Measures of quasisymmetry for stellarators

Abstract

Quasisymmetric stellarators are an attractive class of optimised magnetic confinement configurations. The property of quasisymmetry (QS) is in practice limited to be approximate, and thus the construction requires measures that quantify the deviation from the exact property. In this paper we study three measure candidates used in the literature, placing the focus on their origin and a comparison of their forms. The analysis shows clearly the lack of universality in these measures. As these metrics do not directly correspond to any physical property (except when exactly quasisymmetric), optimisation should employ additional physical metrics for guidance. It is suggested that close to QS minima, one should treat QS metrics through inequality constraints so that additional physics metrics dominate optimisation. The impact of different quasisymmetric measures on optimisation is presented through an example, for which the standard metric that weights the asymmetric Fourier modes of the field magnitude appears to perform best.

Phases and phase-transitions in quasisymmetric configuration space

“…We draw attention to the broken orange lines in Figure 4, which include some estimates for the transitions. In particular, in addition to the expressions for (12), we have considered the other two sign combinations. For k ≥ 3, these are not too bad as guides for the configuration phase.…”

“…When a single harmonic dominates, the symmetric torus unknot is expected to be a good model, and we expect to find s n somewhere between unity and n. In practice, though, one finds multiple harmonics to be relevant, which leads to a growth in complexity (see Figure 4). Although the comparison of s n with unity is still a good first test, a more detailed comparison will be necessary using the analytic conditions (12) or their generalisation (D3). Generally, we shall not need to deal with all the complex, intermediate phases, as strong shaping tends to make them impractical.…”

“…In this appendix, we show the universality of the phase transitions that we found in Section IVC, Eqs. (12). The idea is to prove that these transitions prevail as one includes an arbitrary number of harmonics in the description of the magnetic axis, as well as allow for any choice of Z n with respect to R n .…”

“…Little is known (beyond experience gained by performing numerous optimization studies) of the global spatial structure of the optimization space and how different forms of cost functions shape it. 12 The present paper is the first of a sequence of papers a) Email: eduardor@princeton.edu b) Email: ws3883@princeton.edu c) Email: amitava@princeton.edu where we will attempt to shed some light on the structure of configuration space as it pertains to QS. To do so, we identify configurations approximately with sets of ordered functions and parameters, beginning with the magnetic axis and moving outward.…”

The DESC Stellarator Code Suite Part III: Quasi-symmetry optimization

“…The equilibrium was then optimized for quasi-symmetry on the last closed flux surface (ρ = 1) in this two-dimensional parameter space using each of the three objective functions described previously, and with both first and second-order optimization methods. This simple optimization space was chosen for comparison to a previous study of this problem 24 , and it lends itself well to visualization. Note that in the example used here, quasi-symmetry is only being targeted on a single flux surface and the rotational transform profile is held fixed during optimization.…”