Electron Inertia and Magnetic Reconnection

TL;DR

This work proves that when electron inertia is the sole non-ideality, the three-dimensionally evolving modified magnetic field $\mathcal{B} = \vec{B} + \nabla\times\left( \frac{m_e}{n e^2}\vec{j} \right)$ cannot reconnect, even though $\vec{B}$ itself can; the evolution follows a Faraday-like law with a compact form for $\mathcal{E}$, and a Clebsch-based argument shows field lines of $\mathcal{B}$ are advected without topological change when $\mathcal{E}_r$ vanishes. The paper further shows that electron-inertia–driven reconnection effects on guiding-center trajectories are second-order in $(c/\omega_{pe})^2$ and typically smaller than gyroradius effects, except in extremely large current sheets. In 3D, however, the natural chaotic flow of field lines implies any nonzero diffusive effect leads to large-scale reconnection on a timescale $\tau_u\ln(\tau_d/\tau_u)$, rendering the nonreconnection result less practically relevant for realistic plasmas. The discussion also connects $\mathcal{B}$ to Voigt-regularized field behavior and clarifies how particle trajectories should be analyzed relative to the modified field rather than $\vec{B}$ alone.

Abstract

When electron inertia is the only non-ideal effect in the evolution of a magnetic field $\vec{B}$, the field lines of $\vec{B}$ reconnect, but the lines of a related field $\vec{\mathcal{B}}$ do not. $\vec{\mathcal{B}} \equiv \vec{B} + \vec{\nabla}\times \left( (c/ω_{pe})^2μ_0\vec{j} \right)$ with $ω_{pe}$ the plasma frequency and $\vec{j}$ the current density. Although a full four-dimensional relativistic calculation of $\vec{\mathcal{B}}$ has been made, studies of $\vec{\mathcal{B}}$ have been focused on systems that depend on only two spatial coordinates. Three results are given: (1) A relatively simple demonstration in three dimensional space that the lines of $\vec{\mathcal{B}}$ do not reconnect when electron inertia is the only non-ideal effect. (2) The guiding center motion of charged particles is modified by a term that is proportional to $(c/ω_{pe})^2$, which is smaller than the drifts proportional to the gyroradius unless the current density is extremely large. (3) In three dimensional space, the evolution velocity of $\vec{\mathcal{B}}$ is characteristically chaotic, which means neighboring streamlines separate exponentially on a timescale $τ_u$. $\vec{\mathcal{B}}$ undergoes large scale reconnection on a timescale that is only an order of magnitude or two longer than $τ_u$ unless all diffusive non-ideal effects, such as resistivity, are absolutely zero.

Figures (1)

• Figure 1: A magnetic field $\vec{B}(\vec{x},t)$ can be thought of as consisting of tubes of magnetic flux by placing a gridded surface across the field. Each tube is defined by the magnetic field lines that pass through the perimeters of the grid cells. When the field is chaotic, the perimeter of each cell becomes exponentially longer when the grid is replotted after each line on the perimeters is followed for a distance $\ell$. But, each cell contains exactly the same field lines and has precisely the same neighboring cells. When the magnetic field is evolving ideally with a chaotic velocity $\vec{u}_\bot$, a similar distortion of the grid occurs when the grid is replotted using the location of each line on the perimeters after a time $t$. The figure shows the distortion of a $5\times5$ array. This is Figure 1 of Boozer, Phys. Plasmas 32, 052106 (2025). The distorted grid is part of Figure 5 of Y.-M. Huang and A. Bhattacharjee, Phys. Plasmas 29, 122902 (2022), which was based on a chaotic evolution defined by A. H. Boozer and T. Elder, Phys. Plasmas 28, 062303 (2021). Boozer and Elder illustrated distortions of ideally evolving flux tubes up to a factor $\sim 10^7$.