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Magnetized Waves

Source lecture(s): pc368_lec13_magnetized_wave

Intuition

A magnetic field acts like a plasma prism: it splits electromagnetic waves into different modes depending on their polarization relative to B. Some can pass (whistlers, O-mode), others are reflected at sharp cutoff densities (X-mode, R-mode). This is central to radio communications, ionospheric propagation, and electromagnetic wave heating in fusion devices.

Formal Definition

Magnetized waves are electromagnetic or electrostatic modes in a plasma subject to a background magnetic field \(\mathbf{B}_0\). Their polarization and dispersion are described by the cold-plasma dielectric tensor \(\mathbf{K}\).

Mathematical Formulation

The wave equation in a cold, uniform magnetized plasma is:

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

where \(\mathbf{n} = c\mathbf{k}/\omega\) is the refractive index and

\[\mathbf{K} = \begin{pmatrix} S & -iD & 0 \\ iD & S & 0 \\ 0 & 0 & P \end{pmatrix}\]

with Stix parameters \(S = 1 - \sum_s \frac{\omega_{ps}^2}{\omega^2 - \Omega_{cs}^2}\), \(D = \sum_s \frac{\Omega_{cs}/|\Omega_{cs}|\,\omega_{ps}^2}{\omega^2 - \Omega_{cs}^2}\), \(P = 1 - \sum_s \frac{\omega_{ps}^2}{\omega^2}\).

Derivation

  1. Linearize Vlasov–Maxwell for a uniform magnetized plasma.
  2. Assume exp\((ik_{\parallel}z - i\omega t)\) with \(\mathbf{B}_0 = B_0 \hat{z}\).
  3. Solve the linearized Lorentz force and Ampère’s law to obtain the dielectric tensor \(K_{ij}(\omega, \mathbf{k})\).
  4. Insert into the wave equation. Because of anisotropy, E, B, and k are not necessarily parallel.
  5. Separate into ordinary (O) and extraordinary (X) polarizations.

Worked Example

R-mode cutoff: The right-hand cutoff occurs where \(S + D = 0\), i.e. \(\omega = \omega_{R}\) such that the index \(n^2 \to \infty\). For \(\mathbf{B}_0\) along \(+\hat{z}\), the R-mode sees a resonance at \(\omega = \Omega_{ce}/2 + \sqrt{\Omega_{ce}^2/4 + \omega_{pe}^2}\).

Common Mistakes

  • B-field parity doesn’t matter. The R and L modes depend strongly on the sign of \(\mathbf{B}_0\).
  • All modes propagate everywhere. Cutoffs and resonances block propagation in specific frequency-density windows.
  • Cold plasma is all you need. Finite temperature introduces Bernstein modes and kinetic Alfvén waves.

Quiz Questions

  1. Conceptual: Which mode can reach the Earth’s surface after a solar flare? O-mode, X-mode, R-mode, or L-mode?
  2. Computational: Show that the lower hybrid frequency is \(\omega_{lh} = \sqrt{\Omega_{ci}^2 + \omega_{pi}^2}\).
  3. MCQ: The O-mode (ordinary wave) has its electric field polarized:
  4. A) Parallel to \(\mathbf{k}\)
  5. B) Perpendicular to \(\mathbf{k}\) and \(\mathbf{B}_0\)
  6. C) Perpendicular to \(\mathbf{k}\) but with a component along \(\mathbf{B}_0\)
  7. D) Along \(\mathbf{B}_0\) only

Further Reading

  • T. H. Stix, Waves in Plasmas, Chapter on cold plasma.