First-principles theory of the rate of magnetic reconnection in magnetospheric and solar plasmas

Magnetic reconnection, the process converting magnetic energy into plasma energy, is crucial in various astrophysical and laboratory settings. It governs energy release in solar flares, the Dungey convection cycle in Earth's magnetosphere, and geomagnetic substorms. Numerical simulations and observations suggest a reconnection rate of approximately 0.1 in normalized units, yet a comprehensive theoretical prediction has remained elusive. Previous models, like Petschek's, offer steady-state solutions for open outflow geometries but lack a mechanism for diffusion region localization. The Sweet-Parker solution, while valid for uniform resistivity MHD, predicts a much slower reconnection rate due to its system-size long diffusion region. Invoking anomalous resistivity to explain the observed fast reconnection in kinetic simulations has lacked clear observational evidence. Kinetic simulations beyond MHD show that antiparallel reconnection with open outflow occurs when the current sheet thins to the ion inertial scale, with the Hall term in Ohm's law dominating the electric field in the ion diffusion region (IDR). However, the exact mechanism by which the Hall term localizes the diffusion region and creates an open geometry remains unclear. This work investigates the role of Hall physics in plasma energization and open geometry formation, establishing a first-principles theory of the reconnection rate, without relying on empirical inputs from simulations or observations. This is achieved by focusing on two key points: (1) The Hall term itself doesn't convert energy into plasma (J ⋅ E_Hall = 0), and (2) Pressure depletion at the x-line causes the exhaust to open up.

The paper reviews existing literature on magnetic reconnection, highlighting the discrepancy between observed fast reconnection rates and predictions from MHD models like Sweet-Parker. It discusses previous attempts to explain fast reconnection, including Petschek's model and the invocation of anomalous resistivity, pointing out their limitations. The GEM reconnection challenge study is mentioned to highlight the importance of the Hall term in reproducing fast reconnection in simulations. Several papers are cited discussing energy conversion and particle acceleration during reconnection, underlining the need for a comprehensive theory linking these aspects.

The study employs two-dimensional particle-in-cell (PIC) simulations using the VPIC code to investigate energy conversion in the diffusion region. The simulations model collisionless electron-ion plasmas with a low background beta (β = 0.01). The setup includes a Harris current sheet equilibrium with specified parameters (ion-to-electron mass ratio, temperature ratio, etc.) A localized magnetic perturbation initiates single x-line reconnection. The simulations track various plasma parameters (magnetic field, electric field, pressure, particle distributions) to analyze energy conversion and transport. The generalized Ohm's law is analyzed to identify dominant terms in different regions. The analysis focuses on the Hall term, electron pressure gradient term, and electron inertia. The paper also employs theoretical analysis using force balance and geometrical arguments to link the results of the simulations to a broader theoretical framework, and cross-scale coupling between the EDR, IDR, and upstream MHD region are treated to derive a reconnection rate prediction. The theoretical analysis focuses on estimating the thermal pressure at the x-line, relating it to the magnetic pressure, and linking it to the opening of the outflow exhausts. The approach is used to explain differences in reconnection rates between electron-ion plasmas, electron-positron plasmas, and resistive-MHD systems.

The PIC simulations reveal that the Hall electromagnetic fields divert incoming electromagnetic energy towards the outflow, creating an energy void near the x-line. This energy void leads to pressure depletion at the x-line, a critical factor enabling the opening of the outflow exhaust and consequently faster reconnection. The simulations show that while the Hall term dominates the electric field inside the IDR, it does not directly convert electromagnetic energy into plasma thermal energy (J ⋅ E_Hall ≈ 0). Nevertheless, there is a non-zero energy conversion from the convection electric field, which contributes to pressure buildup, though it is not dominant. The analysis focuses on the zz component of the pressure tensor, showing a significant pressure depletion at the x-line. A first-principles theory is developed that predicts this pressure depletion and the fast reconnection rate, which is consistent with both simulation results and observed reconnection rates in space plasmas. The theory shows that the pressure depletion is a result of limited energy conversion near the x-line due to the nature of the Hall electric field. The theory links the reconnection rate to the slope of the separatrix, providing a first-principles prediction of the reconnection rate of order 0.1. The analysis is extended to explain why Sweet-Parker reconnection is slow, arising from a balanced pressure at the x-line and the absence of energy diversion to the outflow. The framework also explains fast reconnection in electron-positron plasmas, attributing it to the absence of the two-scale structure and the resulting competition between kinetic energy and enthalpy fluxes. The Sweet-Parker scaling is also recovered using this framework.

The findings directly link the Hall effect to diffusion region localization through pressure depletion at the x-line. The theory successfully predicts both the pressure drop at the x-line and the fast reconnection rate observed in simulations and in space plasmas, establishing a fundamental link between the Hall effect and the speed of magnetic reconnection. The explanation of why Sweet-Parker reconnection is slow provides a unified picture of reconnection across different plasma regimes. The results advance our understanding of magnetic reconnection, with implications for various plasma systems, from the Earth's magnetosphere to solar flares and astrophysical plasmas.

This paper provides a first-principles theory explaining the fast rate of magnetic reconnection, linking it to pressure depletion at the x-line caused by the Hall effect. The theory successfully predicts the observed reconnection rate and explains the slower rate of Sweet-Parker reconnection. Future research could focus on extending the theory to three-dimensional systems and investigating the role of secondary instabilities in further enhancing reconnection rates.

The study primarily uses two-dimensional PIC simulations. While many aspects of reconnection are quasi-2D, three-dimensional effects may play a role. The analysis assumes a steady-state reconnection, which might not always be the case in realistic scenarios. The initial current sheet equilibrium might influence the results, and exploring different equilibria is needed. Secondary tearing modes and their influence on the reconnection rate are not fully integrated into the theory.