Direct Computation of Ballooning Stability in the Near-Axis Expansion

The near-axis expansion is a tool that allows for the rapid design of stellarator configurations. The expansion has been used to find a variety of quasisymmetric stellarators, including quasi-axisymmetric (QA), quasi-helical (QH), and quasi-isodynamic (QI) configurations. However, despite their good particle confinement properties, these configurations have no guarantees of stability. Necessary conditions for stability (the Mercier condition and magnetic well) have been computed, but to our knowledge, there are no direct computations of stability for the near-axis expansion.

In this work, we determine ballooning stability on the magnetic axis by directly solving the ideal ballooning equation. Then, using eigenvalue perturbation theory, we compute corrections to the eigenvalue off-axis. These corrections can be used to predict the radius in which a stellarator is ballooning stable, allowing for the trade-off between stability and quasisymmetry to be analyzed near-axis. To test our method, we compare the predicted eigenvalues to those from solving the full 3D problem on known stellarator configurations.