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Plasma Waves

Source lecture(s): pc368_lec09_wave1, pc368_lec10_wave2

Intuition

A plasma can sustain self-reproducing oscillations because inertia (electron mass) and restoring force (space charge) are both built in. The simplest example is the Langmuir wave: electrons slosh back and forth through the stationary ion sea. More complicated geometries (magnetized, multi-species) give rise to a zoo of modes, each with its own polarization and dispersion relation.

Formal Definition

A plasma wave is a self-sustaining disturbance in which the plasma variables (\(n\), \(\mathbf{E}\), \(\mathbf{B}\), \(T\), etc.) oscillate in space and time according to the linearized Vlasov–Maxwell or MHD equations.

Mathematical Formulation

From linearized Vlasov–Poisson for a cold, unmagnetized plasma:

\[1 + \chi_e = 1 - \frac{\omega_{pe}^2}{\omega^2} = 0\]

This yields the Langmuir mode:

\[\omega^2 = \omega_{pe}^2 + 3 k^2 v_{te}^2\]

where \(v_{te} = \sqrt{k_B T_e/m_e}\) is the electron thermal speed (Bohm–Gross correction).

For warm ions, the ion acoustic wave appears:

\[\omega^2 = \frac{k^2 c_s^2}{1 + (k\lambda_D)^2}, \qquad c_s^2 = \frac{\gamma_e k_B T_e + \gamma_i k_B T_i}{m_i}\]

Derivation (Langmuir)

  1. Write continuity, momentum, and Poisson for a perturbed electron fluid.
  2. Assume \(n_e = n_0 + n_1\), \(u_e = u_1\), \(\phi \propto e^{i(kx - \omega t)}\).
  3. Neglect pressure to get cold Langmuir: \(\omega^2 = \omega_{pe}^2\).
  4. Add isothermal pressure \(p_e = n_e k_B T_e\) to get Bohm–Gross.
  5. Verify that \(\omega \gg \omega_{pi}\) justifies the frozen-ion assumption.

Worked Example

Cutoff condition: A radio wave encounters an overdense plasma (\(n_e > n_c\)). The critical density is \(n_c = \varepsilon_0 m_e \omega^2 / e^2\). For \(f = 1\) GHz, \(n_c \approx 1.24\times 10^{16}\,\text{m}^{-3}\). If \(n_e > n_c\), the wave reflects before entering.

Common Mistakes

  • Langmuir waves are not sound waves. They are electrostatic; ions are considered immobile on the electron timescale.
  • Ion acoustic needs \(T_e \gg T_i\). Otherwise Landau damping extinguishes the wave.
  • Dispersion = dissipation. Dispersion changes wave shape; dissipation damps amplitude. They are distinct.

Quiz Questions

  1. Conceptual: If electrons and ions both sloshed equally, would you still have a Langmuir wave?
  2. Computational: Show that for \(n_e = 10^{18}\,\text{m}^{-3}\), the cold Langmuir frequency is \(f_{pe} \approx 9\) GHz.
  3. MCQ: Landau damping is strongest when:
  4. A) \(k\lambda_D \ll 1\)
  5. B) \(k\lambda_D \approx 1\)
  6. C) \(k\lambda_D \gg 1\)
  7. D) Temperature is zero

Further Reading

  • D. J. Stix, Waves in Plasmas.