Theory and discretization of ideal magnetohydrodynamic equilibria with fractal pressure profiles

TL;DR

This paper develops a theoretical framework and numerical methods for modeling ideal magnetohydrodynamic equilibria with fractal pressure profiles, addressing the challenge of infinite currents at rational surfaces.

Contribution

It introduces a novel approach to discretize and analyze fractal pressure profiles in MHD equilibria, combining mathematical classification of irrational numbers with numerical approximation techniques.

Findings

Fractal pressure profiles can be compatible with magnetic field and current density distributions.

Flattening pressure near rational surfaces prevents unphysical infinite currents.

The mathematical classification aids in understanding the support of pressure gradients in fractal profiles.

Abstract

In three-dimensional ideal magnetohydrodynamics, closed flux surfaces cannot maintain both rational rotational-transform and pressure gradients, as these features together produce unphysical, infinite currents. A proposed set of equilibria nullifies these currents by flattening the pressure on sufficiently wide intervals around each rational surface. Such rational surfaces exist at every scale, which characterizes the pressure profile as self-similar and thus fractal. The pressure profile is approximated numerically by considering a finite number of rational regions and analyzed mathematically by classifying the irrational numbers that support gradients into subsets. Applying these results to a given rotational-transform profile in cylindrical geometry, we find magnetic field and current density profiles compatible with the fractal pressure.