Plasma Physics Course

Total: 8 parts covering plasma fundamentals through advanced topics

Part 3, Chapter 6

Two-Fluid Theory

Separate treatment of electrons and ions

6.1 Motivation

MHD treats plasma as a single conducting fluid. Two-fluid theory keeps electrons and ions as separate fluids, capturing physics at scales between MHD and kinetic:

  • Frequencies ω ~ Ωci (ion cyclotron)
  • Scales ~ ρi (ion gyroradius) or c/ωpi (ion skin depth)
  • Hall effect and finite Larmor radius corrections
  • Whistler and ion cyclotron waves

6.2 Two-Fluid Equations

Continuity (each species s)

$$\frac{\partial n_s}{\partial t} + \nabla \cdot (n_s \mathbf{u}_s) = 0$$

Momentum (each species s)

$$m_s n_s \left(\frac{\partial \mathbf{u}_s}{\partial t} + \mathbf{u}_s \cdot \nabla \mathbf{u}_s\right) = q_s n_s (\mathbf{E} + \mathbf{u}_s \times \mathbf{B}) - \nabla p_s + \mathbf{R}_s$$

Maxwell's Equations

$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$
$$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \frac{1}{c^2}\frac{\partial \mathbf{E}}{\partial t}$$

6.3 Generalized Ohm's Law

Combining electron and ion momentum equations gives:

$$\mathbf{E} + \mathbf{u} \times \mathbf{B} = \eta \mathbf{J} + \frac{1}{en}\left(\mathbf{J} \times \mathbf{B} - \nabla p_e + \frac{m_e}{e}\frac{d\mathbf{J}}{dt}\right)$$

Generalized Ohm's Law

Hall Term

$$\frac{\mathbf{J} \times \mathbf{B}}{en}$$

Decouples electron and ion motion

Electron Pressure

$$-\frac{\nabla p_e}{en}$$

Diamagnetic drift contribution

6.4 Hall MHD

Keeping only the Hall term beyond ideal MHD:

$$\frac{\partial \mathbf{B}}{\partial t} = \nabla \times \left(\mathbf{u} \times \mathbf{B} - \frac{\mathbf{J} \times \mathbf{B}}{en}\right)$$

Hall MHD Induction Equation

Key Scale: The ion skin depth di = c/ωpi determines when Hall physics becomes important. For L ≪ di, electrons and ions decouple.

6.5 Two-Fluid Waves

Whistler Waves

$$\omega = \frac{k^2 c^2 \Omega_{ce}}{\omega_{pe}^2} \cos\theta = k^2 d_e^2 \Omega_{ce} \cos\theta$$

Right-hand polarized, electron-dominated, ω ≫ Ωci

Ion Cyclotron Waves

$$\omega \approx \Omega_{ci} \sqrt{1 + k_\perp^2 \rho_i^2}$$

Left-hand polarized, resonant heating of ions

Kinetic Alfvén Wave

$$\omega = k_\parallel v_A \sqrt{1 + k_\perp^2 \rho_s^2}$$

Modified Alfvén wave with k⊥ρs ~ 1, where ρs = cs/Ωci

6.6 Diamagnetic Drifts

Each species has a drift perpendicular to both B and ∇p:

$$\mathbf{u}_{*s} = -\frac{\nabla p_s \times \mathbf{B}}{q_s n_s B^2}$$

Diamagnetic Drift Velocity

Electron Drift

$$\mathbf{u}_{*e} = \frac{\nabla p_e \times \mathbf{B}}{en_e B^2}$$

Ion Drift

$$\mathbf{u}_{*i} = -\frac{\nabla p_i \times \mathbf{B}}{en_i B^2}$$

Note: Electrons and ions drift in opposite directions. The net current j* = en(u*i - u*e) is the diamagnetic current.

Key Takeaways

  • ✓ Two-fluid: separate equations for electrons and ions
  • ✓ Generalized Ohm's law includes Hall, pressure, inertia terms
  • ✓ Hall physics important at scales ~ di = c/ωpi
  • ✓ Whistler, ion cyclotron, kinetic Alfvén waves
  • ✓ Diamagnetic drifts: species drift ⊥ to B and ∇p
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