Quantifying resonant drive in resistive perturbed tokamak equilibria

Abstract: Resonant drive in tokamaks is routinely quantified using a variety of different metrics that target different aspects of a resonant response to an external perturbation. Two of the most direct metrics, $Δ<em>{mn}$ and $b</em>{pen}$, are widely used but their relative behavior was previously uncharacterized. This work examines how these metrics representing the shielding current and penetrated field relate in resistive perturbed tokamak equilibria using asymptotically matched solutions with a resistive MHD inner layer model in GPEC. $b_{pen}$ scales with Lundquist number as $S<sup>{-2/3}$ until saturation at low $S$, and $Δ_{mn}$ remains consistent with its ideal definition but is affected by global kink structure. Both metrics are shown to yield closely similar dominant coupling modes within the same resistive model. However, the resistive physics shifts this dominant mode spectrum to lower poloidal mode numbers $m$ in a low-rotation ITER equilibrium. This alteration is predicted to be observable in experiment in the form of optimal relative phasings of resonant magnetic perturbation coils.

• The paper develops a finite-resistivity definition of the shielding-current metric Δₘₙ and shows it remains relatively stable across resistivity, while the penetrated resonant field scales approximately as S⁻²ᐟ³ before saturation.
• The paper finds that Δₘₙ and the penetrated resonant field are not directly comparable as scalar amplitudes, yet they produce closely matching dominant coupling modes when calculated within the same resistive model.
• The paper predicts that resistivity shifts ITER’s dominant spectrum toward lower poloidal mode numbers and could make ideal coil phasing only 68% effective, motivating experimental tests in low-rotation plasmas.

Motivation and scope

Resonant magnetic perturbations (RMPs) in tokamaks—whether from error fields or intentional ELM-suppression coils—are routinely assessed through a proliferation of "resonant metrics": X-point displacement, the legacy 3-mode metric, island overlap widths and Chirikov parameters, the dominant mode overlap $\delta$, the field derivative discontinuity $\Delta_{mn}$ (equivalently the shielded resonant flux $b_{sh}^{res}$), and the penetrated resonant field $b_{pen}^{res}$. The two most direct of these, $\Delta_{mn}$ and $b_{pen}^{res}$, quantify complementary aspects of the plasma response: $\Delta_{mn}$ is proportional to the shielding current at a rational surface and is well-defined only in ideal MHD, where a singular current sheet enforces zero penetrated field; $b_{pen}^{res}$ is the pitch-resonant surface-normal field that exists only when resistivity permits reconnection and island opening. Prior to this work by Pharr et al. (2603.18267), the relationship between these two metrics in resistive perturbed equilibria had not been characterized, leaving open whether ideal-model error-field-correction (EFC) designs remain valid when resistive physics is included.

The paper addresses this gap using the resistive GPEC framework, which couples force-free outer solutions from Resistive DCON's Galerkin method to a Glasser-Green-Johnson (GGJ) inner layer via asymptotic matching in RMATCH. The growth rate entering the inner-layer equations is taken as the Doppler-shifted frequency $\gamma = i n\Omega$, analogous to MARS-F but with resistivity retained only where it matters—near rational surfaces.

Defining $\Delta_{mn}$ with finite resistivity

With finite resistivity, the ideal surface current $\Delta_{mn}$0 broadens into an integrable distribution over a width $\Delta_{mn}$1, so $\Delta_{mn}$2 must be reinterpreted as an integrated shielding current, denoted $\Delta_{mn}$3, evaluated across a finite jump width. The authors derive this generalization from Boozer's formulation via Ampère's law, showing that the integrated parallel current remains proportional to the finite jump in $\Delta_{mn}$4 across the broadened sheet.

The choice of jump width is physically motivated rather than arbitrary: since the GGJ inner layer width scales as $\Delta_{mn}$5, the authors adopt

$\Delta_{mn}$6

placing the evaluation points outside the region where the missing shielding current distorts the derivative, yet inside any significant global kink structure. A scan in a DIII-D-like equilibrium shows $\Delta_{mn}$7 varies by only about 20% across all resistivities for this choice, confirming robustness. Two caveats are stated plainly: the definition holds only locally near the rational surface under the assumption that perturbed current stays parallel to the equilibrium field, and the optimal constants in $\Delta_{mn}$8 will differ between devices of very different size.

Metric behavior versus Lundquist number

Across four equilibria—an analytic large-aspect-ratio (LAR) case with one rational surface, a two-surface LAR case, an ITER 5 MA low-rotation L-mode from DINA, and a high-rotation DIII-D-like H-mode from TokaMaker—the two metrics behave in distinctly different ways:

• Penetrated field: $\Delta_{mn}$9 increases monotonically with resistivity, following $b_{sh}^{res}$0, until saturation at low $b_{sh}^{res}$1. Outer surfaces saturate first in ITER, plausibly due to lower rotation; in the DIII-D-like case, penetrated fields remain at the numerical floor until $b_{sh}^{res}$2 ($b_{sh}^{res}$3) and $b_{sh}^{res}$4 ($b_{sh}^{res}$5), attributed to rapid rotation advecting the forming island away from the external perturbation.
• Shielding current proxy: $b_{sh}^{res}$6 remains visually indistinguishable from its ideal value down to quite low $b_{sh}^{res}$7 in the LAR cases, deviating only when resistivity becomes large enough to alter global kink structure. In ITER and DIII-D-like equilibria its behavior is non-monotonic, and it does not clearly converge to the ideal value within the numerically accessible range.

The central conclusion of this section is that scalar values of the two metrics are not directly comparable measures of resonant drive—they vary independently with resistivity. However, this does not preclude agreement at the level of mode spectra, examined next. A structural limitation applies throughout: asymptotic matching assumes a thin inner layer, which fails at very low $b_{sh}^{res}$8, and the authors note numerical sensitivity of RDCON outputs, particularly at large rotation.

Dominant mode spectra: metric consistency, resistive modification

Coupling matrices relating energy-normalized edge flux to each resonant metric were constructed and analyzed via SVD, with core-dominant modes computed excluding surfaces beyond $b_{sh}^{res}$9. Two results emerge:

First, within the same resistive model, the dominant modes from $b_{pen}^{res}$0 and $b_{pen}^{res}$1 are closely similar—in the DIII-D-like case both nearly overlap the ideal shielded-field dominant mode; in the ITER case the two resistive dominant modes overlap almost perfectly despite peaks at $b_{pen}^{res}$2 and strong negative-$b_{pen}^{res}$3 sensitivity. This validates the use of $b_{pen}^{res}$4-based coupling matrices in other resistive codes such as M3D-C1 and MARS-F, and implies that EFC coil design decisions are insensitive to the choice between these two metrics.

Second, and more consequentially, resistive physics shifts the ITER dominant mode spectrum toward lower poloidal mode numbers $b_{pen}^{res}$5 relative to the ideal calculation (ideal peak at $b_{pen}^{res}$6–6 versus resistive peak at $b_{pen}^{res}$7). A resistivity scan confirms smooth convergence back toward the ideal spectrum as $b_{pen}^{res}$8. Because tilt and shift coil misalignments preferentially produce low-$b_{pen}^{res}$9 fields, this shift may increase error-field risk in low-rotation ITER L-mode plasmas—a regime particularly susceptible to mode locking.

Experimentally testable coil phasing predictions

The spectral shift translates into concrete, falsifiable predictions for optimal relative phasings of ITER's three EFC coil sets, computed by maximizing the overlap $\Delta_{mn}$0. Both resistive metrics yield identical optimal phasings, but these differ substantially from the ideal prediction: upper–lower phasings differ by roughly 44°, middle–lower by 124°. Critically, applying currents phased per the ideal model would couple at only 68% strength to the resistive dominant mode with a 94° phase mismatch—sufficient to render model-derived EFC ineffective if the resistive description is correct. The authors propose validating this on ITER or comparable multi-coil devices by identifying the phasing that minimizes locking threshold current in a low-rotation, high-resistivity equilibrium, cross-checked against MARS-F and linear M3D-C1.

Limitations and open questions

Several limitations bound the results. The GGJ inner layer neglects drift-kinetic, neoclassical, and two-fluid physics present in more modern inner-layer models, and the assumption $\Delta_{mn}$1 means tearing stability is not determined. Asymptotic matching breaks down when the inner layer ceases to be thin—at the lowest Lundquist numbers explored—and the authors acknowledge that convergence of $\Delta_{mn}$2 to its ideal value at very high $\Delta_{mn}$3 could not be shown due to numerical limits. In the DIII-D-like case, the near-agreement of the two resistive dominant modes may partly reflect the difficulty of resolving $\Delta_{mn}$4 above its numerical floor. Open questions left by the paper include whether the resistively modified dominant-mode spectrum persists with alternative inner-layer models, whether nonlinear simulations confirm the linear predictions, and whether the low-$\Delta_{mn}$5 shift affects construction tolerancing assessments for ITER and pilot plants.

Conclusion

This work establishes that $\Delta_{mn}$6 and $\Delta_{mn}$7, though individually incomparable as scalars in resistive equilibria, yield consistent dominant coupling modes when evaluated within the same resistive model—strengthening confidence in resistive-code-based EFC and RMP design. It further demonstrates, with quantitative phasing predictions for ITER, that resistive physics can materially alter the dominant mode spectrum in low-rotation regimes, providing a specific experimental signature by which the validity of ideal versus resistive modeling of resonant drive can be adjudicated.