Seed-Triggered Onset of Neoclassical Tearing Modes: A Minimal Bifurcation Model and an Open Empirical Test

Neoclassical tearing modes (NTMs) are a principal performance limit in tokamak plasmas: magnetic islands sustained by a helically perturbed bootstrap current, linearly stable yet nonlinearly unstable, and generally requiring a finite "seed" perturbation to trigger sustained growth [1, 2]. This working paper revisits the standard two-term modified Rutherford equation for NTM island width and shows that its equilibrium structure is a textbook saddle-node (fold) bifurcation: a classically stable branch at zero island width, and, above a critical drive threshold, an unstable seed-threshold branch and a stable saturated branch. This is not a new physical result — the qualitative picture is well established in the fusion literature [1–5] — but writing the two equilibria in closed form yields an exact, parameter-independent relation between the seed threshold and the saturated width that, to the author's knowledge, has not been stated explicitly as a directly testable prediction. We state this relation, situate it against published parameter ranges from DIII-D and NSTX, note an existing experimental result that already supports the qualitative bifurcation picture, and pose the precise empirical check that would confirm or falsify the closed-form relation. This is offered as a short, self-contained exercise rather than a comprehensive theory.