Calculations in the theory of tearing instability

TL;DR

This paper reviews the magnetohydrodynamic tearing instability theory, deriving solutions for different profiles, emphasizing the exact inner equation solution, and discussing shear flow effects, to aid plasma turbulence research.

Contribution

It provides a comprehensive derivation of tearing instability theory for arbitrary parameters and profiles, including exact solutions and effects of shear flows, accessible to non-experts.

Findings

Exact solution of the inner equation is central to tearing instability theory.

Derivations for Harris and sine-shaped profiles are provided.

Shear flows influence tearing instability in magnetic structures.

Abstract

Recent studies have suggested that the tearing instability may play a significant role in magnetic turbulence. In this work, we review the theory of the magnetohydrodynamic tearing instability in the general case of an arbitrary tearing parameter, which is relevant for applications in turbulence. We discuss a detailed derivation of the results for the standard Harris profile and accompany it by the derivation of the results for a lesser known sine-shaped profile. We devote special attention to the exact solution of the inner equation, which is the central result in the theory of tearing instability. We also briefly discuss the influence of shear flows on tearing instability in magnetic structures. Our presentation is self-contained; we expect it to be accessible to researchers in plasma turbulence who are not experts in magnetic reconnection.