The nonlinear dynamo effect of tearing modes

TL;DR

This paper derives the nonlinear dynamo effect of tearing modes in resistive MHD, revealing how it influences current density and pressure profiles near rational surfaces, and providing insights into plasma relaxation and magnetic island behavior.

Contribution

It introduces a detailed derivation of the nonlinear dynamo effect of tearing modes, distinguishing between parallel and perpendicular effects and their impact on plasma profiles.

Findings

Parallel dynamo drives opposite current densities, flattening the λ profile near rational surfaces.

Perpendicular dynamo eliminates pressure gradients near rational surfaces.

Plasma tends to relax into the Taylor state due to multiple rational surfaces.

Abstract

The nonlinear dynamo effect of tearing modes is derived with the resistive MHD equations. The dynamo effect is divided into two parts, parallel and perpendicular to the magnetic field. Firstly, the force-free plasma is considered. It is found that the parallel dynamo effect drives opposite current densities at the different sides of the rational surface, making the $\lambda =\boldsymbol{j}\cdot\boldsymbol{B}/|\boldsymbol{B}|^2$ profile completely flattened near the rational surface. There are many rational surfaces for the turbulent plasma, which means the plasma is tending to relax into the Taylor state. In contrast, a bit far from the rational surface, the parallel dynamo effect is much smaller, and the nonlinear dynamo form approximates the quasilinear form. Secondly, the pressure gradient is included. It is found that rather than the $\lambda$ profile, the $\boldsymbol{j}\cdot\boldsymbol{B}$ profile is flattened by the parallel dynamo effect. Besides, the perpendicular dynamo effect of tearing modes is found to eliminate the pressure gradient near the rational surface. In addition, our result also provides another basis for the assumption that current density is flat in the magnetic island for the tearing modes theory.