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Cold Plasma Wave Explorer

Learning goal

Chapter 13 is a wall of algebra: a dielectric tensor, a quartic in the refractive index, and a zoo of named modes. This widget solves that quartic at every pixel so you can see the zoo — and, more usefully, see the gaps between the animals, where no wave propagates at all.

What is being solved

For a cold, magnetised plasma the dispersion relation is a quadratic in \(n^2 = c^2k^2/\omega^2\):

\[A n^4 - B n^2 + C = 0\]
\[A = S\sin^2\theta + P\cos^2\theta,\qquad B = RL\sin^2\theta + PS(1+\cos^2\theta),\qquad C = PRL\]

with the Stix parameters built from every species:

\[S = 1 - \sum_\sigma \frac{\omega_{p\sigma}^2}{\omega^2 - \omega_{c\sigma}^2},\qquad D = \sum_\sigma \frac{\omega_{c\sigma}}{\omega}\frac{\omega_{p\sigma}^2}{\omega^2-\omega_{c\sigma}^2},\qquad P = 1 - \sum_\sigma \frac{\omega_{p\sigma}^2}{\omega^2}\]

and \(R = S+D\), \(L = S-D\). Cutoffs are where \(n^2 = 0\) (the wave reflects); resonances are where \(n^2 \to \infty\), i.e. \(A = 0\) (the wave slows to a halt and is absorbed).

Things to try

  1. θ = 0°, parallel propagation. The two roots collapse to exactly \(n^2 = R\) and \(n^2 = L\) — the right- and left-hand circularly polarised waves. Because they have different refractive indices, a linearly polarised wave sent along \(\mathbf{B}\) has its plane rotated: Faraday rotation, which is how the interstellar magnetic field is measured from pulsar signals.

  2. Stay at θ = 0° and look below \(\omega_{ce}\). The R branch propagates all the way down to zero frequency, with \(n^2\) blowing up as \(\omega \to \omega_{ce}\). That is the whistler — lightning-generated waves that spiral along Earth's field lines to the opposite hemisphere, arriving as a descending tone because low frequencies travel slower. First heard on field telephones in WWI, and unexplained for forty years.

  3. Sweep θ to 90°. Two resonances appear that were not there at θ = 0: the upper hybrid at \(\sqrt{\omega_{pe}^2 + \omega_{ce}^2}\) and the lower hybrid far below. These are the hybrid resonances — they exist only for propagation across the field, and they are where tokamaks deposit megawatts of radio-frequency heating power.

  4. Find the evanescent gap. The grey bands are where neither root has \(n^2 > 0\). A wave launched into that band does not slowly attenuate — it does not propagate at all, and reflects. Every plasma diagnostic and every heating scheme is a plan for routing power around these bands.

  5. Turn the magnetic field off (slide \(\omega_{ce}/\omega_{pe}\) toward 0). All the structure collapses onto one condition: \(n^2 = 1 - \omega_{pe}^2/\omega^2\), propagating only above \(\omega_{pe}\). That single line is the whole unmagnetised story — and the reason the ionosphere reflects some radio and passes the rest.

  6. Switch to the demo ion mass (\(m_i/m_e = 100\)). The ion features — lower hybrid, ion cyclotron resonance — slide up into view instead of hiding four decades below everything else. Real hydrogen is 1836, which is exactly why ion and electron physics live on such stubbornly separated scales.

Magnetised waves · Dielectric tensor · Resonance cones · EM waves in plasma · Cold-plasma dispersion (eq.)