Energetic-particle orbits near rational flux surfaces in stellarators: I. Passing particles

TL;DR

The study analyzes drift islands formed by passing energetic particles near rational flux surfaces in stellarators, establishing that drift islands are governed by a transit adiabatic invariant. It derives the drift-island shape and width, showing a key scaling w ∝ (ρ★ δ / s)^{1/2} a and identifying cyclometric magnetic fields as a sufficient condition for eliminating drift islands for all passing particles. The work extends the theory with higher-order transit invariants and a Hamiltonian perturbation framework, achieving excellent agreement with ASCOT5 simulations up to 3.5 MeV and providing a systematic method to design cyclometric or nearly cyclometric stellarators with favorable confinement. These results offer concrete design principles for minimizing alpha-particle transport while accommodating other optimization goals, including omnigeneity and pseudosymmetry considerations, in reactor-scale devices.

Abstract

Recent simulations have shown that, even when the magnetic field of a stellarator possesses nested toroidal flux surfaces, the orbits of passing energetic particles can exhibit islands. These 'drift islands' arise near rational flux surfaces, where they are likely to enhance alpha-particle transport -- flattening the alpha density profile locally -- unless they can be avoided by suitable design of the stellarator magnetic field. To investigate how this might be achieved, we derive an equation for the drift-island shape in a general stellarator. This result follows from the solution to a more fundamental problem: that of calculating the orbits of passing particles near a rational flux surface. We show that these orbits are determined by conservation of an adiabatic invariant associated with the closed rational-surface field lines. We use this 'transit adiabatic invariant' to prove that there are no drift islands, for all passing particles, if and only if the magnetic field satisfies a weaker version of the Cary-Shasharina condition for omnigeneity; we call such magnetic fields 'cyclometric'. The drift-island width scales as $\sim (ρ_\starδ/s)^{1/2} a$ ($ρ_\star$ is the normalized gyroradius, $δ$ is the deviation from cyclometry, $s$ is the magnetic shear, and $a$ is the minor radius), so large drift islands could arise in low-shear stellarators that are insufficiently cyclometric. To ensure accurate results for very energetic particles, we compute higher-order corrections to the transit invariant. Our calculations agree extremely well with ASCOT5 guiding-centre and full-orbit simulations of alpha particles in reactor-scale equilibria, even at $3.5\text{MeV}$. Finally, we show how our results can also be derived using Hamiltonian perturbation theory, which provides a systematic framework for calculating passing-particle orbits on both rational and irrational surfaces.

Figures (28)

• Figure 1: Poincaré plots, constructed in the $\phi = 0$ plane using cylindrical coordinates $(r, \phi, z)$, for $(a)$ magnetic field lines and $(b)$ guiding-centre orbits of passing alpha particles, in a W7-X equilibrium scaled up to reactor size. The alpha particles in $(b)$ are co-passing relative to the magnetic field and have energy $3.5MeV$ and pitch-angle $\mu/\mathcal{E} = 0.143\per T$, where $\mu$ is their magnetic moment and $\mathcal{E}$ is their kinetic energy per unit mass. The chosen value of $\mu/\mathcal{E}$ coincides with the trapped--passing boundary at the last closed flux surface, so the orbits in $(b)$ are barely passing. The field lines and guiding-centre orbits were computed using the ASCOT5 code; more details about the equilibrium and simulation method are provided in \ref{['subsec:islandplots']}.
• Figure 2: (top row) Illustration of how the rotational transform of a flux surface affects the magnetic field lines in physical space, with colour showing the magnetic field strength. For the flux surface shape, we used a precise-QA equilibrium from Landreman2021. (bottom row) Cartoons showing how the rotational transform affects the field lines in straight-field-line coordinates for a single-field-period stellarator, with grey contours representing the magnetic field strength. ($a$) On an irrational surface, field lines cover the surface densely and never close. ($b$) On a rational surface $\psi_{\rm{r}}$, field lines close on themselves. ($c$) Near the rational surface $\psi_{\rm{r}}$, field lines require many toroidal turns to explore the entire surface.
• Figure 3: Cartoon showing the difference between a closed curve of constant $\theta - (N/M)\mkern1.5mu \zeta$ (shown in red) and a field line (black) in straight-field-line coordinates on a flux surface near the rational surface. In this case, the stellarator has a single field period, $N=1$, and $M=2$. Grey contours represent the magnetic field strength. The change in poloidal angle, $\Delta\theta$, when the field line is followed for one full transit ($\Delta\zeta = 2\pi M$) is indicated.
• Figure 4: Diagram of the coordinate system that we use. The outermost flux surface displayed here is the rational surface $\psi_{\rm{r}}$, on which a magnetic field line is drawn. The constant-$\eta$ surface that coincides, on $\psi_{\rm{r}}$, with this field line is shown in red.
• Figure 5: (top row) Examples of possible $I_{\rm{r}}(\eta,\mathcal{E},\mu)$, normalized by a typical value $I_{\rm{r}0}\sim vL$, plotted against $\eta$. (bottom row) Particle orbits near the rational surface, visualized by plotting level sets of the invariant $\mathcal{I}_{\rm r}(\psi,\eta,\mathcal{E},\mu,\sigma)$ (defined in \ref{['eq:simpleinvariant']}) corresponding to the $I_{\rm{r}}$ functions above. These level sets are plotted against $\psi-\psi_{\rm r}$ (normalized to be dimensionless) on the vertical axis and $\eta$ on the horizontal axis. Co-passing ($\sigma=1$) and counter-passing ($\sigma=-1$) orbits are plotted in red and blue, respectively. For these plots, $\iota_{\rm r}'$ was assumed to be negative.

Theorems & Definitions (1)

• Definition 7.1