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The Interchange Instability

Source lecture(s): PC368 Lec 15

Intuition

This is the Rayleigh–Taylor instability wearing a magnetic costume. Heavy fluid over light fluid falls; plasma pressure supported by a curved magnetic field does the same thing, because field-line curvature acts on the plasma exactly like an effective gravity. The name comes from the mechanism: two flux tubes swap places, and if the swap lowers the total energy it happens spontaneously.

Curvature as gravity

A particle following a curved field line experiences a centrifugal force \(mv_\parallel^2/R_c\). Averaged over the distribution, the plasma feels an effective gravity

\[g_{\rm eff} = \frac{v_{th}^2}{R_c}\]

directed away from the centre of curvature. Then the usual gravitational drift applies:

\[\mathbf{v}_g = \frac{m\mathbf{g}_{\rm eff}\times\mathbf{B}}{qB^2}\]

which is charge-dependent. Perturb the interface and the drift separates ions from electrons on alternate flanks of the ripple; the resulting charge layer makes an electric field; that field's \(\mathbf{E}\times\mathbf{B}\) drift pushes the crest further out and the trough further in. Positive feedback — the ripple grows.

You can build this by hand in the drift orbit lab: the gravitational-drift mode shows step one, the charge separation.

Good and bad curvature

Everything turns on which way the field lines bend relative to the pressure gradient.

  • Bad curvature: field lines curve away from the plasma; \(\mathbf{g}_{\rm eff}\) points down the pressure gradient. Heavy-over-light. Unstable. This is the outboard side of a tokamak, and the outside of any simple mirror.
  • Good curvature: field lines curve toward the plasma. Light-over-heavy. Stable. The inboard side of a tokamak.

Since a torus has both, what matters is the average over a field line — which is what the twist provided by the safety factor delivers. A field line spending enough of its length in good curvature is stable overall, even though parts of it are not. This is the principle of average minimum-B, and it is why cusp and minimum-B mirror geometries were invented.

The growth rate

For a sharp interface the result mirrors Rayleigh–Taylor exactly:

\[\gamma = \sqrt{\frac{g_{\rm eff}}{L_p}} = \frac{v_{th}}{\sqrt{R_cL_p}}\]

with \(L_p\) the pressure-gradient scale length. Short wavelengths grow fastest, so the instability is ultimately limited by finite-Larmor-radius effects — a ripple narrower than a gyroradius is averaged away by the orbit itself, which is a genuine and useful stabilisation.

Where you see it

  • Tokamak ballooning modes — the pressure-driven limit on achievable \(\beta\), localised on the bad-curvature outboard side. The whole point of shaping a plasma cross-section (elongation, triangularity) is to improve this balance.
  • Flute instability in mirrors — so named because the perturbation is constant along \(\mathbf{B}\), fluting the plasma column like a Greek pillar. Killed by minimum-B geometry.
  • The Crab Nebula's finger structures and supernova-remnant filaments — magnetic Rayleigh–Taylor on a parsec scale.
  • Edge localised modes (ELMs) — pressure-gradient-driven eruptions at the tokamak pedestal, a leading engineering problem for ITER.

Common mistakes

  • Assuming a magnetic field always stabilises Rayleigh–Taylor. It suppresses modes that bend field lines, but interchange modes are specifically the ones that do not: flute perturbations swap whole flux tubes without bending them, so magnetic tension never engages.
  • Reading "bad curvature ⇒ unstable" too literally. Only the field-line-averaged curvature matters. A tokamak is stable despite having bad curvature everywhere on its outboard side.
  • Ignoring shear. Magnetic shear couples the mode across surfaces and is a strong stabiliser, independent of curvature.

Knowledge graph position

Prerequisites: MHD, drift motions, MHD equilibrium. Leads to: energy principle, ballooning modes, beta limits, ELMs.

Quiz

Q1 (conceptual). Why does magnetic tension not stabilise the interchange mode?

Answer

Because the interchange (flute) perturbation is constant along \(\mathbf{B}\): it swaps entire flux tubes without bending any field line. Tension only resists bending, so it never engages. That is precisely why this mode is the dangerous one.

Q2 (computational). A mirror has \(R_c = 0.5\) m, \(L_p = 0.1\) m, \(T = 100\) eV for protons. Estimate the growth rate and growth time.

Answer

\(v_{th} = \sqrt{2k_BT/m_p} = \sqrt{2\times1.6\times10^{-17}/1.67\times10^{-27}} \approx 1.4\times10^5\) m/s. \(\gamma = v_{th}/\sqrt{R_cL_p} = 1.4\times10^5/\sqrt{0.05} \approx 6\times10^5\) s⁻¹, so a growth time of about 1.6 µs — instantaneous by any experimental standard.

Q3 (MCQ). Good curvature means field lines curve:

  • (a) away from the plasma, giving effective gravity down the pressure gradient
  • (b) toward the plasma, so the effective gravity points up the pressure gradient — the stable, light-over-heavy arrangement
  • (c) parallel to the pressure gradient
  • (d) not at all
Answer

(b). Curving toward the plasma puts the effective gravity in the stabilising direction. In a tokamak this is the inboard (high-field) side, and twist ensures each field line spends enough length there.