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Magnetic Reconnection in Magnetohydrodynamics

Pedro Caro, Gennaro Ciampa, Renato Lucà

TL;DR

The paper provides analytical constructions of magnetic reconnection in viscous–resistive MHD on the torus, proving that magnetic field-line topology can change over time for smooth global solutions. By perturbing explicit strong reference fields built from Beltrami fields in 3D and Taylor fields in 2D, and leveraging a robust stability theory for the MHD system, it demonstrates both delayed and instantaneous reconnection scenarios with structural stability for the topological transitions. The results cover 3D and 2D settings, include cases with small and large velocities, and reveal that reconnection can occur at arbitrarily small resistivity, highlighting the role of resistivity in breaking topological rigidity. These findings provide rigorous analytical insight into magnetic reconnection mechanisms, complementing numerical and experimental observations and informing reconnection models like Sweet–Parker in a mathematically controlled framework.

Abstract

We provide examples of periodic solutions (in both 2 and 3 dimension) of the Magnetohydrodynamics equations such that the topology of the magnetic lines changes during the evolution. This phenomenon, known as magnetic reconnection, is relevant for physicists, in particular in the study of highly conducting plasmas. Although numerical and experimental evidences exist, analytical examples of magnetic reconnection were not known.

Magnetic Reconnection in Magnetohydrodynamics

TL;DR

The paper provides analytical constructions of magnetic reconnection in viscous–resistive MHD on the torus, proving that magnetic field-line topology can change over time for smooth global solutions. By perturbing explicit strong reference fields built from Beltrami fields in 3D and Taylor fields in 2D, and leveraging a robust stability theory for the MHD system, it demonstrates both delayed and instantaneous reconnection scenarios with structural stability for the topological transitions. The results cover 3D and 2D settings, include cases with small and large velocities, and reveal that reconnection can occur at arbitrarily small resistivity, highlighting the role of resistivity in breaking topological rigidity. These findings provide rigorous analytical insight into magnetic reconnection mechanisms, complementing numerical and experimental observations and informing reconnection models like Sweet–Parker in a mathematically controlled framework.

Abstract

We provide examples of periodic solutions (in both 2 and 3 dimension) of the Magnetohydrodynamics equations such that the topology of the magnetic lines changes during the evolution. This phenomenon, known as magnetic reconnection, is relevant for physicists, in particular in the study of highly conducting plasmas. Although numerical and experimental evidences exist, analytical examples of magnetic reconnection were not known.
Paper Structure (16 sections, 11 theorems, 233 equations, 2 figures)

This paper contains 16 sections, 11 theorems, 233 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Notations and preliminaries
  4. Beltrami fields
  5. Stability of regular solutions of the MHD system in 3D
  6. Magnetic reconnection in 3D
  7. Taylor fields
  8. Building blocks in 2D: Taylor vortices
  9. Stability of the MHD system in 2D
  10. Boundedness of the $H^r$ norms
  11. Boundedness of the $H^r$ norms of the difference equation \ref{['eq:differenceTris']}
  12. Decay of the velocity
  13. Magnetic reconnection in 2D (small velocity)
  14. Construction of the initial datum
  15. Magnetic reconnection in 2D (large velocity)
  16. ...and 1 more sections

Key Result

Theorem 1.1

Consider $d\in\{2,3\}$. Given any viscosity and resistivity $\nu, \eta >0$ and any constants $T>0$ and $M>0$ there exists a zero-average unique global smooth solution $(u,b)$ of eq:mhd on $\mathbb{T}^d$, with initial datum $(0, b_0)$ and $\|b_0\|_{L^2}=M$, such that the magnetic lines at time $t=0$

Figures (2)

  • Figure 1: Phase diagram of the vector field $\mathcal{V}^4_{11}$.
  • Figure 2: Phase diagram of the vector field $V_{1}$.

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Lemma 1.5: Gronwall
  • Theorem 2.1: see Theorem 2.1 in ELP
  • Theorem 3.1
  • Remark 3.2
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}, d=3
  • ...and 26 more