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Landau Dispersion Relation

\[1 + \frac{1}{k^2 \lambda_D^2} \Bigl[ 1 + \zeta Z(\zeta) \Bigr] = 0\]

Source lecture(s): pc368_lec12_landau_damping

Physical Meaning

The Landau dispersion relation is the condition for electrostatic normal modes in a plasma with a given equilibrium distribution \(f_0(v)\). It embeds the physics of resonant wave–particle interactions via the plasma dispersion function \(Z(\zeta)\).

Variable Definitions

Symbol Definition SI Units
\(k\) Wavenumber m\(^{-1}\)
\(\lambda_D\) Debye length m
\(\zeta = \omega/(k\sqrt{2}v_{te})\) Normalized phase speed dimless
\(Z(\zeta)\) Plasma dispersion function dimless
\(v_{te} = \sqrt{k_B T_e/m_e}\) Electron thermal speed m s\(^{-1}\)

Assumptions

  • One-dimensional electrostatic perturbation.
  • Collisionless Vlasov–Poisson system.
  • Maxwellian background \(f_0(v) = n_0/\sqrt{2\pi}v_{te} \exp(-v^2/v_{te}^2)\).

Derivation

  1. Fourier-transform Vlasov in time and space.
  2. Integrate over velocity: \(\chi_e = (e^2/\varepsilon_0 k^2) \int \frac{\partial f_{0e}/\partial v}{\omega - k v} dv\).
  3. Close with Poisson: \(1 + \chi_e = 0\).
  4. The pole at \(v = \omega/k\) is pushed below the real axis (Landau contour \(v \to v - i0^+\)).
  5. Integration yields \(Z(\zeta) = \pi^{1/2} \int_{-\infty}^{\infty} \exp(-t^2)/(t-\zeta) dt\).

Applications

  • Landau damping rate: Weak-damping limit gives \(\gamma/\omega\) formula.
  • Bump-on-tail: \(\partial f_0/\partial v > 0\) gives growth.
  • Dispersion broadening: \(k\lambda_D \sim 1\) regime for satellite plasma wave instruments.

Connections to Other Equations