Influence of higher-order harmonics on the saturation of the tearing mode

TL;DR

This paper investigates how higher-order harmonics affect the nonlinear saturation of the tearing mode in slab geometry, confirming that the standard one-harmonic approach remains valid asymptotically.

Contribution

It develops a formalism including higher-order harmonics in the analysis of tearing mode saturation, demonstrating their higher-order corrections and validating the one-harmonic approximation.

Findings

Higher-order harmonics introduce corrections but are higher order in the perturbation parameter.

The standard one-harmonic approach is asymptotically correct.

The formalism applies to cases with vanishing current gradient at the resonant surface.

Abstract

The nonlinear saturation of the tearing mode is revisited in slab geometry by taking into account higher-order harmonics in the outer solution. The general formalism for tackling this problem in the case of a vanishing current gradient at the resonant surface is derived. It is shown that, although the higher-order harmonics lead to corrections in the final saturation equation, they are of higher order in the perturbation parameter, which provides a formal proof that the standard one-harmonic approach is asymptotically correct.