A stellarator metric tracks simulated alpha losses from tokamak coil misalignment

Discrete coils and assembly errors give tokamaks the ripple that quasisymmetry error quantifies

(Image courtesy of Commonwealth Fusion Systems)

Misaligned toroidal field coils can let more fusion-born alpha particles escape a tokamak, and the ARC orbit-following simulations that quantify this ranged from about 10 to 80 CPU node hours. Commonwealth Fusion Systems researchers report that quasisymmetry error, a metric borrowed from stellarator design, tracked simulated alpha particle losses across 13 SPARC magnetic field cases with a Pearson correlation of 0.97, and takes a few minutes on a single CPU once the field is calculated. The authors say it may be used for rapid toroidal field coil design iteration, but the standard form becomes unreliable with large vertical shifts and tilts of ARC coils.

Why discrete coils make quasisymmetry error relevant to tokamaks

Tokamaks are usually described as axisymmetric, but any machine built from discrete toroidal field coils carries ripple, and assembly errors add to it. The authors argue that real tokamaks are better classed as near-axisymmetric devices that share characteristics with quasi-axisymmetric stellarators, so both can be analysed with similar methods. Quasisymmetry error compares the non-axisymmetric components of the field strength with the axisymmetric ones, and stellarator optimisation already uses it as a proxy for fast ion confinement.

Earlier studies had applied the metric across stellarators and to rippled tokamaks such as ITER, and tokamak researchers had analysed ripple through Fourier components of the field. The study applies the stellarator formulation to randomly generated or selected coil misalignment fields for SPARC and ARC and tests it against ASCOT5 orbit-following simulations. Commonwealth Fusion Systems funded the work and provided the ARC V3A equilibrium and SPARC discharge data it builds on, with support from the US Department of Energy Milestone-Based Fusion Development Program.

Quasisymmetry error against orbit-following simulation for SPARC

The SPARC analysis used 13 magnetic fields, comprising a perfectly aligned set, a median set, a deliberately constructed worst case and ten sets selected from 200 random draws to span the range of ripple. Inner-leg points could shift by up to 2 mm radially and toroidally and the outer point by up to 1 mm, which the authors call a somewhat conservative estimate because newer work points to tighter toroidal constraint on the inner leg.

ASCOT5 gave a ripple-induced alpha power loss for each field, and quasisymmetry error at the plasma edge tracked it with a Pearson correlation of 0.97 in a nearly linear relationship. The worst case sat far beyond every random set in both quasisymmetry error and loss, which the authors treat as a conservative upper limit for alpha losses and power loads to plasma-facing components if assembly tolerances hold. With no ripple, SPARC alpha loss comes out at 1.867 per cent in these runs, against 2.8 per cent reported for an older equilibrium, a difference attributed mainly to changes in the safety factor profile and flux-surface shapes and in how markers were initialised. The authors add that real losses will depend on the equilibrium and profiles SPARC actually realises.

Computing cost anchors the authors’ case. The smallest ARC simulations needed about 10 CPU node hours on the Perlmutter system and the largest about 80, while quasisymmetry error takes a few minutes on a single CPU once the field is calculated. Suggested uses in the conclusions include rapid design iteration of toroidal field coils and assessment of assembly errors, where the metric is faster and simpler than full orbit-following simulation and, the authors say, more accurate than the traditional tokamak ripple metric. In the ARC cases, outer midplane ripple could not account for why radial misalignments caused higher alpha losses than toroidal ones of the same size, while quasisymmetry error at the edge followed the losses in a nearly monotonic relationship, which the authors say likely reflects larger non-axisymmetric components of the field there.

Where the proxy for fast ion losses becomes unreliable in ARC

The ARC simulations used one representative coil set for each misalignment type and sigma, with standard deviations from 0.5 to 8 cm that the authors expect to be significantly larger than actual assembly tolerances. At 8 cm, ASCOT5 returned ripple-induced losses of 1.46 per cent for vertical shifts and 3.46 per cent for vertical tilts, roughly 6 and 14 times the levels the standard metric implied.

The authors suggest the gap comes from the poloidal structure of those perturbations and from field components that resonate with the safety factor near the edge, where Poincaré plots show stochastic regions for every misalignment type except toroidal. An adjusted metric counting only the resonant modes correlated better with losses for these cases but under-predicted toroidal misalignment losses. They frame it as a diagnostic for when the standard metric stops applying, not a replacement proxy.

The worst-case SPARC field was built without tilts, which could understate maximum possible losses, although it still far exceeded any plausible random set. Results from ARC also lead the authors to suggest that excluding tilts from earlier SPARC fast-ion simulations may have been erroneous, because tilts add little vacuum ripple yet can raise alpha losses.

Perfectly aligned SPARC and ARC fields fall between moderately and highly optimised quasi-axisymmetric stellarator designs across much of the plasma, although the most precisely optimised of those designs may beat them across the outer half of the plasma volume, depending on coil length. The authors say the comparison is not like for like, since the SPARC field comes from a close approximation of the engineering coil design while the stellarator cases use theoretical single-filament coils. They also caution that quasisymmetry error is insufficient for predicting losses between significantly different devices, although it correlates well across variations on a specific device when misalignments are not too large, since magnetic shear, size, density and temperature also shape fast ion confinement.

Plasma response and orbit resonance remain open for alpha loss modelling

Every ASCOT5 run combined vacuum coil fields with the axisymmetric equilibrium, so three-dimensional plasma response sits outside the results. Recalculating the equilibrium in three dimensions would likely give more reliable answers for the largest misalignments, the researchers write. Resonant magnetic perturbations, which both SPARC and ARC will contain coils to induce if needed for edge-localised mode suppression, were also left out.

Two further lines of work remain. If those coils prove necessary in ARC and the error they add is large compared with toroidal field coil misalignment, alignment tolerances might relax without significantly worsening alpha losses, although that leaves aside other negative effects of misaligned coils. Metrics tied to the resonance between field asymmetries and fast ion orbit pitch, which need particle energy and pitch as inputs, are left for later work.

For now, the results support the metric for modest misalignments, while fields that turn stochastic at the edge require a more rigorous treatment.

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