# Fast magnetic reconnection and the ideal evolution of a magnetic field

## TL;DR

This paper demonstrates that fast magnetic reconnection occurs naturally in three-dimensional ideal magnetic field evolution, driven by exponential sensitivity to non-ideal effects, with implications for understanding rapid changes in magnetic topology.

## Contribution

It introduces a Lagrangian coordinate framework to analyze ideal magnetic evolution, revealing conditions that lead to fast reconnection driven by exponential sensitivity.

## Key findings

- Fast reconnection requires 3D evolution and exponential sensitivity to non-ideal effects.
- Parallel current density forms exponentially thin and wide ribbons with limited growth.
- Exponential sensitivity explains rapid magnetic reconnection despite small non-ideal effects.

## Abstract

Regardless of how small non-ideal effects may be, phenomena associated with changes in magnetic field line connections are frequently observed to occur on an Alfv\'enic time scale. Since it is mathematically impossible for magnetic field line connections to change when non-ideal effects are identically zero, an ideal evolution must naturally lead to states of unbounded sensitivity to non-ideal effects. That such an evolution is natural is demonstrated using Lagrangian coordinates based on the flow velocity of the magnetic field lines. The Lagrangian representation of an evolving magnetic field is highly constrained when neither the magnetic field strength nor the forces exerted by the magnetic field increase exponentially with time. The development of a state of fast reconnection consistent with these constraints (1) requires a three-dimensional evolution, (2) has an exponentially increasing sensitivity to non-ideal effects, and (3) has a parallel current density, which lies in exponentially thinning but exponentially widening ribbons, with a magnitude that is limited to a slow growth. The implication is that exponential growth in sensitivity is the cause of fast magnetic reconnection when non-ideal effects are sufficiently small. The growth of the non-ideal effect of the resistivity multiplied by the parallel current density is far too slow to be competitive.