On the Relationship between Equilibrium Bifurcations and Ideal MHD Instabilities for Line-Tied Coronal Loops

TL;DR

This paper investigates the connection between bifurcation points in MHD equilibrium sequences and ideal MHD instabilities in line-tied coronal loops, revealing how boundary conditions influence the correspondence between bifurcations and instabilities.

Contribution

It demonstrates that the choice of boundary conditions and equilibrium calculation method affects the identification of bifurcation points related to MHD instabilities in coronal loops.

Findings

Second bifurcation point corresponds to instability with Grad-Shafranov boundary conditions.

Euler potentials reveal the first bifurcation point coincides with the instability threshold.

Boundary conditions influence the bifurcation-instability relationship.

Abstract

For axisymmetric models for coronal loops the relationship between the bifurcation points of magnetohydrodynamic (MHD) equilibrium sequences and the points of linear ideal MHD instability is investigated imposing line-tied boundary conditions. Using a well-studied example based on the Gold-Hoyle equilibrium, it is demonstrated that if the equilibrium sequence is calculated using the Grad-Shafranov equation, the instability corresponds to the second bifurcation point and not the first bifurcation point because the equilibrium boundary conditions allow for modes which are excluded from the linear ideal stability analysis. This is shown by calculating the bifurcating equilibrium branches and comparing the spatial structure of the solutions close to the bifurcation point with the spatial structure of the unstable mode. If the equilibrium sequence is calculated using Euler potentials the first bifurcation point of the Grad-Shafranov case is not found, and the first bifurcation point of the Euler potential description coincides with the ideal instability threshold. An explanation of this results in terms of linear bifurcation theory is given and the implications for the use of MHD equilibrium bifurcations to explain eruptive phenomena is briefly discussed.