Plasma Physics Course

Total: 8 parts covering plasma fundamentals through advanced topics

Part 3, Chapter 4

MHD Waves

Alfvén waves and magnetosonic modes

4.1 Linearized MHD

To study waves, we linearize about a uniform equilibrium with B₀, ρ₀, p₀, and u₀ = 0:

$$\rho = \rho_0 + \rho_1, \quad \mathbf{B} = \mathbf{B}_0 + \mathbf{B}_1, \quad p = p_0 + p_1, \quad \mathbf{u} = \mathbf{u}_1$$

Perturbations ≪ equilibrium quantities

Assuming plane wave solutions ∝ exp(ik·r - iωt):

$$-i\omega\rho_0\mathbf{u}_1 = -i\mathbf{k}p_1 + \frac{1}{\mu_0}(i\mathbf{k} \times \mathbf{B}_1) \times \mathbf{B}_0$$

4.2 Characteristic Speeds

Alfvén Speed

$$v_A = \frac{B_0}{\sqrt{\mu_0 \rho_0}}$$

Speed of magnetic tension wave propagation

Sound Speed

$$c_s = \sqrt{\frac{\gamma p_0}{\rho_0}}$$

Adiabatic sound wave speed

4.3 Alfvén Waves

Incompressible waves with velocity perpendicular to both k and B₀:

$$\omega = k_\parallel v_A = k v_A \cos\theta$$

Shear Alfvén Wave Dispersion

Properties

  • • Incompressible: ∇·u₁ = 0
  • • Transverse oscillation
  • • No density/pressure change
  • • Propagates along B₀

Physical Picture

Field lines act like taut strings under tension B²/μ₀. Perturbations propagate at the Alfvén speed along the field.

4.4 Magnetosonic Waves

Compressible waves involving both pressure and magnetic forces. The dispersion relation is:

$$\left(\frac{\omega}{k}\right)^2 = \frac{1}{2}\left[(c_s^2 + v_A^2) \pm \sqrt{(c_s^2 + v_A^2)^2 - 4c_s^2 v_A^2 \cos^2\theta}\right]$$

Fast (+) and Slow (−) Magnetosonic Waves

Fast Magnetosonic Wave

  • • Compression of both gas and magnetic field in phase
  • • Phase velocity: vf ≥ max(cs, vA)
  • • Propagates in all directions (isotropic for θ = π/2)

Slow Magnetosonic Wave

  • • Compression of gas and field out of phase
  • • Phase velocity: vs ≤ min(cs, vA)
  • • Cannot propagate perpendicular to B₀

4.5 Special Cases

Parallel Propagation (θ = 0)

Alfvén

$$\omega = k v_A$$

Fast

$$\omega = k \cdot \max(c_s, v_A)$$

Slow

$$\omega = k \cdot \min(c_s, v_A)$$

Perpendicular Propagation (θ = π/2)

Fast (Compressional Alfvén)

$$\omega = k\sqrt{c_s^2 + v_A^2}$$

Slow & Alfvén

ω = 0 (no propagation)

4.6 Wave Energy and Group Velocity

Alfvén Wave Energy

$$\mathcal{E} = \frac{1}{2}\rho_0 u_1^2 + \frac{B_1^2}{2\mu_0}$$

Equal kinetic and magnetic energy

Group Velocity

$$\mathbf{v}_g = \frac{\partial \omega}{\partial \mathbf{k}}$$

Alfvén waves: vg = vA along B₀

Key Takeaways

  • ✓ Three MHD wave modes: Alfvén, fast magnetosonic, slow magnetosonic
  • ✓ Alfvén waves: incompressible, transverse, ω = k∥vA
  • ✓ Fast waves: vf ≥ max(cs, vA), propagate all directions
  • ✓ Slow waves: vs ≤ min(cs, vA), guided by B₀
  • ✓ vA = B/√(μ₀ρ) is fundamental Alfvén speed
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