Remarkable connections between extended magnetohydrodynamics models

TL;DR

This paper reveals fundamental connections between various extended magnetohydrodynamics models through variable transformations, Lie-dragged forms, and Poisson brackets, unifying their mathematical structure and deriving their helicities.

Contribution

It establishes the commonality of all extended MHD models via variable transformations and Lie-dragged forms, and analyzes their Poisson brackets and helicities.

Findings

Unified framework for extended MHD models

Explicit Poisson bracket structures verified for all models

Derived helicities for different MHD models

Abstract

Through the use of suitable variable transformations, the commonality of all extended magnetohydrodynamics (MHD) models is established. Remarkable correspondences between the Poisson brackets of inertialess Hall MHD and inertial MHD (which has electron inertia, but not the Hall drift) and extended MHD (which has both effects), are established. The helicities (two in all) for each of these models are obtained through these correspondences. The commonality of all the extended MHD models is traced to the existence of two Lie-dragged 2-forms, which are closely associated with the canonical momenta of the two underlying species. The Lie-dragging of these 2-forms by suitable velocities also leads to the correct equations of motion. The Hall MHD Poisson bracket is analyzed in detail, and the Jacobi identity is verified through a detailed proof and this proof ensures the Jacobi identity for the Poisson brackets of all the models.