Optimized finite-$β$ tokamak-stellarator hybrid configurations achieved by planar dipole-field coils

Abstract

Tokamak--stellarator hybrids seek to combine tokamak-like compactness and confinement with stellarator-like externally generated rotational transform and steady-state operation. In this work, we build on the recent tokamak--stellarator hybrid study using planar dipole-field coils (PDCs) [Yu et al., arXiv:2605.03599], in which the fixed-position, programmable coils on an axisymmetric winding surface generate flexible three-dimensional shaping fields. Using single-stage free-boundary optimization of coil currents and plasma-equilibrium parameters, we construct vacuum and finite-$β$ configurations. The vacuum cases show controllable external transform and magnetic well. The finite-$β$ cases accommodate various density, temperature, and pressure profiles, producing quasi-axisymmetric (QA) equilibria with self-consistent bootstrap current, favorable Mercier stability, and reduced demand for external current drive. Re-optimization enables $β$ ramp-up and access to different field-period QA branches with moderate coil-current changes. At large rotational transform, a toroidally omnigenous (TO)-like configuration exhibits more favorable infinite-$n$ ideal-ballooning behavior than a QA reference with matched profiles, even though ballooning stability is not directly optimized for. These results demonstrate that PDCs provide a flexible platform for achieving optimized finite-$β$ hybrid configurations.