The Maximum Particle Energy Gain During Magnetic Reconnection

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.

💡 Research Summary

The paper investigates what determines the highest energy that individual protons and electrons can attain during magnetic reconnection. Using both analytical arguments and a suite of large‑scale k‑global simulations, the authors focus on the role of system size, quantified through an effective Lundquist number (S_{\nu}=C_A L^3/\nu), where (L) is a macroscopic length and (\nu) a fourth‑order hyper‑resistivity that enables reconnection without resolving kinetic scales.

Previous work showed that a strong guide field suppresses Fermi acceleration by increasing the curvature radius of reconnected field lines, but it also left open why the maximum particle energy grows with the size of the simulation domain. The authors hypothesize that the key factor is the number of magnetic‑island (flux‑rope) mergers that occur as the reconnection layer evolves. In a merger of two equal‑size flux ropes, the total field‑line length shortens by a factor (\sqrt{2}); because the second adiabatic invariant (v_{\parallel}s) is conserved, particle velocities increase by (\sqrt{2}) and energies double. After (N) such mergers the particle energy scales as (W = W_i 2^{N}).

The smallest islands that form early are set by the balance between convective inflow and hyper‑resistive diffusion, giving a characteristic width (w_0\sim L S_{\nu}^{-1/4}). As mergers proceed, island size grows roughly as (\sqrt{2}) per merger, so after (N) mergers the island radius is (w_N\sim 2^{N/2} w_0). When the island reaches the system scale ((w_N\sim L)), one finds (2^{N}\sim (L/w_0)^2\sim S_{\nu}^{1/2}). Substituting into the energy expression yields the central scaling law:

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