The Kruskal-Shafranov Limit

The current-driven external kink threshold requiring the edge safety factor to exceed unity.

The Limit

The Kruskal-Shafranov limit is the classic stability boundary for the current-driven external kink mode. For a current-carrying plasma column, the m=1, n=1 external kink becomes unstable when the safety factor at the edge falls below one, equivalently when the plasma current exceeds a threshold set by the toroidal field and geometry. Stable operation requires q_edge greater than one, and in practice comfortably above two.

Physical Picture

The kink is driven by the parallel current: a helical displacement of the column lets the magnetic energy of the current decrease, so the perturbation grows. The stabilizing agent is the toroidal field, whose tension resists bending of the nearly straight field lines. When the field line's own pitch (measured by q) becomes too tight, the toroidal field can no longer hold the column and it kinks.

Consequence for Current

Because q at the edge is inversely proportional to the plasma current, the Kruskal-Shafranov condition places a hard ceiling on how much current a given toroidal field can carry stably. This ties maximum current to field strength and is one reason high-field devices can run higher current, which in turn improves confinement and raises the density limit.

Relevance

The current limit set by Kruskal-Shafranov, together with the density and beta limits, defines the operational boundary of a tokamak. Spherical tokamaks such as the Hyperion breeder concept carry high current relative to their field, so edge-q margin against the external kink is a key design constraint; the reported 9.86 MA is a design target for a machine in simulation.