Resonant theory of kinetic ballooning modes in general toroidal geometry

The linear theory of the kinetic-ballooning-mode (KBM) instability is extended to capture a weakly driven regime in general toroidal geometry where the destabilization is caused by the magnetic-drift resonance of the ions. Such resonantly destabilized KBMs are characterized by broad eigenfunctions along the magnetic-field line and near-marginal positive growth rates, even well below the normalized-plasma-pressure ( $\beta$ ) threshold of their non-resonant counterparts. This unconventional (or sub-threshold) KBM (stKBM), when destabilized, has been shown to catalyze an enhancement of turbulent transport in the Wendelstein 7-X (W7-X) stellarator (Mulholland et al. 2023 Phys. Rev. Lett. vol. 131, 185101; 2025 Nucl. Fusion vol. 65, 016022). Simplifying the energy dependence of key resonant quantities allows for an analytical treatment of this KBM using the physics-based ordering from the more general equations of Tang et al. (1980 Nucl. Fusion vol. 20, 1439). Results are then compared with high-fidelity gyrokinetic simulations for the (st)KBM in W7-X and the conventional KBM in a circular tokamak at both high and low magnetic shear, where good agreement is obtained in all cases. This reduced KBM model provides deeper insight into (sub-threshold) KBMs and their relationship with geometry, and shows promise for aiding in transport model development and geometry-based turbulence optimization efforts going forward.