Low-temperature plasmas (LTPs) are essential to both fundamental scientific research and critical industrial applications. As in many areas of science, numerical simulations have become a vital tool for uncovering new physical phenomena and guiding technological development. Code benchmarking remains crucial for verifying implementations and evaluating performance. This work continues the Landmark benchmark initiative, a series specifically designed to support the verification of LTP codes. In this study, seventeen simulation codes from a collaborative community of nineteen international institutions modeled a partially magnetized E x B Penning discharge. The emergence of large scale coherent structures, or rotating plasma spokes, endows this configuration with an enormous range of time scales, making it particularly challenging to simulate. The codes showed excellent agreement on the rotation frequency of the spoke as well as key plasma properties, including time-averaged ion density, plasma potential, and electron temperature profiles. Achieving this level of agreement came with challenges, and we share lessons learned on how to conduct future benchmarking campaigns. Comparing code implementations, computational hardware, and simulation runtimes also revealed interesting trends, which are summarized with the aim of guiding future plasma simulation software development.

An axisymmetric hybrid particle/fluid code has been applied to simulate the plasma behavior in the discharge chamber of a radio-frequency ion thruster (RIT). Suitable assumptions have been made to include the coupling with the three other elements of the thruster: the time-averaged deposited power from the radio-frequency (RF) source, the effective transparencies for ion and neutral extraction from the grid system, and the electrical connection with the external neutralizer. A detailed parametric analysis has been carried out in terms of deposited RF power and injected mass flow rate of xenon. As the RF power increases: high propellant utilization and ion beam current are obtained, the maximum plasma density moves away from the chamber center and close to the flow injector, and the beam-to-total energy efficiency decreases. As a result, for a given mass flow rate, there is an optimal RF power that maximizes the thrust efficiency. At these conditions, about 60% of the RF deposited power is lost into the walls due to the lack of magnetic confinement and multiple re-ionizations of wall-recombined neutrals penalize the inelastic losses too. Additional studies have analyzed the influence of: the electrical connections to the neutralizer, which just change the electron current paths; the effective transparency of the grids, where a large sensitivity is found to the ion transparency; the reduction of the energy accommodation of heavy species after wall impact, which penalizes much the maximum thrust efficiency; the presence of metastable atoms and stepwise ionization, which although present, have marginal effects on performances as a result of the multiple wall interactions. A global model, derived from the 2D model and constituted of algebraic expressions, can yield the main performance figures and averaged plasma magnitudes in the chamber with a relative error of up to +/- 15%. Finally, 2D simulations are compared indirectly with existing data for a RIT-4.

Instabilities common to E x B partially magnetized plasmas can ultimately lead to anomalous electron cross-field transport. This work presents a local linear analysis of fluid instabilities driven by the electron E x B drift and gradients in the equilibrium magnitudes of an essentially collisionless plasma. Effects of magnetic curvature, finite electron Larmor radius, and 3D wave propagation are included. Expanding on previous works, the role of gradients both perpendicular and parallel to the magnetic field is analyzed. The analysis of the dispersion relation for the complex frequency of the modes shows that a generalization of the classical lower-hybrid drift instability (LHDI) develops when parallel gradients or parallel propagation are accounted for. Furthermore, this LHDI becomes a generalized modified two-stream instability when perpendicular gradients are absent. Weak collisionality can reduce or enhance the growth rate of the collisionless LHDI, depending on the parametric range of parallel effects. The application of the LHDI dispersion relation to the simulation data of a magnetic nozzle plasma at equilibrium highlights the importance of parallel inhomogeneities in the instability developing in certain nozzle regions. The qualitative agreement with available experimental data on magnetic nozzles is discussed, too. Finally, quasi-linear analysis suggests that the LHDI-induced cross-field transport in the outward direction acts to smooth out the zeroth-order drifts which cause the plasma to destabilize in the first place, and may be an essential mechanism partaking in electron detachment from a magnetic nozzle.

In order to assess the importance of collisional processes in the neutralization of a gridded ion thruster plume this study presents a series of simulations carried out with a planar, full-PIC code provided of a library of electron, ion and neutral collisions. In particular, we find that the inclusion of electron inelastic collisions, such as ionization and excitation, results fundamental in the generation of a population of electrons trapped in the plume's potential well, which play a major role in the neutralization of the plume and in the formation of the electric potential map. A further investigation is therefore carried out on the properties of the electron populations present in the plume, suggesting that the trapped population results mostly insensitive to the emission properties of the cathode, but displays a strong dependence on the inclusion of inelastic collisions and a slow approach to stationary conditions dictated by the timescales of these collisions. Given the symmetry of the plume bulk even in presence of an externally mounted cathode, we further extend the study to an axisymmetric simulation case with an annular cathode, that allows the evaluation of a three-dimensional expansion process. The build up of the trapped electron population in this case is even slower, because of the smaller neutral density observed when expanding the plume in three dimensions, and an acceleration strategy that speeds-up the approach to steady state without altering it is therefore proposed. Nevertheless, the main governing physical processes observed in the planar case remain prominent also in this latter case.

Magnetic arches (MA) (i.e. the magnetic topology that emerges when placing two magnetic nozzles with opposite polarities side by side) are an attractive option for the clustering of multiple electrodeless plasma thrusters, as they are characterized by a zero magnetic dipole moment and thus allow a reduction of perturbing magnetic forces on the spacecraft. This work employs the hybrid code EP2PLUS to simulate and study the plasma expansion for such a magnetic topology in the planar limit. First, a reference simulation is used to analyze the leading physical mechanisms that govern the plume properties. Ions are thus found to be characterized by a double peaked velocity distribution function close to the symmetry plane, where the plasma beams emitted by the two thrusters merge, while the magnetic force acting on electrons is shown to shape both the lateral confinement of the plume, and the thrust profile provided. Second, a parametric sweep on the strength of the magnetic field shows that its influence on the propulsive properties and on the characteristics of the plume saturates for values of the Hall parameter larger than around 10. Beyond this value of the Hall parameter, only the in-plane electron currents are found to be particularly sensitive both to the magnetization levels and the boundary conditions employed, although they are also largely decoupled from the rest of plasma properties. Finally, background pressure effects were considered by including collisions with neutral atoms in the simulations, highlighting the relevance of neutral entrainment in the modification of the plume properties and in the propulsive performance of the MA.

The operation of a 20 kW magnetically shielded Hall effect thruster, with xenon and krypton as propellants, is numerically studied with a 2D (axial-radial) axisymmetric hybrid (particle-in-cell/fluid) code. Thrust efficiency with Kr is 2 to 6% lower than with Xe. Kr exhibits smaller propellant utilization, voltage utilization and divergence efficiencies. Contrary to what is observed in previous works, the current efficiency is higher with Kr. Apart from the dissimilar mass and electronic properties of Xe and Kr atoms, the widening of the ionization and acceleration zones with Kr, with respect to the Xe discharge, plays an important role in the Xe-Kr differences in performance. The level of plume velocity dispersion with both propellants is found similar due to two opposing effects: on the one hand, the stronger overlap of the ionization and acceleration zones with Kr leads to a larger dispersion on a per species basis; and, on the other hand, the smaller fraction of doubly-charged ions with Kr reduces the overall dispersion efficiency. Magnetic shielding is effective with both Xe and Kr as propellants, with slightly higher ion impact energy on the walls in the Kr discharge.

Electrodeless plasma thrusters (EPTs) are a novel technology in which a plasma is generated and heated by radiofrequency fields. The plasma then expands quasineutrally in a magnetic nozzle, without the need of a neutralizer. We present the recent developments of two EPT prototypes at UC3M: a helicon plasma thruster (HPT, operating in the tens of MHz, around 500 W) and an electron-cyclotron resonance thruster (ECRT, GHz range, 80-300 W). We describe the hardware employed for RF generation, amplification, and matching in each case, the mechanisms for propagation and absorption of the electromagnetic waves in the plasma in these devices, the laboratory experiments, and their results. We also report on the latest modeling efforts to solve the plasma-wave interaction problem numerically.

The electron–cyclotron drift instability (ECDI) has been proposed as one of the main actors behind the anomalous transport of electrons in Hall plasmas. In this work, we revisit the classical theory of this instability [Forslund et al., Phys. Rev. Lett. 25, 1266 (1970)] and perform two-dimensional kinetic simulations under several conditions to analyze the non-linear behavior and the induced transport. Fully periodic simulations, with conditions faithful to the linear theory, are analyzed first. In agreement with existing literature, they show the growth of ECDI modes, ion-wave trapping vortexes, and an induced cross field electron current in early simulation times. However, in contrast with similar works, non-linear saturation is observed and the plasma tends, in the long term, to a new equilibrium with mild oscillations and mild anomalous current. This evolution is consistent with what can be expected from energy conservation. The quenching of the oscillations seems to be highly related to the distortion of ion vortexes in phase space after a long-term interaction with the electrostatic wave. This result suggests that sustained oscillations and turbulent current could thrive if ions are renewed by, e.g., removing and injecting particles through axial boundaries instead of applying periodicity. This second type of simulation shows that injection conditions highly impact the late simulation behavior of ECDI oscillations, where we identify several regimes depending on the value of the ion residence time compared to the characteristic saturation time in the fully periodic case. The intermediate regime, where these two times are close, is the only one providing sustained oscillations and electron transport.

A 2D axial-radial particle-in-cell (PIC) model of a Hall thruster discharge has been developed to analyze (mainly) the fluid equations satisfied by the azimuthally-averaged slow dynamics of electrons. Their weak collisionality together with a strong interaction with the thruster walls lead to a non-Maxwellian velocity distribution function (VDF). Consequently, the resulting macroscopic response differs from a conventional collisional fluid. First, the gyrotropic (diagonal) part of the pressure tensor is anisotropic. Second, its gyroviscous part, although small, is relevant in the azimuthal momentum balance, where the dominant contributions are orders of magnitude lower than in the axial momentum balance. Third, the heat flux vector does not satisfy simple laws, although convective and conductive behaviors can be identified for the parallel and perpendicular components, respectively. And fourth, the electron wall interaction parameters can differ largely from the classical sheath theory, based on near Maxwellian VDF. Furthermore, these effects behave differently in the near-anode and near-exit regions of the channel. Still, the profiles of basic plasma magnitudes agree well with those of 1D axial fluid models. To facilitate the interpretation of the plasma response, a quasiplanar geometry, a purely-radial magnetic field, and a simple empirical model of cross-field transport were used; but realistic configurations and a more elaborate anomalous diffusion formulation can be incorporated. Computational time was controlled by using an augmented vacuum permittivity and a stationary depletion law for neutrals.

A non-neutral model (NNM) of the axial plasma discharge in a Hall thruster, including full electron inertia, is presented. In the finite-volume formulation, two types of sheath boundary conditions previously used in the literature are tested and proven to behave practically identically in this model. Both normal and reversed (i.e. electron repelling and attracting, respectively) anode sheaths are admitted. This model is compared with the quasineutral model developed in a previous work, which includes only azimuthal electron inertia and normal anode sheaths. Both models agree excellently within the parametric region where steady-state solutions with a normal anode sheath exist. The NNM shows the absence of steady-state solutions with a reversed anode sheath. Nonetheless, a reversed sheath can appear during the transient to a steady-state solution with a normal sheath and the periodic transition from a normal to a reversed sheath can be observed in the presence of breathing-mode oscillations. In other cases, the reversed sheath leads to the discharge shut-off. Full electron inertia is always important in the presence of a reversed sheath. The parametric threshold of the wall accommodation parameter from a stationary solution to a breathing mode one differs slightly between the non-neutral and the quasi-neutral model.

A fully cylindrical Hall thruster prototype was tested in the power range of 30–300 W with the objective of understanding the behavior of the discharge as a function of input parameters. Various operating conditions were compared, including two magnetic field configurations, a set of propellant mass flow rates, and a range of discharge voltages. Plasma properties were measured in the plume, with a Langmuir probe, a retarding potential analyzer, and a Faraday cup. The experimental results showed that the mass flow rate strongly affects the ionization and, consequently, other related properties such as the plasma density, currents, and propellant utilization. The discharge voltage also appeared to influence the ion energy and propellant utilization. The performance accessible from the measured magnitudes is assessed, resulting in a maximum thrust efficiency of about 18% at 0.35 mg s−1 and 168 W.

Magnetic nozzles are a key component of electrodeless plasma thrusters, acting as their main acceleration stage. Non-stationary phenomena common to the entire range of E × B devices, such as oscillations and instabilities, are likely to exist in the magnetic nozzle, according to the mounting experimental evidence. These mechanisms could lead to anomalous cross-field transport, either enhancing the plasma plume divergence or favoring electron detachment. In this work we present a local linear analysis of fluid instabilities relevant for said devices, expanding on previous works with the addition of plasma inhomogeneities in the direction parallel to the magnetic field, with a rigorous inclusion of the effects of magnetic curvature, finite Larmor radius and 3D wave propagation, allowing for a general formulation of drift-driven instabilities in partially magnetized plasmas. Instability conditions are first studied analytically, and then applied to simulation data of a helicon plasma thruster. Finally, the effect of instabilities on wave-driven cross-field electron transport is assessed by means of quasi-linear analysis. This study predicts the onset of essentially-azimuthal instabilities in the 1 kHz–1 MHz range, in qualitative agreement with some of the available experimental data, and highlights the importance of including parallel inhomogeneities in the formulation of the dispersion relation of an E × B plasma, as these gradients may drive instabilities even in the absence of axial propagation. Lastly, quasi-linear analysis suggests that the induced cross-field transport acts to smooth out the zeroth-order drifts which cause the plasma to destabilize in the first place.

Plasma chemistry of main air components is implemented in a hybrid 2D axisymmetric simulation code to assess the air-breathing concept in an electrodeless plasma thruster. Relevant electron-heavy species collisions for diatomic molecules are included: rotational and vibrational excitation, dissociation and dissociative ionization. Plasma-wall interaction giving rise to associative recombination of atomic species into molecular species is included too. As reference, the plasma thruster is operated with Xe, at a power of 300W and a mass flow of 1mg/s. Simulations are run by injecting 1mg/s of N$_2$ and O independently for powers between 100 and 3000W. The performances and trends of plasma response for these propellants are similar to Xe, but displaced to powers between 1250 and 2000W. At optimum power, the thrust efficiency for N$_2$ and O surpasses that of Xe, due to the excess of re-ionization for Xe. Performances of 50/50 mixtures of N$_2$/O, which are a realistic composition in the ionosphere, are found to be linear combinations of the performances of each propellant. Performances using O$_2$, which could be generated from associative recombination of O at the intake, are very similar to those of the atomic oxygen.

One-dimensional axial models of the plasma discharge of a Hall thruster provide a valuable picture of its physical behavior with a small computational effort. Therefore, they are very suitable for quick parametric analyses or as a support tool for analyzing the impact of modeling decisions. This paper extends a well-known drift-diffusion stationary, quasineutral model by adding electron azimuthal inertia (EAI), a nonzero thickness cathode layer, and the far-plume region where electrons demagnetize and cool down. The EAI dominates on the far plume and affects positively to thrust. For a small ion backstreaming current, EAI modifies much the electron velocities and density near the anode, but has no discernible effect on the electron cross-field transport. Electron axial inertia and azimuthal gyrovisosity are estimated. The thick cathode layer connects quasineutrally the near and far plumes but the coupling between these two regions is weak. The far plume region is sensitive to the decay length of the magnetic field, the downstream boundary conditions on the electron currents, and the stray electric currents.

Electrodeless plasma thrusters consist of a dielectric discharge chamber in which RF electromagnetic fields are used to ionize the propellant gas into a plasma, and an applied magnetic field that (1) enables the propagation and absorption of those RF fields into the plasma, (2) shields the lateral walls of the discharge chamber, and (3) forms an external magnetic nozzle that channels the plasma expansion to generate thrust [1].

Fluid models of the slow-dynamics of magnetized, weakly-collisional electrons lead to build computationally-affordable, long-time simulations of plasma discharges in Hall-effect and electrodeless plasma thrusters. This paper discusses the main assumptions and techniques used in 1D to 3D electron fluid models, and some examples illustrate their capabilities. Critical aspects of these fluid models are the expressions for the pressure tensor, the heat flux vector, the plasma-wall fluxes, and the high-frequency-averaged electron transport and heating caused by plasma waves, generated either by turbulence or external irradiation. The different orders of magnitude of the three scalar momentum equations characterize the electron anisotropic transport. Central points of the discussion are: the role of electron inertia, magnetically-aligned meshes versus Cartesian-type ones, the use of a thermalized potential and the infinite mobility limit, the existence of convective-type heat fluxes, and the modeling of the Debye sheath, and wall fluxes. Plasma plume models present their own peculiarities, related to anomalous parallel cooling and heat flux closures, the matching of finite plume domains with quiescent infinity, and solving fully collisionless expansions. Solutions of two 1D electron kinetic models are used to derive kinetically-consistent fluid models and compare them with more conventional ones.

Experiments and simulations are used to analyze a compact helicon plasma thruster with a cusp in its internal magnetic field. The former rely on a compensated Langmuir probe and a Faraday cup, while the latter employ a hybrid PIC/fluid transport model combined with a frequency-domain electromagnetic field model. Measurements serve to tune the anomalous transport parameters of the model and overall show the same trends as the numerical results, including a secondary peak of electron temperature downstream in the magnetic nozzle, where electron cyclotron resonance conditions for the 13.56 MHz excitation frequency are met. The cusp plays a central role in determining the plasma losses to the walls and the profile of electron temperature, which in turn defines the excitation and ionization losses. While losses to the rear wall are reduced, losses to the lateral wall are increased, which, together with the low production efficiency, limit the performance of the device.

Simulations of energetic plumes from plasma thrusters are of great interest for estimating performances and interactions with the spacecraft. Both in fully fluid and hybrid (particle/fluid) models, the electron population is described by a set of fluid equations whose resolution by different numerical schemes can be strongly affected by convergence and accuracy issues. The case of magnetized plumes is more critical. Here, the numerical discretization of the electron fluid model of a 3D hybrid simulator of plasma plumes was upgraded from a finite-differences (FD) formulation in a collocated grid to a finite-volumes (FV) approach in a staggered grid. Both approaches make use of structured meshes of different resolutions and are compared in two scenarios of interest: 1) an unmagnetized plasma plume around a spacecraft and 2) a magnetized plume expansion in free space. In both physical scenarios, the FD scheme exhibits a global continuity error related to truncation errors that can be reduced only by refining the mesh. The origin of this error is further investigated and explained here. The FV scheme instead can save much computational time using coarser meshes since it is unaffected by these errors due to the conservativeness of its formulation. The physical advantage of the FV scheme over the FD approach is more evident for magnetized plumes with high Hall parameters since it allows us to reach higher anisotropy conditions, here assessed in order to gain insights into the plume magnetization effects, finding that the already foreseen saturation of circulating electric current occurs for Hall parameters of several hundreds.

The operation of a 5 kW-class magnetically shielded Hall effect thruster with sinusoidal modulation of the discharge voltage is investigated through simulations with a 2D axisymmetric hybrid (particle-in-cell/fluid) code. The dynamic response of the thruster for different modulation amplitudes and frequencies is presented and discussed. The analysis of partial efficiencies contributing to thrust efficiency allows identifying counteracting effects limiting net gains in performance figures. Voltage modulation enhances the amplitude of plasma oscillations and can effectively control their frequency when the modulation frequency is close to that of the natural breathing mode (BM) of the thruster. The 2D plasma solution reveals that the dynamics of the ionization cycle are governed by the electron temperature response, enabling a driven BM at the modulation frequency. For modulation frequencies far from the natural BM one, voltage modulation fails to control the plasma production via the electron temperature, and the natural BM of the thruster is recovered. High order dynamic mode decomposition applied to the 2D plasma solution permits analyzing the complex spatio-temporal behavior of the plasma discharge oscillations, revealing the main characteristics of natural and externally driven modes.

Electrostatic probe and thrust balance measurements of a coaxial electron-cyclotron-resonance plasma thruster with magnetic nozzle are compared against numerical simulations of the device that solve self-consistently the plasma transport problem with a hybrid particle-in-cell/fluid approach and the microwave electromagnetic fields using mixed finite elements. A simple phenomenological anomalous transport model similar to those used in Hall thruster modeling is applied. Reasonable average relative errors are reported on the ion current density (8.7%) and plasma density (12.8%) profiles along the plume. Good agreement is found in terms of relative errors on thruster performance parameters as the 90%-current divergence angle (0%–3%), utilization efficiency (3%–10%), peak ion energy (9%–15%), and energy efficiency (2%–17%). The comparison suggests that enhanced cross-field diffusion is present in the plasma. Differences in the experimental and numerical behavior of electron temperature point to the areas of the model that could be improved. These include the electron heat flux closure relation, which must correctly account for the axial electron cooling observed.