Hamiltonian formulation and analysis of a collisionless fluid reconnection model

TL;DR

This paper develops a Hamiltonian framework for a collisionless plasma fluid model, deriving invariants, equilibrium conditions, and analyzing linear stability and tearing mode dynamics.

Contribution

It introduces a noncanonical Hamiltonian formulation with Casimir invariants for collisionless reconnection, extending the Grad-Shafranov equation to include flow.

Findings

Derived new invariants and equilibrium equations for the model.

Analyzed linear stability and thresholds for spectral stability.

Described linear growth and nonlinear saturation of tearing modes.

Abstract

The Hamiltonian formulation of a plasma four-field fluid model that describes collisionless reconnection is presented. The formulation is noncanonical with a corresponding Lie-Poisson bracket. The bracket is used to obtain new independent families of invariants, so-called Casimir invariants, three of which are directly related to Lagrangian invariants of the system. The Casimirs are used to obtain a variational principle for equilibrium equations that generalize the Grad-Shafranov equation to include flow. Dipole and homogeneous equilibria are constructed. The linear dynamics of the latter is treated in detail in a Hamiltonian context: canonically conjugate variables are obtained; the dispersion relation is analyzed and exact thresholds for spectral stability are obtained; the canonical transformation to normal form is described; an unambiguous definition of negative energy modes is given; and thresholds sufficient for energy-Casimir stability are obtained. The Hamiltonian formulation also is used to obtain an expression for the collisionless conductivity and it is further used to describe the linear growth and nonlinear saturation of the collisionless tearing mode.