Magnetohydrodynamic stability and the effects of shaping: a near-axis view for tokamaks and quasisymmetric stellarators

Abstract

How much does the cross-section of a toroidal magnetic field configuration tell us about its magnetohydrodynamic (MHD) stability? It is generally believed that positive triangularity (typically leading to bean-shaped cross-sections with their indentation on the inboard side in stellarators) contributes positively to MHD stability. In this paper, we explore the basis of this statement within a near-axis description for axisymmetric and quasisymmetric magnetic configurations. In agreement with the existing literature, we show that positive triangularity stabilises vertically elongated tokamaks. In quasisymmetric stellarators, the toroidal asymmetry of flux surfaces modifies this relation. The behaviour of stellarator-symmetric, quasisymmetric stellarators can still be described in terms of the shape of one of their up–down symmetric cross-sections. However, we show that for a sample of quasisymmetric configurations, the positive-bean-shaped cross-sections do not contribute positively to stability. Unlike in the axisymmetric case, we also learn that finite $\beta$ can improve stability even without magnetic shear.

Constructing the space of quasisymmetric stellarators through near-axis expansion

“…The other parameter, σ(0), is σ = Y C 11 /Y S 11 , a quantity related to the shaping of flux surfaces at the point of stellarator symmetry ϕ = 0. Beyond these definitions, both of these parameters may be connected to natural elements of the geometry of flux surfaces (see [51] for more details). The shapes of the flux surfaces at the first-order are elliptical and can be characterised on the plane orthogonal to the magnetic axis by an elongation E, and rotation angle, ϑ,…”

“…Although choosing B C 22 this way constitutes a well-posed problem, we should not disregard other aspects of the stellarator that B C 22 also affects. This includes magnetohydrodynamic (MHD) stability near the axis [51,60], the shaping of flux surfaces [51], and as a result, the smallest effective aspect ratio of the configuration [57], ϵ max , all of which can be computed at second-order. Characterising the influence of second-order parameters on geometry in a clear way is non-trivial.…”

“…Characterising the influence of second-order parameters on geometry in a clear way is non-trivial. Detailed analysis on shaping was presented as part of [51], which we refer the reader to for a detailed discussion. However, one can show that in the large |B C 22 | limit, flux surfaces become increasingly shaped to limit their achievable aspect ratio.…”

“…The magnetic shear also presents a sign opposite to that of ι 0 , and thus the rotational transform profiles are tokamak-like: the rotational transform decreases towards the edge of the configuration. The magnetic well criterion [51,60] (via the sign of ⟨B 20 ⟩) shows that the majority of the QA branch gives rise to a magnetic hill, and thus is MHD unstable. This aligns with the conclusions reached in [17], where an additional effort was made to reach a solution with a magnetic well.…”

“…The new-QH branch is particularly interesting. Configurations along it lack the common bean-shaped crosssection and exhibit a natural magnetic well [51]. Nevertheless, most remarkably, no existing QS-optimised design belongs to this class.…”