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MHD Instability

Source lecture(s): pc368_lec15_dynamics_instability

Intuition

An MHD equilibrium is a delicate balance: magnetic pressure bends inward, thermal pressure pushes outward. If the plasma “knows” that there is free energy stored in a non-uniform pressure or current profile, tiny perturbations can grow exponentially, releasing that energy violently. These modes—kink, ballooning, tearing—limit the performance of magnetic confinement devices.

Formal Definition

An MHD configuration is unstable if an infinitesimal perturbation \(\boldsymbol{\xi}(\mathbf{r}, t) = \boldsymbol{\xi}_0 e^{\gamma t}\) grows (\(\gamma^2 > 0\)) in time, converting magnetic or thermal energy into kinetic energy.

Mathematical Formulation

The energy principle states that the potential energy change due to a displacement \(\boldsymbol{\xi}\) is:

\[\delta W = \frac{1}{2}\int_V \Bigl[ \frac{|\mathbf{B}\cdot\nabla\times(\boldsymbol{\xi}\times\mathbf{B})|^2}{\mu_0} + \gamma p |\nabla\cdot\boldsymbol{\xi}|^2 + \dots \Bigr] dV\]

If \(\delta W < 0\) for any admissible \(\boldsymbol{\xi}\), the equilibrium is unstable.

Derivation

  1. Perturb the MHD equations: \(\mathbf{B} \to \mathbf{B} + \delta\mathbf{B}\), \(\mathbf{U} \to \delta\mathbf{U}\), etc.
  2. Combine perturbed induction and Faraday’s laws to get \(\delta\mathbf{B}\).
  3. Derive the equation of motion for \(\boldsymbol{\xi}\).
  4. Multiply by \(\boldsymbol{\xi}^*\) and integrate over the volume.
  5. After boundary-condition integration by parts, obtain the boundary-value problem whose eigenvalues \(\gamma^2\) reveal stability.

Worked Example

Current-driven kink: A Z-pinch with uniform current profile (\(\mathbf{J} = J_0 \hat{z}\)) is unstable to the m=1 kink if the safety factor \(q < 1\). Because \(q = (r B_z)/(R B_\theta) < 1\) implies the field line wraps around the axis less than once per poloidal circuit, the field lines can open into a helical shape that shortens the path, releasing magnetic energy.

Common Mistakes

  • Stability = existence of equilibrium. Many solutions of \(\mathbf{J}\times\mathbf{B}=\nabla p\) are unstable to small perturbations.
  • Only ideal MHD matters. Resistive instabilities (tearing modes) require \(\eta \neq 0\) and are not captured by ideal energy principle.
  • Large-\(n\) modes are always stable. Ballooning modes at high toroidal mode numbers can dominate in high-\(\beta\) tokamaks.

Quiz Questions

  1. Conceptual: Why does the kink instability grow faster at higher current?
  2. Computational: For a step-function current profile, compute the linear growth rate of the m=1 internal kink.
  3. MCQ: The energy principle applies to:
  4. A) Ideal MHD only
  5. B) Resistive MHD only
  6. C) Both ideal and resistive
  7. D) Neither

Further Reading

  • B. Coppi, MHD Stability Theory, review in Phys. Fluids.