On shear Alfvén wave-induced energetic ion transport in optimized stellarators

Abstract

In this work, we investigate prompt ion drift orbit losses caused by shear Alfvén waves (SAW) in quasi-symmetric (QS) and quasi-isodynamic (QI) stellarators optimized for equilibrium confinement of energetic particles (EPs). We use the ideal reduced MHD model for SAW perturbations and study their impact on collisionless EP drift dynamics. We present a semi-analytical model for resonance between the passing EP and SAW, generalized to arbitrary quasi-symmetric configurations including the quasi-poloidal case relevant to QI equilibria. Analysis reveals that an increase in the number of field periods suppresses stochasticity in quasi-helical (QH) and quasi-isodynamic, but not quasi-axissymmetric (QA) stellarators. We show that wave-induced transitions between passing and trapped orbits cause significant losses in QA and QH, but not in QI configurations. For the considered equilibria at scales relevant to fusion power plants (FPPs), we numerically determine SAW amplitudes needed to induce prompt loss of fusion-born alpha particles. Using the weighted Birkhoff averaging technique, we confirm that the onset of prompt losses across all orbit classes occurs with the onset of stochasticity in ion motion. This motivates extending the stochasticity-onset criterion beyond passing orbits in future work.

Figures (5)

• Figure 1: Example of AE3D wave classification based on STELLGAP continuum, for finite $\beta$ QH equilibrium landreman2022optimization considered in § \ref{['sec:analysis']}. The left panel shows the Alfvén continuum computed with STELLGAP, overlaid with AE3D results. Each black dot corresponds to a SAW from the AE3D calculation, placed at the radial location of the peak $|\Phi_{m,n}|$ amplitude across all harmonics (\ref{['eq:deltaPhi']}). Most of these dots follow the continuum curves and identify continuum-damped SAW. The top-right panel shows one such SAW, with its peak at $s=0.86$ (blue circle) at the continuum crossing. Because the vorticity model (\ref{['eq:vorticity']}) does not resolve continuum damping, the wave has abrupt radial structure on the scale of the AE3D radial grid. In contrast, the bottom-right panel shows a radially global SAW with frequency $\omega=34.3\;{\rm kHz}=0.777\omega_{\rm A}^{\rm axis}$, with its peak at $s=0.52$ (red square), in the gap of the Alfvén continuum. In both right panels, colored lines show the five Fourier harmonics (\ref{['eq:deltaPhi']}) with the largest peak $\Phi_{m,n}$ amplitude, with $m$ and $n$ mode numbers labeled in the legend. Gray lines show the remaining lower-amplitude harmonics from the simulation.
• Figure 2: Alfvén continuum (left) and radially global SAW eigenfunctions (right) for QA (top, 30.3 kHz) and QI (bottom, 46.8 kHz) configurations from Table \ref{['tab:equilibria']}, computed with STELLGAP and AE3D. Black dots indicate mode amplitude maxima; red squares mark the highlighted global modes. As in figure \ref{['fig:STELLGAP_AE3D']}, colored lines show the five Fourier harmonics (\ref{['eq:deltaPhi']}) with the largest peak $\Phi_{m,n}$ amplitude, with $m$ and $n$ mode numbers labeled in the legend. Gray lines show the remaining lower-amplitude harmonics from the simulation. The narrow gaps in QA lead to strong continuum damping in the presence of the density shear, whereas the wide helical gaps in QI support radially extended global SAW that persist despite the edge continuum shear near $s \sim 1$ caused by the $\rho_0^{\rm axis}(1-s^5)$ density profile.
• Figure 3: The figure compares equilibrium fields and losses for QA (dotted), QH (solid), and QI (dashed) stellarators from Table \ref{['tab:equilibria']}. Left panel shows (black) dominant Boozer harmonics $B_{M,N}$ and (red) mirror ratio $B_{\max}/B_{\min}$. The right three panels show prompt losses in QA, QH, and QI equilibria, for passing (blue circles), trapped (green squares), and barely-trapped (red triangles) orbit classes. Black diamond lines show total losses across orbit classes. Gray dash-dotted lines show losses without SAW.
• Figure 4: The figure shows interaction between a single harmonic SAW and passing $\mu=0$ alpha particles co-propagating along the quasi-poloidal approximation to the QI field. The QI equilibrium is described in Table \ref{['tab:equilibria']}. The perturbation is the most energetic harmonic of the $\omega=46.8\;{\rm kHz}$ SAW in the QI equilibrium, with poloidal $m=15$ and toroidal $n=16$ mode numbers. Left panel demonstrates resonance analysis (\ref{['eq:poloidal_resonance']}), showing that the mode resonates with the co-propagating trajectory of helicity $h=0.933$ near $s=0.905$ flux surface. This agrees with the Poincaré cross-section shown on the right panel, where there is an island chain at $s=0.905$. The middle panel shows the radial profile $\Phi_{15,16}(s)$ of the mode, with zero crossings $\Phi_{15,16}=0$ creating "island-like" structure on the Poincaré map. Because of the wide spacing between $l=0$ and $l=1$ resonances, the island overlap is avoided for this perturbation.
• Figure 5: Three panels show fraction of low digit accuracy ${\rm DA}<{\rm DA_{threshold}}=3$ fusion-born alpha particles in QA (left), QH (center), and QI (right) equilibria, for passing (blue circles), trapped (green squares), and barely-trapped (red triangles) orbit classes. Black diamond lines show total fraction across orbit classes.