How Does The Magnetic Gradient Scale Length Influence Complexity of Filamentary Coils in Stellarators?

Abstract

The distance between the last closed flux surface (LCFS) and the nearest electromagnetic coils is a dominating factor in the cost, size, and engineering difficulty of stellarators. The smallest magnetic gradient scale length on the LCFS - denoted L_gradB - has been shown to be a good proxy for minimum coil-surface distance in optimizations of a current potential on a winding surface, such as through the REGCOIL method. However, it has not been shown the same is true for filament coils, or that the magnetic gradient scale length is an effective objective function in optimization. In this paper, we explore examples in which min(L_gradB) is correlated with the minimum coil-surface distance for filament coils. First, we analyze a subset of the single-stage-optimized equilibria from the QUASR dataset [Giuliani et al. JPP (2024)]. We find that the majority of configurations have min(L_gradB) located nearby the point of closest coil-surface distance. Second, we optimize quasihelically symmetric equilibria to have improved min(L_gradB), and optimize coils via a continuation method. We then traced alpha particles to test confinement. Finally, we compare min(L_gradB) to the minimum coil-surface distance with filament coils optimized for a set of finite beta equilibria with random boundary shapes. For all datasets, we find that min(L_gradB) is correlated with both the minimum coil-surface and coil-coil distances if sufficient coil length is allowed. Even when there is a trade-off with proxies for confinement, optimizing for improved min(L_gradB) can result in better confinement in the presence of coils, up to a point. This is because - when holding coil-coil distance constant - equilibria with lower min(L_gradB) have a larger normal field error dominated by coil ripple causing particle loss. Both can be reduced by increasing coil-surface distance for equilibria with a high min(L_gradB).

Figures (24)

• Figure 1: Properties of plasma equilibria and associated coil sets for a subset of 3027 QUASR configurations. While initially the correlation between $\mathrm{min}(L_{ \nabla \mathbf{B}})/a$ and $\mathrm{min}(d_{cs})$ seems strong, some of the correlation can be explained by the variation in the aspect ratio. Generally, equilibria with larger aspect ratio have larger $\mathrm{min}(d_{cs})/a$.
• Figure 2: Properties of plasma equilibria and associated coil sets for a subset of 3027 QUASR configurations. This is the same dataset shown in figure \ref{['fig:QUASR_lgradb_over_a_full']} normalized by $R_0$ instead of $a$. When normalized by major radius, $\mathrm{min}(L_{ \nabla \mathbf{B}})$ remains correlated with $\mathrm{min}(d_{cs})$.
• Figure 3: Properties of plasma equilibria and associated coil sets for a subset of 3027 QUASR configurations. This is the same dataset shown in figure \ref{['fig:QUASR_lgradb_full']}. When the length scale on the X axis has been changed from equation \ref{['eq:LgradB']} to equation \ref{['eq:LgradgradB']}, a stronger correlation is found.
• Figure 4: This plot depicts the LCFS for an example equilibrium from the QUASR dataset in terms of $\theta$ and $\phi$. The colorbar represents $L_{ \nabla \mathbf{B}}$, where the smallest values are in yellow. Black dotted lines depict planes of symmetry. In red, the locations of $\mathrm{min}(L_{ \nabla \mathbf{B}})$ are shown. In green, the locations of the smallest $d_{cs}$ are shown. The closest distance between $\mathrm{min}(L_{ \nabla \mathbf{B}})$ and $\mathrm{min}(d_{cs})$ is measured by $\Delta$ and is shown by the blue lines. For this example, $\Delta$ is 0.1.
• Figure 5: A histogram of distances from points of $\mathrm{min}(d_{cs})$ to points of $\mathrm{min}(L_{ \nabla \mathbf{B}})$ or $\mathrm{min}(L_{ \nabla \nabla \mathbf{B}})$ for 3027 equilibria in QUASR. $\Delta$ measures the distance between the locations of the smallest coil-surface distance and the lowest $L_{ \nabla \mathbf{B}}$ or $L_{ \nabla \nabla \mathbf{B}}$. For $L_{ \nabla \mathbf{B}}$, 44.5% of equilibria have $\Delta$ less than 0.1. For $L_{ \nabla \nabla \mathbf{B}}$. 62.7% of equilibria have $\Delta$ less than 0.1.