Hamiltonian formulation of X-point collapse in an extended magnetohydrodynamics framework

TL;DR

This paper formulates the Hamiltonian structure of X-point collapse in extended magnetohydrodynamics, providing a foundation for future structure-preserving numerical methods and deeper theoretical understanding.

Contribution

It derives the noncanonical Hamiltonian structure, Poisson bracket, and Casimir invariants for X-point collapse in extended MHD, a novel theoretical framework.

Findings

Derived Hamiltonian and Poisson bracket for extended MHD X-point collapse

Proved the Poisson bracket satisfies antisymmetry and Jacobi identity

Presented governing equations and a Casimir invariant solution

Abstract

The study of X-point collapse in magnetic reconnection has witnessed extensive research in the context of space and laboratory plasmas. In this paper, a recently derived mathematical formulation of X-point collapse applicable in the regime of extended magnetohydrodynamics (XMHD) is shown to possess a noncanonical Hamiltonian structure composed of five dynamical variables inherited from its parent model. The Hamiltonian and the noncanonical Poisson bracket are both derived, and the latter is shown to obey the requisite properties of antisymmetry and the Jacobi identity (an explicit proof of the latter is provided). In addition, the governing equations for the Casimir invariants are presented, and one such solution is furnished. The above features can be harnessed and expanded in future work, such as developing structure-preserving integrators for this dynamical system.