The increasing need to demonstrate the correctness of computer simulations has highlighted the importance of benchmarks. We define in this paper a representative simulation case to study low-temperature partially-magnetized plasmas. Seven independently developed particle-in-cell codes have simulated this benchmark case, with the same specified conditions. The characteristics of the codes used, such as implementation details or computing times and resources, are given. First, we compare at steady-state the time-averaged axial profiles of three main discharge parameters (axial electric field, ion density and electron temperature). We show that the results obtained exhibit a very good agreement within 5% between all the codes. As E × B discharges are known to cause instabilities propagating in the direction of electron drift, an analysis of these instabilities is then performed and a similar behaviour is retrieved between all the codes. A particular attention has been paid to the numerical convergence by varying the number of macroparticles per cell and we show that the chosen benchmark case displays a good convergence. Detailed outputs are given in the supplementary data, to be used by other similar codes in the perspective of code verification.

In this paper we propose a representative simulation test-case of E × B discharges accounting for plasma wall interactions with the presence of both the electron cyclotron drift instability and the modified-two-stream-instability. Seven independently developed particle-in-cell (PIC) codes have simulated this benchmark case, with the same specified conditions. The characteristics of the different codes and computing times are given. Results show that both instabilities were captured in a similar fashion and good agreement between the different PIC codes is reported as main plasma parameters were closely related within a 5% interval. The number of macroparticles per cell was also varied and statistical convergence was reached. Detailed outputs are given in the supplementary data, to be used by other similar groups in the perspective of code verification.

The large differential drift motion between electrons and ions that is created by the E × B current can produce different instabilities, such as the electron cyclotron drift instability, perpendicular to the magnetic field, and the Modified Two-Stream Instability (MTSI), with a component along the magnetic field. In this paper, we derive and validate a stability condition for the apparition of the MTSI modes in 2D particle-in-cell simulations of E × B discharges in the radial-azimuthal plane of a Hall thruster. We verify that, by choosing properly the domain dimensions, it is possible to capture correctly the MTSI growth and its corresponding number of azimuthal periods. In particular, we show that an azimuthal length that is smaller than a certain threshold prevents the MTSI from growing. Moreover, we show that the MTSI growth does not depend on the plasma density, but is affected by the axial electric field (perpendicular to the simulation domain). Additionally, we show that during its linear growth in the early times of the simulations, the MTSI produces an enhanced heating of the electrons in the magnetic field direction as well as an increased cross field mobility. For longer times, in the nonlinear regime, the system evolves toward a more chaotic state with the presence of structures that mostly exhibit large azimuthal wavelengths.

Understanding anomalous electron transport in $\mathbf{E}\times \mathbf{B}$ discharges remains a key challenge in the development of self-consistent models of these systems. It has been shown that short-wavelength, high-frequency instabilities in the azimuthal $\mathbf{E}\times \mathbf{B}$ direction may be responsible for increased electron transport due to an enhanced electron-ion friction force. Although a theoretical model based on quasi-linear kinetic theory has previously been proposed to describe this friction force, it has so far only undergone limited validation testing. Here, we rigorously assess this theoretical model by comparison with the friction force self-consistently obtained from 2D axial-azimuthal particle-in-cell simulations. The simulation geometry is based on a recently established benchmark configuration for $\mathbf{E}\times \mathbf{B}$ discharges, and a broad parametric study is performed by varying the magnetic field strength, the discharge current density, and the presence of different neutral collisional processes. Overall, the theory is found to be in very good agreement with the simulation results for all cases studied, verifying the underlying physical mechanisms leading to enhanced electron transport. We demonstrate, however, that the friction force depends sensitively on the shape of the electron velocity distribution function, thus posing significant challenges to fully self-consistent, first principles modeling of anomalous transport in fluid simulations.

Recent simulations and experiments have observed a transition from short to long-wavelength azimuthal instabilities that leads to enhanced electron transport in Hall thrusters. Here we make the hypothesis that this phenomenon stems directly from the interaction between the axial ion transit-time instabilities (ITTI), and the azimuthal electron drift instability (EDI). This interaction is studied using 2D axial-azimuthal self-consistent particle-in-cell simulations which include a 1D neutral dynamics solver. It is found that a short to long-wavelength transition only occurs if the breathing-mode and ITTI are captured in the simulation, and two distinct instability regions can be distinguished depending on the local ion Mach number. Upstream of the ion sonic point the EDI exhibits an ion-acoustic behaviour, and the associated instability-enhanced electron transport is well described by a previously developed model based on kinetic theory. Downstream of the ion sonic point however, the ITTI significantly changes the local plasma parameters, and this modifies the EDI while increasing the electron transport.

Even after half a century of development, many phenomena in Hall Effect Thrusters are still not well-understood. While numerical studies are now widely used to study this highly non-linear system, experimental diagnostics are needed to validate their results and identify specific oscillations. By varying the cathode heating current, its emissivity is efficiently controlled and a transition between two functioning regimes of a low power thruster is observed. This transition implies a modification of the axial electric field and of the plasma plume shape. High-speed camera imaging is performed and the data are analysed using a Proper Orthogonal Decomposition method to isolate the different types of plasma fluctuations occurring simultaneously. The low-frequency breathing mode is observed, along with higher frequency rotating modes that can be associated to rotating spokes or gradient-induced instabilities. These rotating modes are observed while propagating outside the thruster channel. The reduction of the cathode emissivity beyond the transition comes along with a disappearance of the breathing mode, which could improve the thruster performance and stability.