Plasma Physics Course

Total: 8 parts covering plasma fundamentals through advanced topics

Part 3, Chapter 5

MHD Instabilities

Stability analysis and common instability modes

5.1 Energy Principle

A plasma equilibrium is stable if and only if the potential energy change δW is positive for all perturbations:

$$\delta W = \frac{1}{2}\int \left[\frac{|\mathbf{Q}|^2}{\mu_0} + \gamma p|\nabla \cdot \boldsymbol{\xi}|^2 + (\boldsymbol{\xi} \cdot \nabla p)(\nabla \cdot \boldsymbol{\xi}) + \mathbf{J} \cdot (\boldsymbol{\xi} \times \mathbf{Q})\right] d^3r$$

Where Q = ∇ × (ξ × B) is the perturbed field

Stabilizing Terms

  • • Field line bending: |Q|²/μ₀
  • • Compression: γp|∇·ξ|²

Destabilizing Terms

  • • Pressure gradient
  • • Current-driven (kink)

5.2 Interchange Instability

Occurs when plasma pressure gradient opposes effective gravity (or field curvature):

$$\nabla p \cdot \boldsymbol{\kappa} > 0 \quad \Rightarrow \quad \text{Unstable}$$

κ = field line curvature vector, pointing toward center of curvature

Physical Picture: Like Rayleigh-Taylor instability. Plasma on the "outside" of curved field lines (bad curvature) is unstable to interchange of flux tubes.

5.3 Sausage and Kink Instabilities

Sausage Instability (m = 0)

Axisymmetric pinching of plasma column:

$$\gamma^2 = \frac{k^2 v_A^2}{1 + k^2 a^2}\left(\frac{B_z^2}{B_\theta^2} - 1 - k^2 a^2\right)$$

Stability requires Bz > Bθ (strong axial field)

Kink Instability (m = 1)

Helical displacement of plasma column:

$$\text{Kruskal-Shafranov: } q(a) = \frac{2\pi a B_z}{\mu_0 I} > 1$$

Stability requires safety factor q > 1 at the edge

5.4 Tearing Mode

Resistive instability that tears and reconnects magnetic field lines:

$$\gamma \sim \left(\frac{\eta}{\mu_0}\right)^{3/5} \left(\frac{k B_0'}{\sqrt{\mu_0 \rho}}\right)^{2/5}$$

Growth rate scales with fractional power of resistivity

Conditions

  • • Resonant surface where k·B = 0
  • • Current sheet forms
  • • Requires finite resistivity

Consequences

  • • Magnetic island formation
  • • Energy release (flares)
  • • Confinement degradation

5.5 Ballooning Modes

Pressure-driven instabilities localized to bad curvature regions:

$$\alpha = -\frac{2\mu_0 R q^2}{B^2}\frac{dp}{dr} > \alpha_{\text{crit}}$$

Normalized pressure gradient (α) must exceed critical value

Ballooning modes set the β-limit in tokamaks. They "balloon" out on the low-field side where curvature is unfavorable.

5.6 Rayleigh-Taylor Analog

A plasma supported against gravity by a magnetic field is unstable when:

$$\gamma^2 = g \frac{-\rho'}{\rho} - \frac{(k \cdot B)^2}{\mu_0 \rho}$$

The magnetic field provides stabilization through field line tension proportional to (k·B)². Modes with k ⊥ B are most unstable.

Key Takeaways

  • ✓ Energy principle: δW > 0 for all ξ implies stability
  • ✓ Interchange: unstable when ∇p·κ > 0 (bad curvature)
  • ✓ Kink: requires q > 1 (Kruskal-Shafranov limit)
  • ✓ Tearing: resistive reconnection at resonant surfaces
  • ✓ Ballooning: sets β-limit in toroidal devices
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