Kinetic ballooning mode turbulence in low-magnetic-shear 3D equilibria*

Electromagnetic flux-tube simulations of the HSX stellarator using the gyrokinetic code \textsc{Gene} show that the kinetic ballooning mode (KBM) threshold $\beta^\mathrm{KBM}$ is an order of magnitude smaller than the MHD ballooning limit when a strong ion temperature gradient is present. As the ion temperature gradient becomes weaker, $\beta^\mathrm{KBM}$ approaches the MHD ballooning limit. $\beta^\mathrm{KBM}$ is also sensitive to locally-self-consistent modifications of the magnetic shear. Simulations of Heliotron-J also display behavior similar to HSX with respect to $\beta^\mathrm{KBM}$. Finite-$\beta$ ($\approx 0.5 \%$) simulations of HSX exhibit significant nonlinear finite-$\beta$ stabilization when saturation is achieved. We also introduce a fluid model that expands upon a three-field model [C.C.~Hegna et al., Phys.~of Plasmas {\bf 25}, 022511] by including finite-$\beta$ effects. We employ this reduced model to investigate KBM turbulence saturation in 3D magnetic equilibria both when strong ion temperature gradients are present and as the magnetic shear is varied.