_i}{\omega_r} \sim \frac{\sqrt{\pi} \omega_{pe}^2}{\omega_r k_z v_e \left[ 2 + \frac{\omega_{pe}^2\omega_{ce}}{\omega_r (\omega_r - \omega_{ce})^2}\right]} \exp\left[ - \left( \frac{\omega_r - \omega_{ce}}{k_z v_x} \right)^2 \right]\]The damping shrinks exponentially away from resonance, but the denominator also grows. Far enough away from resonance, the growing denominator would break the \( |\omega_i| / |\omega_r| \ll 1 \) condition. Away from resonance, we do see weak damping, and the cold plasma dispersion relation is valid.
Expanding to higher order in \( Z(\zeta_{-1e}) \):
\[\textcolor{cadetblue}{n_R ^2 = 1} \textcolor{darkgoldenrod}{- \frac{\omega_{pe}^2}{\omega (\omega - \omega_{ce})}} \left[\textcolor{darkgoldenrod}{1} + \textcolor{darkviolet}{\frac{k_z ^2 v_e ^2}{2(\omega - \omega_{ce})^2}}\right] + \textcolor{tomato}{\frac{i \sqrt{\pi} \omega_{pe}^2}{\omega k_z v_e} \exp\left[- \left( \frac{\omega - \omega_{ce}}{k_z v_e} \right)^2 \right]}\] \[\textcolor{cadetblue}{\text{EM waves in vacuum}}\] \[\textcolor{darkgoldenrod}{\text{Cold Plasma Dispersion Relation}}\] \[\textcolor{darkviolet}{\text{Fluid Warm Plasma Dispersion Relation}}\] \[\textcolor{tomato}{\text{Kinetic Model}}\]
Finite Larmor Orbit Effects: Physical Interpretation of Cyclotron Damping #
Why does the wave lose energy and the particles gain energy due to cyclotron damping? What is the mechanism of energy transfer between the particles and the exciting wave?
Consider a particle with velocity \( v_z \) traveling parallel to the wave with wavenumber \( k_z \), and \( \vec B = B_0 \vu z \).
For certain values of \( v_z \), the particle will traverse an integer number of wavelengths during a cyclotron orbit:
\[\omega - k_z v_z = \pm n \omega_c\]