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It has long been understood that alpha particle equilibrium in tokamaks relies on the existence of approximate symmetries in time, gyroangle and toroidal angle. Violation of these symmetries in tokamaks leads to energetic particle losses observed in simulation and experiment. This behaviour is well modelled in terms of collisionless particle trajectories.
In stellarators, a chief concern for confinement is to adequately substitute for these symmetries when nominal axisymmetry is not available. A number of schemes have been proposed, but underlying them all is the realization that the modulus of B is the most important determinant of the guiding centre motion, and the quantity upon which a symmetry must be constructed. Quasi-symmetric solutions to the force balance equation in stellarators have been found by Landreman and co-authors which promise integrable guiding-centre motion. A key question is whether the full-orbit motion remains in the neighbourhood of the perfectly confined guiding-centre orbits or whether a secular component to the trajectory accumulates to deconfine the particles. Expensive Monte-Carlo simulations can always give a definitive answer to this question for any given case under scrutiny, but insights into the reasons can be elusive, and the optimizations required for improvement are not obvious.
Exploiting the similarity between quasi-axisymmetric motion and tokamak motion, we have constructed a novel method to capture the full orbit corrections as a fictitious magnetic perturbation. All deviations due to ripples, error fields, Alfven waves, and FLR effects may therefore be treated in a unified manner. The most important component of this fictitious field is revealed to be one that does not depend on gyroangle and relates to the torsion of the magnetic field 𝜏≡(𝑏̂⋅∇𝑒̂2)⋅𝑒̂1. Accumulating guiding centre trajectory error can be expected for resonant orbits in the usual manner once drift-island overlap occurs. The magnitude of the fictious ripple is approximately 𝛿𝐵𝐵∼𝜏𝐿𝜌𝛼∗ 2 suggesting a crude estimate of 𝜌𝛼∗ ∼10% for island overlap resulting in alpha loss. Details of the torsion and how the resonance width depends on shear are likely to lower this estimate in specific cases.
This work has been funded by the EPSRC Fusion Grant 2022/27 [grant number EP/W006839/1]. To obtain further information on the data and models underlying this paper please contact PublicationsManager@ukaea.uk*.