Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hamiltonian formulation and matrix discretization for axisymmetric magnetohydrodynamics

Michael Roop

Abstract

Equations of ideal magnetohydrodynamics (MHD) play an important role in the studies of turbulence, astrophysics, and plasma physics. These equations possess remarkable geometric structures and symmetries. Indeed, they admit a geodesic formulation in the sense of Arnold, as a Lie--Poisson flow on the dual of an infinite-dimensional Lie algebra. Zeitlin's model, previously developed for MHD on the flat torus and the two-sphere, is a matrix approximation of MHD consistent with the underlying geometric structures. In this paper, we derive the reduced model of axially symmetric magnetohydrodynamics on the three-sphere and give its Hamiltonian formulation. We further extend finite dimensional Zeitlin's matrix model for MHD from 2D to axially symmetric 3D flows of magnetized fluids, yielding the first discrete model for 3D magnetohydrodynamics compatible with the underlying Lie--Poisson structure.

Hamiltonian formulation and matrix discretization for axisymmetric magnetohydrodynamics

Abstract

Equations of ideal magnetohydrodynamics (MHD) play an important role in the studies of turbulence, astrophysics, and plasma physics. These equations possess remarkable geometric structures and symmetries. Indeed, they admit a geodesic formulation in the sense of Arnold, as a Lie--Poisson flow on the dual of an infinite-dimensional Lie algebra. Zeitlin's model, previously developed for MHD on the flat torus and the two-sphere, is a matrix approximation of MHD consistent with the underlying geometric structures. In this paper, we derive the reduced model of axially symmetric magnetohydrodynamics on the three-sphere and give its Hamiltonian formulation. We further extend finite dimensional Zeitlin's matrix model for MHD from 2D to axially symmetric 3D flows of magnetized fluids, yielding the first discrete model for 3D magnetohydrodynamics compatible with the underlying Lie--Poisson structure.
Paper Structure (13 sections, 12 theorems, 97 equations)

This paper contains 13 sections, 12 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Hamiltonian formulation of MHD
  3. Hamiltonian structures for 3D MHD
  4. Hamiltonian structures for 2D MHD
  5. Axisymmetric MHD equations
  6. Derivation of reduced equations
  7. Hamiltonian formulation of reduced equations
  8. Conservation laws for reduced equations
  9. Matrix discretizations for axisymmetric MHD
  10. Zeitlin's model for 2D spherical MHD
  11. Zeitlin's model for 3D axisymmetric MHD
  12. Convergence of Casimirs
  13. Derivation of reduced equations

Key Result

Theorem 2.1

The equations of ideal magnetohydrodynamics MHD are a Lie--Poisson system on the dual $\mathfrak{f}^{*}$ of the semidirect product Lie algebra $\mathfrak{f}=\mathfrak{X}_{\mu}(M)\ltimes\left(\Omega^{1}(M)/\mathrm{d}\Omega^{0}\right(M))$. The Hamiltonian function is given by hamiltonfstar.

Theorems & Definitions (18)

  • Theorem 2.1
  • Definition 3.1
  • Proposition 3.1
  • proof
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Remark 3.1
  • Theorem 4.1: Charles, Polterovich
  • Theorem 4.2
  • ...and 8 more