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

Abstract

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 and test it against the open data from [9] for DIII-D discharge #186472. Using the reported small-island width $ w_1 \approx 1.9\ \text{cm} $ (an upper bound on the true seed threshold) and saturated width $ w_2 \approx 6.4\ \text{cm} $, we find $ w_1 \cdot w_2 \approx 12.16\ \text{cm}^2 $, implying an effective threshold scale $ w_d \approx 3.5\ \text{cm} $. This value is consistent with the expected polarization scale when profile corrections and finite transport effects are included [2, 9, 10]. The result suggests that the saddle-node bifurcation model is quantitatively consistent with the observed NTM onset dynamics in this discharge. The author invites independent verification using the full dataset, including the profile-corrected calculation of $ w_d $.