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Relative magnetic helicity under turbulent relaxation

Sauli Lindberg, David MacTaggart

TL;DR

The paper addresses how relative magnetic helicity behaves under turbulent relaxation for line-tied magnetic fields, extending Taylor relaxation concepts to non-tangent boundary conditions. By adopting Onsager's ideal turbulence framework, it develops a rigorous Leray-Hopf-based theory for resistive MHD with line-tied boundaries and a carefully chosen reference field, ensuring a gauge-invariant relative helicity ${\mathcal H}$. The main results show that in the ideal MHD limit the relative helicity is conserved during turbulent relaxation, while the velocity field decays to zero and the magnetic field asymptotically approaches a weak magnetohydrostatic balance. This provides a mathematically solid connection between turbulent relaxation, helicity conservation, and end-state magnetohydrostatic configurations, with implications for both astrophysical and laboratory settings.

Abstract

Magnetic helicity is a quantity that underpins many theories of magnetic relaxation in electrically conducting fluids, both laminar and turbulent. Although much theoretical effort has been expended on magnetic fields that are everywhere tangent to their domain boundaries, many applications, both in astrophysics and laboratories, actually involve magnetic fields that are line-tied to the boundary, i.e. with a non-trivial normal component on the boundary. This modification of the boundary condition requires a modification of magnetic helicity, whose suitable replacement is called relative magnetic helicity. In this work, we investigate rigorously the behaviour of relative magnetic helicity under turbulent relaxation. In particular, we specify the normal component of the magnetic field on the boundary and consider the \emph{ideal limit} of resistivity tending to zero in order to model the turbulent evolution in the sense of Onsager's theory of turbulence. We show that relative magnetic helicity is conserved in this distinguished limit and that, for constant viscosity, the magnetic field can relax asymptotically to a magnetohydrostatic equilibrium.

Relative magnetic helicity under turbulent relaxation

TL;DR

The paper addresses how relative magnetic helicity behaves under turbulent relaxation for line-tied magnetic fields, extending Taylor relaxation concepts to non-tangent boundary conditions. By adopting Onsager's ideal turbulence framework, it develops a rigorous Leray-Hopf-based theory for resistive MHD with line-tied boundaries and a carefully chosen reference field, ensuring a gauge-invariant relative helicity . The main results show that in the ideal MHD limit the relative helicity is conserved during turbulent relaxation, while the velocity field decays to zero and the magnetic field asymptotically approaches a weak magnetohydrostatic balance. This provides a mathematically solid connection between turbulent relaxation, helicity conservation, and end-state magnetohydrostatic configurations, with implications for both astrophysical and laboratory settings.

Abstract

Magnetic helicity is a quantity that underpins many theories of magnetic relaxation in electrically conducting fluids, both laminar and turbulent. Although much theoretical effort has been expended on magnetic fields that are everywhere tangent to their domain boundaries, many applications, both in astrophysics and laboratories, actually involve magnetic fields that are line-tied to the boundary, i.e. with a non-trivial normal component on the boundary. This modification of the boundary condition requires a modification of magnetic helicity, whose suitable replacement is called relative magnetic helicity. In this work, we investigate rigorously the behaviour of relative magnetic helicity under turbulent relaxation. In particular, we specify the normal component of the magnetic field on the boundary and consider the \emph{ideal limit} of resistivity tending to zero in order to model the turbulent evolution in the sense of Onsager's theory of turbulence. We show that relative magnetic helicity is conserved in this distinguished limit and that, for constant viscosity, the magnetic field can relax asymptotically to a magnetohydrostatic equilibrium.

Paper Structure

This paper contains 13 sections, 4 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Motivation of "ideal turbulence"
  3. Anomalous dissipation
  4. Fast magnetic reconnection
  5. Magnetic helicity conservation
  6. Problem setup
  7. Relative helicity
  8. Leray-Hopf solutions
  9. Function spaces
  10. Main definitions
  11. Existence of Leray-Hopf solutions
  12. Ideal turbulent MHD
  13. Summary

Key Result

Theorem 3

For every ${\bm u}_0 \in L^2_\sigma(\Omega)$ and ${\bm B}_0 \in L^2(\Omega)$ with $\nabla \cdot {\bm B}_0 = 0$ there exists a Leray-Hopf solution $({\bm u},{\bm B})$ of mhd_1--mhd_5. Furthermore,

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • proof : Proof of Theorem \ref{['t:Leray-Hopf']}
  • Theorem 4
  • proof