The Scaling of Vortical Electron Acceleration in Thin-current Magnetic Reconnection and Its Implications in Solar Flares

To investigate how magnetic reconnection (MR) accelerates electrons to a power-law energy spectrum in solar flares, we explore the scaling of a kinetic model proposed by Che & Zank (CZ) and compare it to observations. Focusing on thin current sheet MR particle-in-cell (PIC) simulations, we analyze the impact of domain size on the evolution of the electron Kelvin-Helmholtz instability (EKHI). We find that the duration of the growth stage of the EKHI ( tG similar to omega e-1 ) is short and remains nearly unchanged because the electron gyrofrequency omega e is independent of domain size. The quasi-steady stage of the EKHI (t MR) dominates the electron acceleration process and scales linearly with the size of the simulations as L/v A0, where v A0 is the Alfven speed. We use the analytical results obtained by CZ to calculate the continuous temporal evolution of the electron energy spectra from PIC simulations and linearly scale them to solar flare observational scales. For the first time, an electron acceleration model predicts the sharp two-stage transition observed in typical soft-hard-harder electron energy spectra, implying that the electron acceleration model must be efficient with an acceleration timescale that is a small fraction of the duration of solar flares. Our results suggest that we can use PIC MR simulations to investigate the observational electron energy spectral evolution of solar flares if the ratio t MR/t G is sufficiently small, i.e., less than or similar to 10%.