Computing Flux-Surface Shapes in Tokamaks and Stellarators

TL;DR

The paper introduces a general method to characterize flux-surface shapes in both axisymmetric and non-axisymmetric MHD equilibria by defining symmetry-aligned cross-sections and expressing the cross-section minor radius as a Fourier series, $\rho(z,\eta)=\rho_{\rm eff}(z)\left[1+\sum_{\ell\ge1}\rho_{\ell}(z)\cos(\ell\eta)\right]$, enabling a direct link between shaping parameters (e.g., elongation, triangularity, squareness) and higher-order modes. In non-axisymmetric equilibria, an additional degree of freedom—rotation of shaping modes about the magnetic axis—emerges and is analyzed through the toroidal spectra $\hat{\rho}_{\ell}(k_{\phi})$, with $\rho_{\ell}(\phi_{ma})=\rho_{\ell}(\phi_{ma}+2\pi/n_{fp})$. Analyses of precise QA and QH equilibria, alongside QUASR database samples, reveal a consistent spatial resonance: quasi-symmetry tends to occur when shape complexity and axial rotation align along a low-dimensional line in the FTSS, with the line slope ~$3/1$ for QA and ~$2/1$ for QH, modulated by the rotational transform and number of field periods. The framework provides a pathway to systematically connect flux-surface geometry to quasi-symmetry and other figures of merit, guiding more robust stellarator optimization and future analytic developments.

Abstract

There is currently no agreed-upon methodology for characterizing a stellarator magnetic field geometry, and yet modern stellarator designs routinely attain high levels of magnetic-field quasi-symmetry through careful flux-surface shaping. Here, we introduce a general method for computing the shape of an ideal-MHD equilibrium that can be used in both axisymmetric and non-axisymmetric configurations. This framework uses a Fourier mode analysis to define the shaping modes (e.g. elongation, triangularity, squareness, etc.) of cross-sections that can be non-planar. Relative to an axisymmetric equilibrium, the additional degree of freedom in a non-axisymmetric equilibrium manifests as a rotation of each shaping mode about the magnetic axis. Using this method, a shaping analysis is performed on non-axisymmetric configurations with precise quasi-symmetry and select cases from the QUASR database spanning a range of quasi-symmetry quality. Empirically, we find that quasi-symmetry results from a spatial resonance between shape complexity and shape rotation about the magnetic axis. The quantitative features of this resonance correlate closely with a configuration's rotational transform and number of field periods. Based on these observations, it is conjectured that this shaping paradigm can facilitate systematic investigations into the relationship between general flux-surface geometries and other figures of merit.

Figures (18)

• Figure 1: The magnetic field strength is plotted on the $\psi/\psi_\mathrm{edge}=0.5$ flux surface as a function of symmetry and geometric coordinates in the top and bottom rows, respectively. Each column, from left to right, corresponds to a circular tokamak, a precise QA, and a precise QH configuration, respectively. Included in each panel are the $\eta$ (black) and $\xi$ (white) contours. This shows that while the symmetry-aligned contours are uniformly distributed and orthogonal in symmetry coordinates, they are neither in geometric coordinates. Moreover, significant geometric deformations are observed in the non-axisymmetric configurations.
• Figure 2: The algorithm for identifying a symmetry-defined cross-section is shown graphically in an axisymmetric configuration. Panel (a) shows the $\mathbf{r}_\mathrm{ax}$ vector field over a portion of a flux surface. Panel (b) shows the symmetry-aligned basis vector field $\mathbf{e}_{\xi}$ over the same domain. Then, panel (c) shows the projection of these two fields over that domain. Lastly, panel (d) shows the cross-section described by the roots of the projection. In each panel, the magnetic axis is shown as a green curve, while the point along the magnetic axis where the cross-section is defined is indicated with a green sphere.
• Figure 3: The algorithm for identifying a symmetry-defined cross-section is shown graphically in the precise QA configuration. Panel (a) shows the $\mathbf{r}_\mathrm{ax}$ vector field over a portion of a flux surface. Panel (b) shows the symmetry-aligned basis vector field $\mathbf{e}_{\xi}$ over the same domain. Then, panel (c) shows the projection of these two fields over that domain. Lastly, panel (d) shows the cross-section described by the roots of the projection. In each panel, the magnetic axis is shown as a green curve, while the point along the magnetic axis where the cross-section is defined is indicated with a green sphere.
• Figure 4: The algorithm for identifying a symmetry-defined cross-section is shown graphically in the precise QH configuration. Panel (a) shows the $\mathbf{r}_\mathrm{ax}$ vector field over a portion of a flux surface. Panel (b) shows the symmetry-aligned basis vector field $\mathbf{e}_{\xi}$ over the same domain. Then, panel (c) shows the projection of these two fields over that domain. Lastly, panel (d) shows the cross-section described by the roots of the projection. In each panel, the magnetic axis is shown as a green curve, while the point along the magnetic axis where the cross-section is defined is indicated with a green sphere.
• Figure 5: The $\ell=1,\, 2,\ 3,$ and $4$ shaping modes are shown for how they weight different segments of a cross-section in panels (a), (b), (c), and (d), respectively. Cross-section segments weighted positively are are shown in red while segments weighted negatively are shown in blue. The magnitude of the weights is represented by the distance of each curve from the magnetic axis, located at $(0,\ 0)$ and the blue/red shading. This provides a pictorial interpretation of how the shaping modes relate to familiar concepts like the Shafranov shift and shaping parameters like elongation, triangularity, and squareness.