Hall Resistive Tearing Mode: A Variational Formulation

TL;DR

This paper introduces a unified linear tearing-mode formulation incorporating resistivity and Hall effects, using a variational method to derive an analytical dispersion relation that aligns with previous results and extends understanding of tearing mode dynamics.

Contribution

It presents a novel variational approach to derive a comprehensive analytical dispersion relation for Hall resistive tearing modes, unifying resistive and Hall regimes.

Findings

Recovers the classical Furth-Killeen-Rosenbluth result for resistive tearing modes.

Provides a growth rate for the Hall branch consistent with electron-inertia driven tearing modes.

Derives a scaling relation for the transition from resistive to Hall regimes in reconnection.

Abstract

A unified linear tearing-mode formulation is given incorporating both resistivity and Hall effects. A variational method is used that appears to be best suited to deal with the difficulties peculiar to the {\it triple-deck} structure associated with the Hall resistive tearing mode but also to lead to a convenient analytical dispersion relation for the Hall resisitive tearing mode. This analytical dispersion relation - * recovers the Furth-Killeen-Rosenbluth[15] result for the resistive branch; * gives a growth rate for the Hall branch which appears to be consistent with the growth rate of the electron-inertia driven tearing mode given previously (Coppi [19]); * recovers the scaling relation for the transition from the resisitive regime to the Hall regime numerically established by Fitzpatrick[20] in a driven Hall resistive reconnection situation.