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Cold Plasma Dispersion Tensor

\[\mathbf{n}\times(\mathbf{n}\times\mathbf{E}) + \mathbf{K}\cdot\mathbf{E} = 0\]

Source lecture(s): pc368_lec13_magnetized_wave

Physical Meaning

The cold-plasma wave equation describes electromagnetic wave propagation in a magnetized, collisionless plasma. The refractive index tensor \(\mathbf{K}\) determines cutoff, resonance, and mode polarization.

Variable Definitions

Symbol Definition SI Units
\(\mathbf{n} = c\mathbf{k}/\omega\) Refractive-index vector dimless
\(\mathbf{K}\) Dielectric (Stix) tensor dimless
\(S, D, P\) Stix parameters dimless
\(\omega_{ps}, \Omega_{cs}\) Species plasma/cyclotron frequencies rad s\(^{-1}\)

Assumptions

  • Uniform \(\mathbf{B}_0\).
  • Cold fluid species (finite temperature adds kinetic corrections).
  • Linear waves.

Derivation

Linearize the cold-fluid momentum equation for each species, include Lorentz force, and couple to Maxwell’s equations. The susceptibility tensor is:

\[\chi_{ij} = -\sum_s \frac{\omega_{ps}^2}{\omega^2} \Bigl[ \mathbf{I} + \frac{\omega}{\Omega_{cs}}\mathbf{M} + \Bigl(\frac{\omega}{\Omega_{cs}}\Bigr)^2 \mathbf{b}\mathbf{b} \Bigr]\]

where \(\mathbf{M}\) encodes the magnetization. In their standard form this yields the \(S, D, P\) tensor.

Applications

  • RF heating: Ion cyclotron range of frequencies (ICRF).
  • Ionospheric propagation: HF radio windows and cuts.
  • ECRH: Electron cyclotron resonance heating.

Connections to Other Equations