Spectrally accurate, reverse-mode differentiable bounce-averaging algorithm and its applications

TL;DR

The paper presents a spectrally accurate, reverse-mode differentiable bounce-averaging algorithm implemented in DESC to optimize stellarator performance. By deriving and differentiating the banana-regime neoclassical transport proxy $\epsilon_{\mathrm{eff}}^{3/2}$, the authors enable gradient-based optimization of finite-$\beta$ equilibria while preserving ideal MHD force balance. The approach combines inverse MHD solving, moving-grid spectral mappings, and advanced quadrature to achieve high accuracy and scalability, demonstrated on an OH equilibrium. This work provides a scalable framework for multi-objective stellarator optimization with potential extensions to other transport proxies and stability metrics.

Abstract

We present a fast, spectrally accurate, automatically differentiable bounce-averaging algorithm implemented in the DESC stellarator optimization suite. Using this algorithm, we can perform efficient optimization of many objectives to improve stellarator performance, such as the $ε_{\mathrm{eff}}^{3/2}$ proxy for the neoclassical transport coefficient in the $1/ν$ (banana) regime. By employing this differentiable approximation, for the first time, we optimize a finite-beta stellarator to directly reduce neoclassical ripple transport using reverse-mode differentiation. This ensures the cost of differentiation is independent of the number of controllable parameters.

Figures (14)

• Figure 1: A schematic categorizing neoclassical transport is shown. Most fusion stellarator designs lie in the banana regime where the effective ripple quantifies transport best.
• Figure 2: This figure shows bounce points within $\openint{\zeta_1}{\zeta_2} = (0, 4 \mathrm{\pi})$ on the field line $(\psi, \alpha) = (1, 0)$ for a mesh of $\varrho$ values on a W7-X stellarator. For a given $\varrho$ marked by a horizontal line, $\lvert v_{\parallel}\rvert = 0$ at the bounce points marked by triangles. The plasma distribution vanishes in the hypograph of $\lvert B\rvert$. The collocation nodes to minimize the force balance residual include $\zeta = 0$, so the higher frequency oscillations at $\zeta = 2 \mathrm{\pi}$ are not noise.
• Figure 3: This figure shows $\iota \omega - \Lambda$ at toroidal angles $\phi \in \clopenint{0}{2\mathrm{\pi} /N_{\text{FP}}}$ for a W7-X stellarator in the lab frame. $\iota \omega - \Lambda$ is $(2 \mathrm{\pi}, 2\mathrm{\pi} / N_{\text{FP}})$ periodic in $(\theta \text{ or } \vartheta, \zeta \text{ or } \phi)$.
• Figure 4: This figure shows $\alpha$ at toroidal angles $\phi \in \clopenint{0}{2\mathrm{\pi}}$ for a W7-X stellarator in the lab frame. The discontinuities shown above demarcate the next branch cut of $\alpha$ where $\theta$ crosses $2 \mathrm{\pi}$.
• Figure 5: This figure shows $\theta, \zeta \mapsto \alpha$ and $\alpha, \zeta \mapsto \theta - \alpha$ at the plasma boundary for a W7-X stellarator. $\theta - \alpha$ is $(2 \mathrm{\pi}, \infty)$ periodic in $(\theta \text{ or } \vartheta \text{ or } \alpha, \zeta \text{ or } \phi)$.