Extended theory of nonlinear saturation of ballooning modes

It was conjectured that the saturated state of ballooning modes are isolated flux tubes that moves on a constant $\alpha$ surface [Ham, et al. Plasma Phys. Control. Fusion 60 075017 (2018)]. With this model, the study of nonlinear stability of ballooning modes is made possible analytically. The nonlinear ballooning equation has been implemented in tokamak and stellarator geometry. Metastable states are found, indicating possible link to the hard stability limit like ELMs. Comparisons with flux tube structure in a W7X soft ballooning limit calculation [Zhou, Phys. Rev. L. 133, 135102 (2024)] shows good agreements.

To further study the structure of the displaced flux tube, the nonlinear ballooning theory [Ham, et al. Plasma Phys. Control. Fusion 60 075017 (2018)] is extended to describe flux tubes of finite size in this work. The bending of the magnetic field outside the flux tube due to finite sized tube is balanced by the the deformation of the fields inside the flux tube. A closed set of equations is derived to describe the magnetic field inside and outside the finite-sized flux tube and is solved numerically using Picard iteration method. Total energy release can be calculated. Comparison with 3D ideal MHD simulations in simple geometries is underway. This model can be potentially used to give an analytical estimate of ELM filament size and the induced transport to the divertor target from the ELM filaments.