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Stability of Chandrasekhar's nonlinear force-free fields

Ken Abe

Abstract

Chandrasekhar's nonlinear force-free field, discovered in 1956, is a rare example of a nonlinear force-free field describing a large coherent structure of axisymmetric magnetic field lines with swirls in the presence of Alfvén waves. We demonstrate that his traveling wave solution is orbitally stable with the energy norm in ideal MHD regarding weak ideal limits of axisymmetric Leray-Hopf solutions.

Stability of Chandrasekhar's nonlinear force-free fields

Abstract

Chandrasekhar's nonlinear force-free field, discovered in 1956, is a rare example of a nonlinear force-free field describing a large coherent structure of axisymmetric magnetic field lines with swirls in the presence of Alfvén waves. We demonstrate that his traveling wave solution is orbitally stable with the energy norm in ideal MHD regarding weak ideal limits of axisymmetric Leray-Hopf solutions.
Paper Structure (57 sections, 60 theorems, 409 equations, 1 figure)

This paper contains 57 sections, 60 theorems, 409 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Taylor states
  3. Chandrasekhar's nonlinear force-free fields
  4. The main result
  5. Stability: Euler and ideal MHD
  6. Taylor state stability
  7. Stability of Chandrasekhar's nonlinear force-free fields
  8. Vortex and magnetic fields
  9. Remarks on magnetic relaxation
  10. Ideas and main ingredients
  11. Taylor state stability: the toy model
  12. The new ingredients
  13. The difficulties of the proof
  14. The strict subadditivity: the first main idea
  15. The global flux convergence: the second main idea
  16. ...and 42 more sections

Key Result

Theorem 1.1

Let ${\mathcal{S}}_{h}$ be a set of eigenfunctions of the rotation operator on $L^{2}_{\sigma}(\Omega)$ associated with the least positive (resp. largest negative) eigenvalue with magnetic helicity $h>0$ (resp. $h<0$). Let ${\mathcal{S}}_0=\emptyset$. The set ${\mathcal{S}}_{h}=\{U_j\}_{j=1}^{N}$ is there exists a weak ideal limit $(u,B)$ of Leray--Hopf solutions to (1.1)--(1.2) for $(u_0,B_0)$ su

Figures (1)

  • Figure 1: The (8,11)-torus knots: the field lines have 8 circuits in the toroidal direction and 11 circuits in the poloidal direction

Theorems & Definitions (136)

  • Theorem 1.1: Taylor state stability
  • Remark 1.2
  • Remark 1.3: Three Taylor states
  • Theorem 1.4: Stability of Chandrasekhar's nonlinear force-free fields
  • Remark 1.5
  • Remark 1.6: Vortex rings with swirls
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 126 more