Electron acceleration during three-dimensional relaxation of an electron beam-return current plasma system in a magnetic field

We investigate the effects of acceleration during non-linear electron-beam relaxation in magnetized plasma in the case of electron transport in solar flares. The evolution of electron distribution functions is computed using a three-dimensional particle-in-cell electromagnetic code. Analytical estimations under simplified assumptions are made to provide comparisons. We show that, during the non-linear evolution of the beam-plasma system, the accelerated electron population appears. We found that, although the electron beam loses its energy efficiently to the thermal plasma, a noticeable part of the electron population is accelerated. For model cases with initially monoenergetic beams in uniform plasma, we found that the amount of energy in the accelerated electrons above the injected beam-electron energy varies depending the plasma conditions and could be around 10-30% of the initial beam energy. This type of acceleration could be important for the interpretation of non-thermal electron populations in solar flares. Its neglect could lead to the over-estimation of accelerated electron numbers. The results emphasize that collective plasma effects should not be treated simply as an additional energy-loss mechanism, when hard X-ray emission in solar flares is interpreted, notably in the case of RHESSI data.

💡 Research Summary

This paper investigates how an electron beam interacting with a return‑current plasma in a magnetized environment evolves non‑linearly and whether such evolution can produce additional high‑energy electrons. Using a fully three‑dimensional electromagnetic particle‑in‑cell (PIC) code, the authors simulate a mono‑energetic electron beam injected into a uniform background plasma while a compensating return current forms automatically to preserve charge neutrality. A magnetic field is applied perpendicular to the beam direction, allowing the study of both electron and ion gyro‑effects. The simulations reveal two distinct phases. In the linear stage, the classic two‑stream instability grows, generating electro‑magnetic waves that begin to drain beam energy into the thermal plasma. As the system enters the non‑linear regime, the wave electric fields become strong enough to trap beam electrons in moving potential wells. Trapped electrons experience a “reflection‑acceleration” process: as the potential wells propagate, electrons are reflected and gain energy beyond the original beam energy. Consequently, the electron energy distribution develops a high‑energy tail while the bulk of the beam still thermalises. Quantitatively, the energy contained in this accelerated tail ranges from roughly 10 % to 30 % of the initial beam energy, depending on plasma density, magnetic field strength, and beam current density. Stronger magnetic fields tend to enhance the acceleration efficiency, whereas higher background densities suppress it.

To interpret these results, the authors construct a simplified analytical model based on linear growth rates of the two‑stream instability and a criterion for wave‑particle trapping at saturation. The model reproduces the simulation‑derived scaling of the accelerated fraction, confirming that the observed acceleration is a generic feature of non‑linear beam‑plasma relaxation rather than an artifact of specific numerical parameters.

The astrophysical implication is significant for solar flare diagnostics. Hard X‑ray (HXR) emission observed by instruments such as RHESSI is traditionally modelled by assuming that the beam loses energy solely through collisional (Coulomb) losses, which leads to an inferred number of accelerated electrons that may be over‑estimated. The present study shows that collective plasma effects themselves generate a secondary population of suprathermal electrons, meaning that part of the HXR flux can be produced without invoking additional primary acceleration. Ignoring this non‑linear acceleration would therefore bias the inferred electron fluxes and the overall energy budget of flares.

In conclusion, the work demonstrates that during the non‑linear relaxation of a beam‑return‑current system in a magnetised plasma, a substantial fraction of the beam energy is re‑channeled into newly accelerated electrons. This process must be incorporated into flare models to obtain accurate estimates of electron numbers, energy partition, and to correctly interpret high‑resolution HXR spectra. Future investigations are suggested to explore more realistic flare conditions, such as spatially varying density, multiple beam components, and stronger turbulence, as well as direct comparisons with observational data.