On the non-existence of stepped-pressure equilibria far from symmetry

TL;DR

This paper investigates the limitations of stepped-pressure equilibria in 3D magnetic confinement systems, revealing that interfaces can become fractal and support limited pressure jumps, especially near break-up conditions, challenging assumptions of equilibrium existence.

Contribution

The study demonstrates that in non-symmetric 3D configurations, certain interfaces become fractal and cannot support large pressure jumps, indicating non-existence of equilibria under these conditions.

Findings

Interfaces can become fractal near break-up.

Maximum pressure jump is limited by interface smoothness.

Convergence studies are essential near interface break-up.

Abstract

The Stepped Pressure Equilibrium Code (SPEC) [Hudson et al., Phys. Plasmas 19, 112502 (2012)] has been successful in the construction of equilibria in 3D configurations that contain a mixture of flux surfaces, islands and chaotic magnetic field lines. In this model, the plasma is sliced into sub-volumes separated by ideal interfaces, and in each volume the magnetic field is a Beltrami field. In the cases where the system is far from possessing a continuous symmetry, such as in stellarators, the existence of solutions to a stepped-pressure equilibrium with given constraints, such as a multi-region relaxed MHD minimum energy state, is not guaranteed but is often taken for granted. Using SPEC, we have studied two different scenarios in which a solution fails to exist in a slab with analytic boundary perturbations. We found that with a large boundary perturbation, a certain interface becomes fractal, corresponding to the break up of a Kolmogorov-Arnold-Moser (KAM) surface. Moreover, an interface can only support a maximum pressure jump while a solution of the magnetic field consistent with the force balance condition can be found. An interface closer to break-up can support a smaller pressure jump. We discovered that the pressure jump can push the interface closer to being non-smooth through force balance, thus significantly decreasing the maximum pressure it can support. Our work shows that a convergence study must be performed on a SPEC equilibrium with interfaces close to break-up. These results may also provide insights into the choice of interfaces and have applications in finding out the maximum pressure a machine can support.