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Landau Damping

Source lecture(s): pc368_lec12_landau_damping

Intuition

A wave propagating in a collisionless plasma can decay even without collisions. Resonant particles—those moving at the wave’s phase velocity—exchange energy with the wave. If the slope of the equilibrium distribution at \(v_\phi\) is negative, the wave damps; if positive, it is driven. This was Landau’s famous insight in 1946: collisions are not needed for dissipation.

Formal Definition

Landau damping is the collisionless damping of a plasma wave due to resonant wave–particle interactions. The dielectric function is evaluated on the Landau contour in the complex \(v\)-plane.

Mathematical Formulation

For a one-dimensional electrostatic perturbation, the dielectric function is:

\[\varepsilon(\omega, k) = 1 + \frac{1}{k^2 \lambda_D^2} \Biggl[ 1 + \zeta Z(\zeta) \Biggr] = 0\]

where \(\zeta = \frac{\omega/k}{\sqrt{2} v_{te}}\) and \(Z(\zeta)\) is the plasma dispersion function. For small \(k\lambda_D\), the damping rate is:

\[\gamma \approx -\frac{\pi}{2} \frac{\omega}{k \lambda_D} \left(\frac{\omega/k}{v_{te}}\right)^2 \exp\!\Bigl[-\Bigl(\frac{\omega/k}{\sqrt{2} v_{te}}\Bigr)^2\Bigr]\]

provided \(\partial f_0/\partial v < 0\) at \(v_\phi = \omega/k\).

Derivation

  1. Laplace transform the Vlasov equation: assume \(\propto e^{-i\omega t}\) with \(\operatorname{Im}(\omega) > 0\) for causality.
  2. The velocity integral of the perturbed distribution picks up contributions from the pole at \(\omega - kv = 0\).
  3. To avoid the pole on the real axis, deform the integration contour below the pole (Landau contour).
  4. The integral gives the \(Z(\zeta)\) function.
  5. The imaginary part yields the physical damping/growth rate.

Worked Example

A Langmuir wave with \(k\lambda_D = 0.2\) propagates through a Maxwellian plasma. Estimate the Landau damping rate. Using the weak-damping formula: \(\zeta = \omega/(k\sqrt{2}v_{te}) \approx 1\). \(Z(\zeta) \approx -\pi^{1/2} + i\sqrt{\pi}\). The imaginary part gives \(\gamma/\omega \sim -(\pi/2)(k\lambda_D)^2 \approx -0.06\). The wave loses ~6% of its energy per oscillation period.

Common Mistakes

  • Landau damping needs collisions. No—it is purely collisionless, arising from resonant trapping in \(v\)-space.
  • Damping means energy is lost to heat. It is transferred to resonant particles, not randomized by collisions.
  • All Langmuir waves are damped. Only if \(\partial f_0/\partial v < 0\) at \(v_\phi\); otherwise the wave is driven.

Quiz Questions

  1. Conceptual: Explain why a perturbation with equal numbers of resonant particles on either side of \(v_\phi\) sees no net Landau damping.
  2. Computational: For a bump-on-tail distribution with a local maximum, explain why Landau predicts growth.
  3. MCQ: Landau damping is fundamentally:
  4. A) A collisional effect
  5. B) A resonant wave–particle effect
  6. C) A numerical artifact
  7. D) Caused by resistivity

Further Reading

  • E. Centroni & P. P., Landau’s 1946 Paper Revisited.