A Dynamical Bridge from Tearing-Mode Torque Balance to Forced Kuramoto Dynamics: Coherent Forcing Geometry and Mode Locking

Abstract

Mode locking of tearing modes is a major precursor to tokamak disruptions. We show that the overdamped torque balance for a tearing-mode island interacting with an error field is exactly the Adler phase-locking equation and, for multiple resonant surfaces, has the form of a heterogeneously forced Kuramoto network. This bridge gives a native interpretation to the complex error-field profile. For normalized surface drives, we prove that the coherent projection F equals the maximum common-phase drive amplitude per surface divided by the root-mean-square drive scale, while F squared is the coherent fraction of the squared forcing budget. Extending the source model's matched-forcing theorem, we prove a sharp result for the reduced phase model: over unconstrained unit-root-mean-square complex surface drives, an exact all-mode common-phase equilibrium exists if and only if the drive-rate budget is at least the root-mean-square detuning. A closed-form detuning-matched profile attains the threshold independently of the phase-difference coupling matrix; stability under general heterogeneous coupling is not claimed. Separately, a prospectively specified clean-room study of 2,124 finite Kuramoto simulations found that the tested unmatched low-F profiles strongly reduced coherence and locking relative to uniform forcing, with the strongest cell changing from 30/30 locked runs to at most 4/30. These source-model results motivate, but do not prove, the plasma hypothesis that the weighted coherent projection across resonant surfaces correlates with tearing-mode locking susceptibility. The paper distinguishes exact reduced-model statements, imported numerical evidence, and plasma-facing hypotheses, and identifies few-mode heterogeneous-coupling and nonlinear-MHD validation as open tests. This version incorporates corrections from an adversarial multi-model audit: a strict-inequality statement of the Adler stability condition with the saddle-node marginality at equality made explicit; an exact sign-convention statement relating the net physical torque to the normalized common-phase drive; and a corrected DOI for one reference.