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Sausage & Kink: Current-Driven Instabilities

Source lecture(s): PC368 Lec 15

Intuition

Interchange modes are driven by pressure. These are driven by current — and since a tokamak's confining field comes from the plasma current itself, they are unavoidable by construction. They are classified by the azimuthal mode number \(m\) of the perturbation \(\xi \propto e^{i(m\theta + kz)}\).

\(m = 0\): the sausage

Squeeze the column locally. The azimuthal field \(B_\theta = \mu_0 I/2\pi r\) increases where \(r\) decreases, so the pinch force \(B_\theta^2/\mu_0 r\) gets stronger exactly where the column is already thinnest.

Positive feedback → the column necks off in microseconds, producing the sausage-link morphology that killed the early Z-pinch programme. The disruption at the neck generates enormous inductive voltages — this is the mechanism behind the intense neutron and X-ray bursts from a dense plasma focus.

Cure: an axial field \(B_z\) trapped inside the plasma. Compressing the neck now costs magnetic pressure \(B_z^2/2\mu_0\), and since flux is conserved \(B_z \propto 1/r^2\) rises faster than the destabilising term. Stability requires roughly \(B_z^2 > B_\theta^2/2\).

\(m = 1\): the kink

Bend the column sideways. Field lines bunch up on the concave (inner) side of the bend where they are compressed, so magnetic pressure is higher there, pushing the bend further out.

Positive feedback → the column writhes into a helix. In a tokamak an \(m=1\) kink that reaches the axis produces the sawtooth crash: the core temperature collapses in tens of microseconds via reconnection at the \(q = 1\) surface, then rebuilds and does it again. Sawteeth are usually tolerated — they even help flush impurities — but they seed more dangerous modes.

The Kruskal–Shafranov limit

Kink stability requires the field line to twist less than once around the torus over its full toroidal circuit:

Kruskal–Shafranov condition

\[q(a) = \frac{aB_\phi}{RB_\theta} > 1\]

with \(q\) the safety factor. Since \(B_\theta\) comes from the plasma current, this is a hard ceiling on current at fixed toroidal field:

\[I < \frac{2\pi a^2 B_\phi}{\mu_0 R}\]

In practice machines stay above \(q(a) \approx 2\)–3, because \(q < 2\) invites \(m = 2\) tearing modes even when the ideal kink is stable. And that ceiling on current is a ceiling on \(B_\theta\), hence on confinement — one of the central design tensions in tokamak physics: the current that makes confinement possible is also the free energy that destroys it.

Internal, external, and the wall

  • Internal kink (\(m=1\), inside \(q=1\)): mild, gives sawteeth.
  • External kink: displaces the plasma boundary; violent, and a leading cause of disruptions — the sudden loss of the entire plasma current, dumping megajoules into the wall and driving runaway electrons. Disruption mitigation is one of ITER's key engineering challenges.
  • Resistive wall mode: a nearby conducting wall stabilises the external kink by opposing flux change — but only while wall currents persist. On the wall's resistive timescale the mode grows back, slowly enough to be caught by active feedback coils. Real machines fly this way deliberately.

Astrophysical kinks

The helical wiggles in astrophysical jets — most famously in M87 — are widely interpreted as \(m=1\) kinks on a current-carrying column. The same instability, at a scale of kiloparsecs rather than centimetres, from the same free energy.

Common mistakes

  • Confusing current-driven with pressure-driven. Kinks persist at \(\beta \to 0\); the free energy is in the current, not the pressure.
  • Assuming \(q > 1\) everywhere is enough. It is necessary, not sufficient. Resistive modes (tearing, neoclassical tearing) grow at rational surfaces even where ideal MHD is stable.
  • Thinking a conducting wall solves the external kink. It converts a fast instability into a slow one. Slow is a great deal better, but it is not stability.

Knowledge graph position

Prerequisites: MHD equilibrium, magnetic stress tensor, pinch equilibria. Leads to: energy principle, disruptions, tearing modes, magnetic reconnection.

Quiz

Q1 (conceptual). Why does an internal \(B_z\) stabilise the sausage mode but a purely azimuthal field cannot?

Answer

Flux conservation makes the trapped \(B_z \propto 1/r^2\) as the neck compresses, so its magnetic pressure rises faster than the destabilising \(B_\theta^2/\mu_0r\). The azimuthal field alone only reinforces the squeeze, since \(B_\theta \propto 1/r\) grows in the same direction as the collapse.

Q2 (computational). A tokamak has \(a = 0.5\) m, \(R = 1.7\) m, \(B_\phi = 3\) T. What is the maximum current allowed by Kruskal–Shafranov?

Answer

\(I < 2\pi a^2B_\phi/\mu_0R = 2\pi(0.25)(3)/(4\pi\times10^{-7}\times1.7) \approx 2.2\) MA. Operating at \(q(a) = 3\) instead of 1 would cut this to about 0.7 MA.

Q3 (MCQ). The sawtooth crash in a tokamak is associated with:

  • (a) an external kink at the plasma boundary
  • (b) an internal \(m=1\) kink and reconnection at the \(q=1\) surface
  • (c) the sausage instability
  • (d) an interchange mode on the outboard side
Answer

(b). The internal \(m=1\) mode grows inside \(q=1\), reconnects at that surface, flattens the core profiles in tens of microseconds, and the cycle repeats as the current profile peaks again.