Existence and numerical convergence of stellarator equilibria with nested flux surfaces

The paper develops a nonlinear flux-tube model for ballooning-mode saturation in stellarators and shows that direct energy calculations are contaminated by force-balance errors in numerical stellarator equilibria. This sensitivity is tied to the fact that 3D equilibria with nested flux surfaces lack a general existence proof, in contrast to axisymmetric tokamak equilibria, and numerical solutions may not converge to zero force error.

To mitigate this, the authors introduce a variational method to compute flux-tube energy that is robust against force-balance errors. However, the broader mathematical question of the existence of stellarator equilibria with integrable magnetic fields (nested flux surfaces) and the guaranteed convergence of numerical solvers to force-balanced states remains unresolved, which directly impacts the reliability of nonlinear stability predictions.