Feasibility of Negative Triangularity Equilibria in the SPARC Tokamak

Abstract: We investigate the feasibility of negative triangularity (NT) plasma configurations in the SPARC tokamak, a compact, high-field device (up to 12.2 T) designed for positive triangularity (PT) operation. Using the FreeGS free-boundary equilibrium solver, we systematically explore the parameter space through 600+ equilibrium calculations spanning triangularity (-0.7-0.7) and elongation (1.0-2.1) while respecting SPARC's coil current limits and first-wall constraints and selecting a reduced-field 8 T operating point. We find that moderate NT double-null plasmas (delta = -0.35, kappa = 1.68) are achievable at 8 T with plasma currents of 2.1 MA. However, NT operation requires 43% plasma volume reduction (from 20.0 to 11.4 m$<sup>3$) compared to the baseline PT design due to vessel walls optimized for PT conformity, and connection lengths are reduced by 40% due to geometric mismatch with the PT-optimized divertor. Central solenoid current requirements decrease by 54% in NT configurations, though specific shaping coils (PF3) require a factor of 5.5 higher currents. All equilibria satisfy fundamental MHD stability criteria with comfortable margins. Thus, while performance is degraded relative to baseline design, these results demonstrate SPARC's potential as a high-field experimental bridge between contemporary NT experiments and proposed NT reactor concepts, capable of testing whether NT's operational advantages (ELM-free operation, favorable impurity transport) persist under reactor-relevant conditions provided by SPARC.

• The paper demonstrates that moderate negative-triangularity double-null equilibria with δ = −0.35 to −0.37 and κ = 1.68 can fit within SPARC’s coil-current and wall constraints at 8 T and about 2.1 MA.
• The analysis shows that NT operation reduces plasma volume by roughly 42–47% and connection length by about 40% versus positive-triangularity cases, while increasing PF3 current demand by approximately 5.5 times.
• The paper finds a 54% reduction in central-solenoid amp-turns for NT relative to matched 8 T positive-triangularity operation, but emphasizes that transport, divertor performance, vertical stability, and pulse-length benefits remain untested.

Motivation and scope

Negative triangularity (NT) tokamak configurations have attracted renewed attention because sufficiently negative edge triangularity suppresses pedestal formation, yielding ELM-free operation, enhanced outward impurity transport, and improved confinement in L-mode-like regimes, as demonstrated on TCV, DIII-D, and ASDEX Upgrade. In parallel, several reactor-scale NT concepts have been proposed, including NT variants of the ARC family such as MANTA. This paper by Yüksek and Golfinopoulos addresses an intermediate question: can SPARC—a compact ($R_0 = 1.85$ m), high-field (12.2 T on-axis) device whose vessel, divertor, and poloidal field (PF) coil system were all optimized for positive triangularity (PT) double-null operation—support NT equilibria at all, and within what limits? To the authors' knowledge, this is the first quantitative assessment of NT feasibility for SPARC.

The study is purely computational: axisymmetric free-boundary Grad–Shafranov equilibria are computed with FreeGS, using Miller parameterization to specify target LCFS shapes (inner/outer midplane points and both x-points), with current limits enforced on all PF coils, copper divertor coils, and central solenoid (CS) circuits.

Methods

Wall and coil coordinates were extracted from published SPARC V2 device footprints. The plasma current was constrained via the Uckan estimate for $q^*$, and pressure profiles were set from a fixed $\beta$ set point of 0.007 (corresponding to the 12 T L-mode case in the SPARC design reports), giving a core pressure scaling $p_0 \approx 10^6 \beta B_\phi^2$. Because SPARC's tight-fitting first wall ($R_{min} = 1.27$ m to $R_{max} = 2.43$ m at the midplane) and PT-optimized divertor would impose severe wall loading at full performance, the authors deliberately targeted a reduced-field operating point: $B_0 = 8$ T, $R_0 = 1.75$ m, conservative $q_{Uckan} = 5.5$, yielding $I_p \approx 2.1$ MA. A matched 8 T PT case with identical field, major radius, and current scaling was computed so that differences between NT and PT can be attributed to triangularity sign rather than to field strength or current—avoiding conflation with the 12.2 T baseline.

Convergence robustness dictated a minimal constraint set: adding strike-point specifications alongside x-point and isoflux constraints frequently caused convergence failure or excessive coil currents. The parameter scan spanned $q^*$0 and $q^*$1 over more than 600 equilibrium calculations, many of which failed to converge; reliable convergence occurred mostly for $q^*$2 and $q^*$3. Success criteria were: convergence, coil currents within engineering limits, x-points and shaping parameters near targets, and the boundary inside the first wall.

Results

Three representative cases are compared:

Parameter	PRD (12.2 T, PT)	PT 8 T	NT 8 T
$q^*$4 / $q^*$5	0.52 / 1.94	0.35 / 1.68	−0.35 / 1.68
$q^*$6 [MA]	8.7	1.99	2.1
Volume [m³]	~20.0	10.30	11.37
$q^*$7	0.29%	0.39%	0.36%
$q^*$8	2.83	4	3.9
Connection length [m]	27.5	29.4	17.5
Total PF amp-turns [MA-turns]	21.66	19.16	22.34
Total CS amp-turns [MA-turns]	51.3	26	11.98

The headline finding is that moderate NT double-null plasmas ($q^*$9 to $\beta$0, $\beta$1) are achievable within SPARC's engineering envelope at 8 T, but only after substantial downsizing: the plasma volume shrinks by roughly 40–47% relative to the PRD baseline because the X-points must move to the low-field side while remaining inside a vessel contoured for high-field-side x-points, with non-vanishing divertor leg lengths. The accessible shape space is narrow—excessive elongation causes wall contact or single-null transitions, while excessive shrinkage inflates shaping-coil demands beyond limits.

Two quantitative trade-offs stand out. First, CS amp-turn requirements drop by 54% in NT relative to the matched PT 8 T case (11.98 vs. 26 MA-turns) and by 77% relative to the PRD, which could translate into extended pulse capability or relaxed solenoid design margins in purpose-built NT reactors. Second, this advantage is offset by a factor-of-~5.5 increase in PF$\beta$2 current (−5.3 vs. +0.96 MA-turns), reflecting the migration of x-points from the high-field to the low-field side; total PF amp-turns remain comparable across cases, but their distribution changes materially.

Connection lengths computed by field-line tracing 1 mm outside the LCFS at the outboard midplane fall by ~40% in NT (17.5 m vs. 27.5–29.4 m), a direct consequence of the geometric mismatch between NT topology and the PT-optimized divertor. Shorter connection lengths are unfavorable for detachment, though the authors note that NT's lack of an edge pedestal may lower edge and divertor temperatures enough to compensate—an assertion that remains untested here.

All three equilibria satisfy fundamental MHD stability criteria with large margins: the Troyon limit ($\beta$3) is exceeded by factors of roughly four to eight in allowed $\beta$4 (e.g., $\beta$5 vs. $\beta$6 for the NT case), and kink safety factor thresholds ($\beta$7) are comfortably met. The reduced currents in the 8 T scenarios provide even greater margin against beta-limited disruptions than the PRD itself.

Limitations and open questions

Several caveats bound these results. The study is limited to fixed-boundary-shape equilibrium solutions at a deliberately low performance point ($\beta$8, $\beta$9); no transport, turbulence, or power-balance modeling is performed, so whether NT's claimed physics benefits (ELM suppression, impurity exhaust, detachment behavior with shortened legs) actually materialize in SPARC geometry is not established. The choice of a minimal constraint set sacrifices fine control of strike points, and the authors report that including strike-point constraints often prevented convergence—meaning divertor targeting under NT has not been demonstrated even computationally. Vertical stability and controllability of the higher-elongation NT configurations are not assessed. The volume penalty (42–47%) and the PF$p_0 \approx 10^6 \beta B_\phi^2$0 overloading are consequences of retrofitting a PT-optimized machine; they quantify the cost of non-optimality but do not generalize to purpose-built NT devices. Finally, the claim that reduced CS demand could enable extended pulses assumes flux consumption scales directly with CS amp-turns, which is asserted rather than analyzed.

Conclusion

This work establishes, through systematic free-boundary equilibrium calculations, that negative triangularity double-null plasmas are feasible in SPARC within its coil current limits and tight-fitting first wall, provided the toroidal field is reduced to 8 T, the plasma is shrunk (minor radius 0.4–0.45 m, $p_0 \approx 10^6 \beta B_\phi^2$1), and operation is held at conservative current (~2.1 MA) and beta. The configuration trades a ~42% volume reduction and a 40% reduction in scrape-off connection length against a 54% reduction in central solenoid demand, with all MHD stability criteria satisfied comfortably. If realized experimentally, such discharges would test whether NT's operational advantages persist at reactor-relevant field strength, providing an experimental bridge between present-day NT devices and proposed NT reactor concepts—and quantifying, by contrast, the design freedoms that purpose-built NT reactors should exploit.