Resonant theory of kinetic ballooning modes in general toroidal geometry

TL;DR

This work tackles the kinetic-ballooning mode (KBM) in general toroidal geometry, emphasizing sub-threshold KBMs destabilized by ion-magnetic-drift resonances in weakly-driven regimes. It develops a reduced, gyrokinetic-based KBM equation in field-aligned geometry, retaining FLR stabilization and resonant ion-drift effects while neglecting trapped particles, $\delta B_\parallel$, and Landau damping, and reformulates it as a nonlinear eigenvalue problem. A fast solver (the Key code) computes self-consistent KBM eigenvalues and eigenvectors across tokamak and stellarator geometries, validated against high-fidelity gyrokinetic simulations (e.g., W7-X, Cyclone-base-case tokamak). The results reveal that stKBMs exist only when ion-temperature gradient-driven resonances are present and are strongly shaped by local curvature and magnetic shear; resonant physics lowers $\beta$-thresholds relative to non-resonant predictions, yielding broad field-line eigenfunctions and near-marginal growth. Together, these findings enable geometry-aware transport modeling and turbulence optimization for high-$\beta$ devices, while outlining future work to incorporate neglected kinetic effects and fully general equilibria.

Abstract

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 beta threshold of their non-resonant counterparts. This unconventional (or sub-threshold) KBM, when destabilized, has been shown to catalyze an enhancement of turbulent transport in the Wendelstein 7-X (W7-X) stellarator [1, 2]. 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, Connor, and Hastie [3]. 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.

Figures (6)

• Figure 1: (st)KBM eigenvalues (growth rates $\gamma$ and real frequencies $\omega_\mathrm{r}$) with increasing $\beta$ in W7-X KJM geometry at wavenumber $k_y \rho_\mathrm{s} = 0.2$, comparing Gene and Key. Key curves include results from non-resonant (green triangles, dotted lines), analytic-resonant (partial toroidal resonance; red squares, dashed lines), and numeric-resonant (full toroidal resonance; magenta diamonds, dashed lines) eigenvalue (EV) solutions. Results from Gene (blue circles, solid lines) include effects of trapped particles and $\delta B_\parallel$, which are absent in Key. For the non-resonant approach, stable KBMs are obtained until reaching the predicted conventional-KBM regime at $\beta_\mathrm{crit}^\mathrm{KBM} \approx 3.3\%$, beyond which the mode grows rapidly; both resonant approaches detect low-$\beta$ destabilization of the stKBM at $\beta_\mathrm{crit}^\mathrm{stKBM} \approx 1\%$ and the mode's approximately constant growth rate until reaching $\beta_\mathrm{crit}^\mathrm{KBM} \approx 3.3\%$. All solutions return similar frequencies with $\beta$; beyond $\beta_\mathrm{crit}^\mathrm{KBM} \approx 3.3\%$, the non-resonant approach recovers the theoretically predicted frequency for strongly driven KBMs of $\omega_\mathrm{r} = \omega_\mathrm{*pi}/2$.
• Figure 2: (st)KBM eigenfunctions $\mathrm{Re}\,\Phi$ versus ballooning angle $\theta$ at different $\beta$ in W7-X KJM geometry at wavenumber $k_y \rho_\mathrm{s} = 0.2$, comparing Gene (blue solid lines) and Key (red dashed lines), where Key eigenfunctions correspond to the analytic-resonant eigenvalues in Fig. \ref{['fig:stkbm_evals_w7x_kjm_key_kinetic_evp_fluid_evp_vs_gene']}. Results from Gene include effects of trapped particles and $\delta B_\parallel$, which are absent in Key. Key captures the broad mode structure of the stKBM for $\beta \approx 1\%-3\%$ in the complex geometry of W7-X, as well as its progressive narrowing with increasing drive as it transitions to the conventional KBM for $\beta > 3\%$.
• Figure 3: KBM eigenvalues (growth rates $\gamma$ and real frequencies $\omega_\mathrm{r}$) as functions of $\beta$ in tokamak geometry with $\hat{s} \approx 0.8$ at wavenumber $k_y \rho_\mathrm{s} = 0.1$, comparing Gene and Key. Key curves include results from non-resonant (green triangles, dotted lines), analytic-resonant (partial toroidal resonance; red squares, dashed lines), and numeric-resonant (full toroidal resonance; magenta diamonds, dashed lines) eigenvalue (EV) solutions. Growth rates from Key show fair but not perfect agreement with Gene (blue circles, solid lines); resonant results follow Gene more consistently than non-resonant results. For the frequencies, the resonant approaches follow the trend of Gene, while the non-resonant approach recovers the theoretically predicted value for strongly driven KBMs of $\omega_\mathrm{r} = \omega_\mathrm{*pi}/2$. Resonant results from Key recover $\beta_\mathrm{crit}^\mathrm{KBM} \approx 0.5\%$, in agreement with Gene. The increasing stabilization seen for $\beta > 2\%$ is well captured in Key.
• Figure 4: KBM eigenfunctions $|\Phi|$ versus ballooning angle $\theta$ at different $\beta$ in tokamak geometry with $\hat{s} \approx 0.8$ at wavenumber $k_y \rho_\mathrm{s} = 0.1$, comparing Gene (blue solid lines) with Key (red dashed lines), where Key eigenfunctions correspond to the numeric-resonant eigenvalues in Fig. \ref{['fig:kbm_evals_salpha_shat_0.796_key_evp_kinetic_vs_gene']}. Good agreement is found for all $\beta$.
• Figure 5: KBM eigenvalues (growth rates $\gamma$ and real frequencies $\omega_\mathrm{r}$) as functions of $\beta$ in tokamak geometry with $\hat{s} \approx 0.1$ at wavenumber $k_y \rho_\mathrm{s} = 0.1$, comparing Gene and Key. Key curves include results from non-resonant (green triangles, dotted lines), analytic-resonant (partial toroidal resonance; red squares, dashed lines), and numeric-resonant (full toroidal resonance; magenta diamonds, dashed lines) eigenvalue (EV) solutions. Growth rates from Key show qualitative agreememnt with Gene (blue circles, solid lines); resonant results follow Gene more consistently than non-resonant results. For the frequencies, the resonant approaches follow the trend of Gene, while the non-resonant approach recovers the theoretically predicted value for strongly driven KBMs of $\omega_\mathrm{r} = \omega_\mathrm{*pi}/2$. Key slightly overestimates $\beta_\mathrm{crit}^\mathrm{KBM}$ at $0.1\%-0.3\%$, compared with Gene's $\beta_\mathrm{crit}^\mathrm{KBM} \approx 0.05\%$. The modulating $\gamma$ with increasing $\beta$ is well captured in Key.