Plasma Physics Course

Total: 8 parts covering plasma fundamentals through advanced topics

Part 3, Chapter 7

The Closure Problem

Truncating the moment hierarchy

7.1 The Hierarchy Problem

Taking moments of the kinetic equation generates an infinite hierarchy:

$$\text{0th moment (continuity):} \quad \frac{\partial n}{\partial t} \rightarrow \text{depends on } \mathbf{u}$$
$$\text{1st moment (momentum):} \quad \frac{\partial \mathbf{u}}{\partial t} \rightarrow \text{depends on } \mathbf{P}$$
$$\text{2nd moment (energy):} \quad \frac{\partial \mathbf{P}}{\partial t} \rightarrow \text{depends on } \mathbf{q}$$
$$\text{3rd moment (heat flux):} \quad \frac{\partial \mathbf{q}}{\partial t} \rightarrow \text{depends on } \ldots$$

The Problem: Each moment equation introduces the next higher moment. To close the system, we must truncate this hierarchy with an approximation.

7.2 Common Closures

Cold Plasma (T → 0)

$$p = 0, \quad \mathbf{P} = 0, \quad \mathbf{q} = 0$$

Valid for high-frequency waves where thermal motion is negligible

Isothermal Closure

$$p = nk_BT, \quad T = \text{const}, \quad \mathbf{q} = 0$$

Infinitely fast heat conduction maintains constant temperature

Adiabatic Closure

$$\frac{d}{dt}\left(\frac{p}{\rho^\gamma}\right) = 0, \quad \mathbf{q} = 0$$

No heat flux; Îł = (N+2)/N where N = degrees of freedom (Îł = 5/3 for 3D)

Double Adiabatic (CGL)

$$\frac{d}{dt}\left(\frac{p_\parallel B^2}{\rho^3}\right) = 0, \quad \frac{d}{dt}\left(\frac{p_\perp}{\rho B}\right) = 0$$

Chew-Goldberger-Low; separate parallel and perpendicular pressures

7.3 Braginskii Transport

For collisional plasmas (λmfp â‰Ș L), Chapman-Enskog expansion gives:

Heat Flux

$$\mathbf{q} = -\kappa_\parallel \nabla_\parallel T - \kappa_\perp \nabla_\perp T + \kappa_\wedge \mathbf{b} \times \nabla T$$

Parallel

$$\kappa_\parallel \propto T^{5/2}$$

Dominant; along B

Perpendicular

$$\kappa_\perp \sim \frac{\kappa_\parallel}{(\Omega\tau)^2}$$

Suppressed by B

Diamagnetic

$$\kappa_\wedge \sim \frac{\kappa_\parallel}{\Omega\tau}$$

Cross-field drift

7.4 Viscosity Tensor

The pressure tensor splits into isotropic and viscous parts:

$$P_{ij} = p\delta_{ij} + \pi_{ij}$$

The viscous stress tensor π has five independent components in a magnetized plasma:

$$\boldsymbol{\pi} = -\eta_0 \mathbf{W}_0 - \eta_1 \mathbf{W}_1 - \eta_2 \mathbf{W}_2 - \eta_3 \mathbf{W}_3 - \eta_4 \mathbf{W}_4$$

Wi are rate-of-strain tensors; ηi are viscosity coefficients

7.5 Gyrokinetic Closure

For low-frequency turbulence with k⊄ρi ~ 1, gyrokinetics provides a closure:

Ordering

$$\frac{\omega}{\Omega_i} \sim \frac{k_\parallel}{k_\perp} \sim \frac{\delta B}{B} \sim \frac{e\phi}{T_e} \sim \epsilon \ll 1$$

Gyrokinetics averages over the fast gyromotion, retaining finite Larmor radius effects while reducing the kinetic equation to 5D (position + v∄ + magnetic moment ÎŒ).

7.6 Landau Closure

For collisionless plasmas, Landau damping must be captured. Hammett-Perkins closure:

$$q_\parallel = -n\chi_\parallel \frac{\partial T}{\partial z} \cdot \text{sgn}(k_\parallel)$$

Non-local closure mimics Landau damping

This "Landau fluid" model captures the essential kinetic physics of parallel phase mixing within a fluid framework.

7.7 Validity Conditions

ClosureValid WhenApplication
Coldω ≫ kvthHigh-frequency waves
Isothermalτheat â‰Ș τdynFast heat conduction
Adiabaticτheat ≫ τdynIsolated evolution
Braginskiiλmfp â‰Ș LCollisional plasmas
Landau fluidCollisionless, k∄vth ~ ωKinetic effects needed

Key Takeaways

  • ✓ Moment hierarchy is infinite; must truncate with closure
  • ✓ Common closures: cold, isothermal, adiabatic, CGL
  • ✓ Braginskii: collisional transport with Îș∄ ≫ Îș⊄
  • ✓ Gyrokinetics: low-frequency, k⊄ρi ~ 1 ordering
  • ✓ Landau fluids capture collisionless damping
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