Kinetic Ballooning Mode turbulence in small-average-magnetic-shear equilibria*

Kinetic ballooning mode (KBM) turbulence is studied in small-average-magnetic-shear equilibria, namely HSX, Heliotron-J, and a circular tokamak, to understand stellarator transport at finite $\beta$ and to identify configurations with improved confinement. Electromagnetic flux-tube simulations of HSX using the gyrokinetic turbulence code GENE show that the onset of KBM instability at low $k_y$ occurs at a value of normalized plasma pressure $\beta$ that is an order of magnitude smaller than the MHD ballooning limit $\beta^\mathrm{MHD}$. This small $\beta^\mathrm{KBM}$ is sensitive to modifications of the magnetic shear. Heliotron-J and an axisymmetric geometry exhibit behavior similar to HSX. Regardless, saturation of nonlinear simulations of HSX with $\beta^\mathrm{MHD} > \beta > \beta^\mathrm{KBM}$ is achievable and results in lower heat fluxes than the electrostatic case. A fluid model which expands upon an electrostatic model [C.C. Hegna, 2018] by including finite-$\beta$ effects is introduced; it allows for ITG-KBM saturation in stellarators to be dominated by the transfer of energy from unstable to stable modes at similar scales via nonlinear coupling and will be used to build a physical understanding for the relationship between geometry and $\beta^\mathrm{KBM}$.