Part 2, Chapter 2

Landau Damping

Collisionless damping of plasma waves through wave-particle resonance

2.1 Introduction

Landau damping is one of the most remarkable phenomena in plasma physics: the exponential damping of electrostatic waves in a collisionless plasma. Discovered theoretically by Lev Landau in 1946, it seemed paradoxical—how can a wave decay without any dissipative mechanisms like collisions?

The answer lies in wave-particle resonance: particles moving with velocities close to the wave's phase velocity (v ≈ ω/k) can exchange energy coherently with the wave. If there are slightly more particles moving slower than the wave than faster, the wave loses energy to accelerate the slower particles, leading to damping.

💡 Key Insight: Landau damping is a kinetic effect that cannot be described by fluid models (MHD, two-fluid). It's a reversible process (entropy-conserving) arising from phase mixing in velocity space, not collisional dissipation!

2.2 Physical Picture

Resonant Particles

Consider a plasma wave with electric field E = E₀ cos(kx - ωt). Particles with different velocities experience this field differently:

v < ω/k (Slower)

These particles see the wave overtaking them. They experience a net acceleration from the wave electric field, gaining energy from the wave.

v ≈ ω/k (Resonant)

Resonant particles move with the wave. They stay in phase with the wave electric field, experiencing maximum energy exchange. This is where Landau damping occurs!

v > ω/k (Faster)

These particles overtake the wave. They see the wave receding and experience a net deceleration, giving energy to the wave.

The Distribution Function Slope

The key is the slope of the distribution function f₀(v) at the resonant velocity v_φ = ω/k:

• If ∂f₀/∂v < 0 at v = v_φ (normal Maxwellian): More slower particles than faster → Wave loses energy → Landau damping
• If ∂f₀/∂v > 0 at v = v_φ (inverted population, bump-on-tail): More faster particles → Wave gains energy → Instability!
• If ∂f₀/∂v = 0 at v = v_φ: No net energy exchange → Undamped wave

Important: Landau damping is exponential (γ_L ∝ e^-something) and very weak when k λ_D ≪ 1, but becomes strong when k λ_D ~ 1 (short wavelengths comparable to Debye length).

2.3 Mathematical Derivation

Plasma Dispersion Function

From the linearized Vlasov equation (see previous chapter), the dispersion relation for electrostatic waves is:

$$1 + \frac{\omega_{pe}^2}{k^2} \int_{-\infty}^{\infty} \frac{k \, \partial f_0/\partial v}{\omega - kv} \, dv = 0$$

Electrostatic dispersion relation for electrons

The integral has a pole at v = ω/k. We must specify how to handle this singularity. Landau showed that the correct prescription (from causality, initial value problem) is to integrate along a contour in the complex v-plane that passes above the pole (Landau contour):

$$\int_{-\infty}^{\infty} \frac{g(v)}{\omega - kv} \, dv \equiv \mathcal{P} \int \frac{g(v)}{\omega - kv} \, dv + i\pi \frac{g(\omega/k)}{k}$$

Plemelj formula (Landau prescription)

where 𝒫 denotes the principal value integral. The imaginary part comes from the residue at the pole and gives rise to damping (or growth).

Plasma Dispersion Function Z(ζ)

For a Maxwellian distribution f₀(v) = (n₀/√(2πv_th²)) exp(-v²/2v_th²), the dispersion relation can be written in terms of the plasma dispersion function:

$$Z(\zeta) = \frac{1}{\sqrt{\pi}} \int_{-\infty}^{\infty} \frac{e^{-x^2}}{x - \zeta} \, dx$$

$$\text{where } \zeta = \frac{\omega}{k v_{th}}$$

The dispersion relation becomes:

$$1 + \frac{1}{k^2 \lambda_D^2} \left(1 + \zeta Z(\zeta)\right) = 0$$

where λ_D = v_th/ω_pe is the Debye length

Asymptotic Expansion for |ζ| ≫ 1

For plasma oscillations (ω ≈ ω_pe ≫ kv_th), we have |ζ| ≫ 1. Using the asymptotic expansion:

$$Z(\zeta) \approx -\frac{1}{\zeta} - \frac{1}{2\zeta^3} - i\sqrt{\pi} e^{-\zeta^2} \quad (\text{for } \text{Im}(\zeta) > 0)$$

Substituting into the dispersion relation and solving to first order in 1/ζ²:

$$\omega = \omega_{pe}\left(1 + \frac{3}{2}k^2\lambda_D^2\right) - i \sqrt{\frac{\pi}{8}} \frac{\omega_{pe}}{k^3\lambda_D^3} e^{-1/(2k^2\lambda_D^2)}$$

Complex frequency: Real part = oscillation, Imaginary part = damping

2.4 Landau Damping Rate

The imaginary part of ω gives the damping rate:

$$\gamma_L = -\text{Im}(\omega) = \sqrt{\frac{\pi}{8}} \frac{\omega_{pe}}{k^3\lambda_D^3} \exp\left(-\frac{1}{2k^2\lambda_D^2} - \frac{3}{2}\right)$$

Landau damping rate (for Maxwellian plasma)

Key Properties

1. Exponentially Small for Long Wavelengths

When k λ_D ≪ 1, the damping rate γ_L ∝ exp(-1/(2k²λ_D²)) is exponentially small. This is why fluid models (which neglect kinetic effects) work reasonably well for long wavelengths.

2. Strongest for k λ_D ~ 1

Maximum damping occurs when the wavelength λ ~ 2π λ_D is comparable to the Debye length. For shorter wavelengths, the damping becomes strong and quickly kills the wave.

3. Depends on Distribution Function Slope

The damping rate is proportional to ∂f₀/∂v evaluated at v = v_φ = ω/k. For a Maxwellian, this slope is negative, giving damping. For a bump-on-tail distribution with positive slope, you get instability instead!

4. Collisionless but Irreversible (in practice)

Although the Vlasov equation is time-reversible, Landau damping appears irreversible because fine-scale structures in phase space (filamentation) become unobservable. Energy is transferred from coherent wave motion to incoherent particle kinetic energy.

2.5 Energy Transfer Mechanism

Wave Energy Decay

The wave energy decays exponentially:

$$W_{wave}(t) = W_0 e^{-2\gamma_L t}$$

Where does this energy go? It goes into kinetic energy of resonant particles. Specifically:

Particles with v slightly less than v_φ are accelerated by the wave
Particles with v slightly greater than v_φ are decelerated
Because ∂f₀/∂v < 0 (for Maxwellian), there are more slower particles, so net energy flows from wave to particles
The distribution function develops a plateau near v = v_φ (flattening of f)

Phase Space Evolution

In phase space (x, v), the distribution function undergoes phase mixing:

t = 0: Smooth, Maxwellian distribution f₀(v)
Early times: Wave perturbation creates oscillations in f(x,v,t)
Later times: Different velocity particles have different phase velocities → filamentation in phase space
Long times: Fine-scale structure becomes unmeasurably small → appears as heating

This is a reversible process in principle (Vlasov equation conserves entropy), but irreversible in practice because the fine structure is below measurement resolution. This is similar to Poincaré recurrence in statistical mechanics.

2.6 Experimental Verification

Landau damping was first experimentally verified by Malmberg and Wharton (1964) using electron plasma oscillations in a pure electron plasma column. They measured:

Measurement Technique

• Launch plasma wave with grid excitation
• Measure wave amplitude vs distance (or time)
• Observe exponential decay: E(x) ∝ e^-γ_Lx/v_φ
• Vary plasma parameters (T_e, n_e) to test theory

Results

✓ Exponential decay observed as predicted
✓ Damping rate agrees with Landau formula
✓ Temperature dependence confirmed
✓ Wave-particle resonance directly measured

More recently (2019), Mouhot & Villani won the Fields Medal for rigorous mathematical proof of Landau damping in the nonlinear regime!

2.7 Related Phenomena

Ion Landau Damping

Ion acoustic waves can be Landau damped by resonant ions and electrons. For k λ_D ≪ 1, the damping is weak unless T_e ≈ T_i (comparable temperatures).

→ Important for ion heating in fusion plasmas

Transit-Time Damping

Analogous to Landau damping but for waves propagating along a magnetic field. Particles bounce between mirror points and can resonate with the wave.

→ Relevant for whistler waves, Alfvén waves in magnetized plasmas

Cyclotron Damping

Resonance occurs when ω - k_∥v_∥ = nΩ_c (n = 0, ±1, ±2, ...). Particles gyrating at cyclotron frequency can exchange energy with waves.

→ Used for RF heating (ICRH, ECRH) in fusion devices

Inverse Landau Damping

If ∂f₀/∂v > 0 at resonance (bump-on-tail, two-stream), the sign flips and you get growthinstead of damping → Kinetic instabilities!

→ Next chapter: Two-stream instability

2.8 Applications

🌌 Space Plasmas

• Solar wind turbulence (wave cascade termination)
• Magnetospheric wave damping (chorus, hiss)
• Auroral electron acceleration
• Planetary bow shock heating

🔥 Fusion Plasmas

• Wave heating (ICRH, LHCD damping mechanism)
• Alpha particle slowing down
• Anomalous resistivity (weak)
• Current drive efficiency limits

💥 Laser-Plasma

• Electron plasma wave damping (SRS)
• Hot electron generation limits
• Wave-breaking thresholds
• Inertial fusion energy coupling

⚡ Accelerators

• Beam-plasma interactions
• Wakefield accelerator damping
• Collective instability suppression
• Coherent synchrotron radiation

📚 Key Takeaways

✓ Landau damping is collisionless wave damping via wave-particle resonance at v = ω/k
✓ Damping occurs when ∂f₀/∂v < 0 at resonance (Maxwellian); growth when ∂f₀/∂v > 0
✓ Damping rate γ_L ∝ exp(-1/(2k²λ_D²)) is exponentially small for k λ_D ≪ 1
✓ Energy flows from wave to resonant particles through phase mixing in velocity space
✓ Experimentally verified (Malmberg & Wharton 1964) and mathematically proven (Mouhot & Villani 2011)
✓ Critical for understanding wave damping in fusion, space, laser-plasma, and accelerator physics
✓ Cannot be described by fluid models—requires kinetic (Vlasov) description

← Vlasov Equation Next: Two-Stream Instability →

Plasma Physics Course