Hamiltonian structure of single-helicity, incompressible magnetohydrodynamics and application to magnetorotational instability

TL;DR

This paper develops a Hamiltonian framework for a simplified model of incompressible magnetohydrodynamics in cylindrical geometry, enabling accurate analysis of magnetorotational instability and stability properties.

Contribution

It introduces a Hamiltonian structure for a four-field reduced MHD model, clarifies the Lie-Poisson bracket and Casimir invariants, and applies this to analyze MRI stability.

Findings

Reproduces the local dispersion relation of axisymmetric MRI

Performs linear stability analysis including negative-energy MRI

Provides a rigorous Hamiltonian formulation for reduced MHD models

Abstract

A four-field reduced model of single helicity, incompressible MHD is derived in cylindrical geometry. An appropriate set of noncanonical variables is found, and the Hamiltonian, the Lie-Poisson bracket and the Casimir invariants are clarified. Detailed proofs of properties of the Lie-Poisson bracket, (i) antisymmetry, (ii) Leibniz rule, and (iii) Jacobi identity, are given. Two applications are presented: the first is that the local dispersion relation of axisymmetric magnetohydrodynamics (MRI) is properly reproduced, and the second is that linear stability analyses including negative-energy MRI were successfully performed.