The Maximum Particle Energy Gain During Magnetic Reconnection

TL;DR

This paper tackles what sets the upper limit on particle energization during magnetic reconnection in macroscale systems. Using the kglobal model to span a wide range of effective Lundquist numbers $S_\nu$, it isolates the role of magnetic-island mergers and shows that maximum energy follows $W_{ m max} \sim W_i 2^N$ and scales as $W_{ m max} \propto S_\nu^{1/2}$, with $N$ determined by the merger history. A key finding is that the initiation energy $W_i$ differs by species ($W_i \sim m_i C_A^2$ for protons vs $\sim 0.2 m_i C_A^2$ for electrons), and merger-normalized spectra collapse to a common upper endpoint, providing a quantitative bound on reconnection-driven acceleration. The results link microscopic merger-driven Fermi processes to macroscopic system properties, with broad implications for particle acceleration in heliospheric and astrophysical plasmas and for interpreting PIC/hybrid model limitations.

Abstract

The factors that control the maximum energy attained by protons and electrons during magnetic reconnection are investigated analytically and using large-scale simulations with the \textit{kglobal} model. Previous work revealed that a strong ambient guide field strongly impacts particle energy gain during reconnection, suppressing energy gain from Fermi reflection by increasing the radius of curvature of reconnected field lines. However, previous simulations have also shown that the maximum energy gain increases with the system size. The physical basis for this result has not been explored. We perform simulations that vary the effective system size over a large range to isolate the processes determining the maximum energy gain. The maximum energy $W_{max}$ is regulated by the number of magnetic-island mergers that occur, as multiple flux ropes that form at early time repeatedly merge until the largest approaches the system scale. Fermi reflection in these repeated mergers dominates particle energy gain. The number of mergers is linked to the effective system size -- larger systems produce a larger number of flux ropes and more mergers. That $W_{max}$ is linked to the number of flux rope mergers has implications for understanding why particle-in-cell simulations only produce powerlaw distributions of energetic particles with a limited range in energy.

Figures (3)

• Figure 1: Proton temperature evolution in a reconnecting current sheet. (a) Early-time two-dimensional proton temperature at $t \simeq 1\,\tau_A$, together with magnetic-flux contours, showing the onset of reconnection along the current sheet and the formation of multiple small magnetic islands. (b) Space--time evolution of the proton temperature along the center of the current sheet. The formation, growth, and merging of magnetic islands produce filamentary heating structures and progressively higher temperatures at later times. (c) Late-time proton temperature with magnetic-flux contours. A large system-scale island has formed through successive mergers.
• Figure 2: Proton (top) and electron (bottom) energy spectra for different effective system sizes $S_{\nu}$, shown using the standard normalized energy $W/(m_i C_{A0}^2)$. The high-energy cutoff increases systematically with $S_{\nu}$, reflecting the stronger particle acceleration that takes place in larger simulation domains.
• Figure 3: Proton (top) and electron (bottom) energy spectra for different effective system sizes $S_{\nu}$, shown using the merger-scaled energy normalization $W/(m_i C_{A0}^2 2^{N})$. Here $N$ denotes the number of island mergers characteristic of each system size. Division by $2^{N}$ collapses the upper energy limit to a common initial energy (see Eq. (\ref{['eq:Wmax']})). This scaling reduces systems with differing effective sizes to a common normalized energy frame.