Plasma Physics Course

Total: 8 parts covering plasma fundamentals through advanced topics

Part 2, Chapter 5

Quasilinear Theory

Self-consistent evolution of waves and particles through resonant interactions

5.1 Introduction

Quasilinear theory describes the self-consistent evolution of wave fields and particle distribution functions when they interact through resonant wave-particle interactions. It goes beyond linear theory (which assumes fixed f₀) to account for how waves modify the distribution function, which in turn affects wave growth/damping.

The theory is called "quasilinear" because it retains terms that are second-order in the wave amplitude (δf₁ · E₁) while still assuming the perturbations are small. This allows us to describe:

  • Wave-induced diffusion: Resonant particles diffuse in velocity space due to waves
  • Plateau formation: Distribution function flattens at resonant velocities
  • Instability saturation: How growing waves eventually stabilize by modifying f₀
  • Turbulent heating: Energy transfer from waves to particles via diffusion

💡 Key Insight: Quasilinear theory bridges linear kinetic theory (Vlasov, Landau damping) and fully nonlinear phenomena (particle trapping, turbulence). It's valid when wave amplitudes are small but interactions persist long enough to change f₀.

5.2 Ordering and Assumptions

Expansion Scheme

We expand the distribution function and fields in powers of a small parameter ε (wave amplitude):

$$f(\mathbf{r}, \mathbf{v}, t) = f_0(\mathbf{v}, t) + f_1(\mathbf{r}, \mathbf{v}, t) + f_2(\mathbf{r}, \mathbf{v}, t) + \cdots$$
$$\mathbf{E}(\mathbf{r}, t) = \mathbf{E}_1(\mathbf{r}, t) + \mathbf{E}_2(\mathbf{r}, t) + \cdots$$

f₀ ~ O(1), f₁ ~ O(ε), f₂ ~ O(ε²) | E₁ ~ O(ε), E₂ ~ O(ε²)

Time Scale Separation

The theory assumes a separation of time scales:

Fast Time Scale

tfast ~ ω⁻¹, k⁻¹vth

• Wave oscillations
• Particle gyration
• Linear response (f₁ to E₁)

Slow Time Scale

tslow ~ ε⁻² ω⁻¹

• Evolution of f₀
• Diffusion in velocity space
• Plateau formation

Validity Condition:

$$\frac{eE_1}{m\omega v_{th}} \ll 1 \quad \text{(weak turbulence)}$$

Wave electric field is small compared to particle thermal energy

5.3 Quasilinear Diffusion Equation

Derivation from Vlasov Equation

Starting from the Vlasov equation and averaging over fast oscillations, we obtain the quasilinear diffusion equation for the slowly evolving background distribution:

$$\frac{\partial f_0}{\partial t} = \frac{\partial}{\partial \mathbf{v}} \cdot \left( \mathbf{D} \cdot \frac{\partial f_0}{\partial \mathbf{v}} \right)$$

Quasilinear diffusion equation

where D is the diffusion tensor in velocity space:

$$D_{ij} = \frac{\pi q^2}{m^2} \sum_{\mathbf{k}} \delta(\omega_{\mathbf{k}} - \mathbf{k} \cdot \mathbf{v}) \, k_i k_j \, |E_{\mathbf{k}}|^2$$

Sum over all wave vectors; δ-function enforces resonance ω = k·v

Physical Interpretation

  • • Resonance condition: Only particles with v satisfying ω = k·v contribute to diffusion
  • • Random walk: Particles undergo random kicks from wave electric fields, leading to velocity diffusion
  • • Second-order effect: D ~ |E₁|², arises from correlations ⟨E₁ · δf₁⟩
  • • Irreversibility: Unlike linear Landau damping, quasilinear diffusion is irreversible (entropy increases)

1D Electrostatic Case

For electrostatic waves propagating in one direction (longitudinal waves along x):

$$\frac{\partial f_0}{\partial t} = \frac{\partial}{\partial v} \left( D(v) \frac{\partial f_0}{\partial v} \right)$$
$$D(v) = \frac{\pi e^2}{m^2} \sum_{k} k^2 |E_k|^2 \delta(\omega_k - kv)$$

Diffusion coefficient peaks at resonant velocities vres = ωk/k

5.4 Wave Energy Evolution

Wave Kinetic Equation

The waves also evolve due to their interaction with particles. The spectral energy density Wk = ε₀|Ek|²/2 satisfies:

$$\frac{\partial W_{\mathbf{k}}}{\partial t} = 2\gamma_{\mathbf{k}} W_{\mathbf{k}}$$

γk is the growth/damping rate from linear theory

The growth rate depends on the current distribution function f₀(v,t):

$$\gamma_{\mathbf{k}} = -\frac{\pi \omega_{pe}^2}{2k} \left. \frac{\partial f_0}{\partial v} \right|_{v=\omega/k}$$

Growth rate proportional to slope of f₀ at resonance (Landau formula)

Energy Conservation

Total energy (particles + waves) is conserved in quasilinear theory:

$$\frac{d}{dt}\left( \int \frac{1}{2}mv^2 f_0 \, d^3v + \sum_{\mathbf{k}} W_{\mathbf{k}} \right) = 0$$

Energy lost by waves → gained by particles (or vice versa)

Physical Picture: Growing waves extract energy from gradients in f₀. The diffusion flattens f₀ at resonant velocities, reducing ∂f₀/∂v → reducing growth rate → saturation. The wave energy is eventually absorbed by particles through the diffusive heating process.

5.5 Plateau Formation

Relaxation to Marginal Stability

Consider an unstable initial distribution f₀(v,0) with ∂f₀/∂v > 0 in some velocity range (e.g., bump-on-tail). Waves grow and cause diffusion at resonant velocities. The system evolves toward a state where:

$$\frac{\partial f_0}{\partial v} \approx 0 \quad \text{at resonant velocities}$$

Plateau formation (marginal stability)

This is called a plateau in velocity space. Once formed, the plateau ismarginally stable: any small perturbation that creates ∂f₀/∂v ≠ 0 immediately drives diffusion to restore the plateau.

Time Scale for Plateau Formation

$$t_{plateau} \sim \frac{1}{\gamma_{QL}} \sim \frac{m^2 v_{th}^2}{\pi e^2 k^2 |E_k|^2}$$

γQL ~ D/vth² is the quasilinear relaxation rate

Before Plateau

• Positive slope ∂f₀/∂v > 0
• Waves grow exponentially
• Wave energy Wk increases
• Strong diffusion at resonance

After Plateau

• Flat distribution ∂f₀/∂v ≈ 0
• Waves saturated (γ → 0)
• Wave energy constant
• Marginal stability maintained

5.6 Applications and Limitations

Applications

  • • Bump-on-tail instability: Saturation via plateau formation
  • • Current-driven instabilities: Quasilinear diffusion limits runaway electrons
  • • Solar wind heating: Wave-induced diffusion heats minor ions
  • • Radiation belts: Pitch-angle diffusion by whistler waves
  • • Anomalous transport: Enhanced collision rate in fusion plasmas

When Quasilinear Theory Fails

  • • Large amplitudes: eE/(mωvth) ~ 1 → particle trapping
  • • Wave-wave coupling: Nonlinear three-wave interactions
  • • Coherent structures: BGK modes, phase-space holes
← Previous: Fokker-PlanckNext: Particle Trapping →