SPARC Tokamak Error Field Expectations and Physics-Based Correction Coil Design

Abstract: Non-axisymmetric magnetic field coils have been designed to provide efficient error field correction and suppress edge localized modes in SPARC - a compact high-field tokamak that is presently under construction at Commonwealth Fusion Systems. These designs utilize the Generalized Perturbed Equilibrium Code's (GPEC's) representation of the multi-modal, non-axisymmetric plasma response to optimize the geometric coupling between 3D coil arrays and the desired core or edge plasma response. Error field correction coils are designed to couple to the plasma-amplified kink that dominates the drive of core resonances. The maximum allowable error field is projected to SPARC using an empirical scaling that is consistent with linear and nonlinear MHD modeling expectations. Asymmetric construction and assembly tolerances are then balanced against the corresponding kA-turns needed for correction to levels below the allowable limit. These physics-driven coil designs provide confidence in our ability to operate SPARC in new high field tokamak regimes without error field induced locked modes.

• The paper presents a physics-based method using SVD and empirical scaling laws to quantify the dominant error fields that drive locked-mode disruptions in SPARC.
• It introduces a probabilistic risk assessment framework with Monte Carlo sampling to evaluate intrinsic error field distributions and guide assembly tolerance decisions.
• The study demonstrates an optimized EF correction coil design, achieving over 57% coupling efficiency with midplane coils to ensure robust spectral control.

Physics-Based Error Field Correction and Risk Assessment in SPARC Tokamak

Introduction

The study presents a comprehensive physics-based framework for the quantification and mitigation of intrinsic non-axisymmetric error fields (EFs) in the SPARC tokamak, emphasizing projection of core locked-mode thresholds and the corresponding design and optimization of error field correction coils (EFCCs). By leveraging ideal MHD response calculations, empirical multi-device scaling laws, and probabilistic risk metrics, the work provides a detailed technical basis for assembly tolerances and EFCC system specification in SPARC, targeting the operational robustness in high-field, compact tokamak scenarios.

Modeling Error Fields: Metrics and Risk Quantification

Dominant Mode Metric

Error fields are quantified using the dominant core resonant mode overlap, computed via singular value decomposition (SVD) of the linear coupling matrix between external non-axisymmetric fields and resonant plasma surfaces. This metric isolates the spectral component of the EF that is maximally coupled to the core $n=1$ tearing mode, which is the principal driver for locked-mode disruptions. The normalized overlap parameter $\delta$ serves as the central quantity for both empirical threshold projection and EFCC coupling optimization.

Empirical Scalings for Locked-Mode Thresholds

Empirical scaling laws derived from a multi-device ITPA database relate the critical EF overlap threshold $\delta_c$ to machine and scenario parameters (e.g., $n_e$, $B_T$, $R_0$, $\beta_N/\ell_i$). The analysis reports that SPARC's projected thresholds are notably below those of ITER at comparable operating points, indicating higher sensitivity due to increased magnetic field and density coupled with compact geometry. Monte Carlo uncertainty propagation across scaling exponents yields probabilistic locking thresholds, enabling non-deterministic risk assessment rather than worst-case bounding.

Nonresonant Error Field Effects

While dominant mode alignment addresses the primary locked-mode risk, residual EF spectral components can degrade performance through NTV torque and edge coupling. Past empirical evidence suggests that even with optimized EFC, practical efficacy is limited to halving the intrinsic EF amplitude, a conservative assumption adopted throughout the risk assessment framework. The study models NTV-imposed limits on correctable EF, showing that for practical coil designs and projected intrinsic rotation in SPARC, NTV does not present a dominant limitation relative to the risk posture.

Assembly Tolerances and Intrinsic Error Field Distributions

Projected EFs from as-built coil asymmetries, winding errors, and installation-induced shifts/tilts are evaluated using detailed coil system models and CAD-based winding descriptions. The dominant EF contributions arise from small, millimeter-scale misalignments, with total EF probability distributions constructed via Monte Carlo sampling over prescribed assembly tolerances. The framework incorporates coherent (common-mode) and individual tolerances (e.g., for central solenoid and PF/VS/DIV coils), mapping tolerance allocations directly to operational risk of locked-mode occurrence.

By integrating the PDF of intrinsic EF with the CDF of locking thresholds, the probabilistic risk is calculated for any tolerance scheme. A 1-in-1000 risk posture for locking in both L- and H-mode scenarios supports tolerances of 1 mm for CS coils and up to 4.3 mm for other coils, balancing cost, engineering feasibility, and operational safety. This risk-driven scheme contrasts with the more conservative, worst-case approach employed in ITER, which would necessitate substantially tighter tolerances.

Physics-Driven Design and Optimization of EF Correction Coils

Spectral Coupling Optimization

Initial EFCC design explorations employ SVD-based coupling metrics to maximize the overlap with the dangerous dominant mode while minimizing nonresonant spectral pollution (i.e., NTV-driving components). Parametric scans of coil cross-sectional length, distance from plasma, and poloidal placement indicate that midplane coils, positioned as close as feasible to the plasma, maximize core-resonant efficiency ($>57\%$ overlap), with further optimization feasible through multi-array phasing.

SPARC EFCC System Specification

The resultant EFCC system comprises three poloidally distributed arrays (upper, midplane, lower), each containing six coils. This configuration aligns closely with the optimal placement and length derived from the physics scans, delivering sufficient flexibility for multi-mode ($n=1$, $n=2$, $n=3$) spectral control and the secondary objective of ELM suppression. The system is capable of exceeding the maximum projected correctable EFs by an order of magnitude, providing robust margin for both anticipated and potential future physics needs.

Multi-array phasing allows spectral purity optimization, with theoretical maximum overlap in excess of 81% achievable, and the midplane coils alone sufficing for most practical scenarios. The conservative design basis, combined with flexible phasing and significant current capability, mitigates the risk posed by NTV and imperfect spectral alignment.

Implications and Future Directions

The methodology and system design demonstrated here establish a technically rigorous framework for balancing assembly tolerances, EFCC authority, and operational risk in high-field, compact tokamaks. The scalable risk-metric approach is transferable to next-generation devices (e.g., ARC), where proximity of EFCCs to the plasma will be further constrained, and rotation profiles become increasingly uncertain due to high power alpha heating.

SPARC’s operational experience will directly inform future empirical scaling development, specifically in testing the applicability of present scalings to high-field, wave-heated scenarios and refining models of nonresonant torque and rotation braking. The probabilistic risk quantification enables dynamic tolerance allocations, supporting accelerated assembly schedules without compromising physics objectives.

Conclusion

The study establishes an authoritative physics-based workflow for quantifying, assessing, and mitigating locked-mode risk due to intrinsic EFs in SPARC, leveraging SVD-based dominant mode metrics, probabilistic scaling law projection, and real-world coil system modeling. Adoption of a 1-in-1000 risk posture supports practical assembly tolerances, and the EFCC system is engineered for robust, flexible spectral control. The framework is directly extensible to future high-field power plant designs, where EF and EFC management will be critical for high reliability and operational performance.