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Simulating streamer discharges in 3D can computationally be very expensive, which is why 2D Cartesian simulations are sometimes used instead, especially when dealing with complex geometries. Although 2D Cartesian simulations can only be used to obtain qualitative results, it is nevertheless interesting to understand how they differ from their 3D or axisymmetric counterparts. We therefore compare 2D Cartesian and axisymmetric simulations of positive streamers in air, using a drift-diffusion-reaction fluid model with the local field approximation. With the same electrode length and width, inception voltages are found to be about a factor two higher in the 2D Cartesian case. When compared at the same applied voltage, the 2D Cartesian streamers are up to four times thinner and slower, their maximal electric field is about 30% lower and their degree of ionization is about 65% lower, with the largest differences occurring at the start of the discharge. When we compare at a similar ratio of applied voltage over inception voltage, velocities become rather similar, and so do the streamer radii at later propagation times. However, the maximal electric field in the 2D Cartesian case is then about 20-30% lower, and the degree of ionization is about 40-50% lower. Finally, we show that streamer branching cannot qualitatively be modeled in a 2D Cartesian geometry.

We study how external magnetic fields from 0 to 40 T influence positive streamers in atmospheric pressure air, using 3D PIC-MCC (particle-in-cell, Monte Carlo collision) simulations. When a magnetic field $\vec{B}$ is applied perpendicular to the background electric field $\vec{E}$, the streamers deflect towards the $+\vec{B}$ and $-\vec{B}$ directions which results in a branching into two main channels. With a stronger magnetic field the angle between the branches increases, and for the 40 T case the branches grow almost parallel to the magnetic field. Due to the $\vec{E}\times\vec{B}$ drift of electrons we also observe a streamer deviation in the opposite $-\vec{E}\times\vec{B}$ direction, where the minus sign appears because positive streamers propagate opposite to the electron drift velocity. The deviation due to this $\vec{E}\times\vec{B}$ effect is smaller than the deviation parallel to $\vec{B}$. In both cases of $\vec{B}$ perpendicular and parallel to $\vec{E}$, the streamer radius decreases with the magnetic field strength. We relate our observations to the effects of electric and magnetic fields on electron transport and reaction coefficients.

We investigate the propagation of positive streamers in CO$_2$ through 3D particle-in-cell simulations, which are qualitatively compared against experimental results at 50\,mbar. The experiments show that CO$_2$ streamers are much more stochastic than air streamers at the same applied voltage, indicating that few electrons are available in front of the streamer head. In the simulations, we include a photoionization model for CO$_2$. The computational results show that even a small amount of photoionization can sustain positive streamer propagation, but this requires a higher electric field than in air. We discuss the uncertainties in CO$_2$ photoionization and provide an estimate for the quenching pressure, which is based on the radiative lifetime of emitting states and the collision frequency of the gas. Furthermore, a criterion for self-sustained streamer growth in CO$_2$ is presented and compared against simulation results.

Streamer discharges are the primary mode of electric breakdown of air in lightning and high voltage technology. Streamer channels branch many times, which determines the developing tree-like discharge structure. Understanding these branched structures is for example important to describe streamer coronas in lightning research. We simulate branching of positive streamers in air using a 3D fluid model where photoionization is included as a discrete and stochastic process. The probability and morphology of branching are in good agreement with dedicated experiments. This demonstrates that photoionization indeed provides the noise that triggers branching, and we show that branching is remarkably sensitive to the amount of photoionization. Our comparison is therefore one of the first sensitive tests for Zheleznyak's photoionization model, confirming its validity.

In this paper, we address two main topics: steady propagation fields for positive streamers in air and streamer deceleration in fields below the steady propagation field. We generate constant-velocity positive streamers in air with an axisymmetric fluid model, by initially adjusting the applied voltage based on the streamer velocity. After an initial transient, we observe steady propagation for velocities of $3\times10^4$ m/s to $1.2\times10^5$ m/s, during which streamer properties and the background field do not change. This propagation mode is not fully stable, in the sense that a small change in streamer properties or background field eventually leads to acceleration or deceleration. An important finding is that faster streamers are able to propagate in significantly lower background fields than slower ones, indicating that there is no unique stability field. We relate the streamer radius, velocity, maximal electric field and background electric field to a characteristic time scale for the loss of conductivity. This relation is qualitatively confirmed by studying streamers in N2-O2 mixtures with less oxygen than air. In such mixtures, steady streamers require lower background fields, due to a reduction in the attachment and recombination rates. We also study the deceleration of streamers, which is important to predict how far they can propagate in a low field. Stagnating streamers are simulated by applying a constant applied voltage. We show how the properties of these streamers relate to the steady cases, and present a phenomenological model with fitted coefficients that describes the evolution of the velocity and radius. Finally, we compare the lengths of the stagnated streamers with predictions based on the conventional stability field.

Both fluid and particle models are commonly used to simulate streamer discharges. In this paper, we quantitatively study the agreement between these approaches for axisymmetric and 3D simulations of positive streamers in air. We use a drift-diffusion-reaction fluid model with the local field approximation and a PIC-MCC (particle-in-cell, Monte Carlo collision) particle model. The simulations are performed at 300 K and 1 bar in a 10 mm plate-plate gap with a 2 mm needle electrode. Applied voltages between 11.7 and 15.6 kV are used, which correspond to background fields of about 15 to 20 kV/cm. Streamer properties like maximal electric field, head position and velocity are compared as a function of time or space. Our results show good agreement between the particle and fluid simulations, in contrast to some earlier comparisons that were carried out in 1D or for negative streamers. To quantify discrepancies between the models, we mainly look at streamer velocities as a function of streamer length. For the test cases considered here, the mean deviation in streamer velocity between the particle and fluid simulations is less than 4%. We study the effect of different types of transport data for the fluid model, and find that flux coefficients lead to good agreement whereas bulk coefficients do not. Furthermore, we find that with a two-term Boltzmann solver, data should be computed using a temporal growth model for the best agreement. The numerical convergence of the particle and fluid models is also studied. In fluid simulations the streamer velocity increases somewhat using finer grids, whereas the particle simulations are less sensitive to the grid. Photoionization is the dominant source of stochastic fluctuations in our simulations. When the same stochastic photoionization model is used, particle and fluid simulations exhibit similar fluctuations.

We compare simulations and experiments of single positive streamer discharges in air at 100 mbar, aiming towards model validation. Experimentally, streamers are generated in a plate-plate geometry with a protruding needle. We are able to capture the complete time evolution of reproducible single-filament streamers with a ns gate-time camera. A 2D axisymmetric drift-diffusion-reaction fluid model is used to simulate streamers under conditions closely matching those of the experiments. Streamer velocities, radii and light emission profiles are compared between model and experiment. Good qualitative agreement is observed between the experimental and simulated optical emission profiles, and for the streamer velocity and radius during the entire evolution. Quantitatively, the simulated streamer velocity is about 20% to 30% lower at the same streamer length, and the simulated radius is about 1 mm (20% to 30%) smaller. The effect of various parameters on the agreement between model and experiment is studied, such as the used transport data, the background ionization level, the photoionization rate, the gas temperature, the voltage rise time and the voltage boundary conditions. An increase in gas temperature due to the 50 Hz experimental repetition frequency could probably account for some of the observed discrepancies.

We investigate the dynamics of negative surface discharges in air through numerical simulations with a 2D fluid model. A geometry consisting of a flat dielectric embedded between parallel-plate electrodes is used. Compared to negative streamers in bulk gas, negative surface streamers are observed to have a higher electron density, a higher electric field and higher propagation velocity. On the other hand, their maximum electric field and velocity are lower than for positive surface streamers. In our simulations, negative surface streamers are slower for larger relative permittivity. Negative charge accumulates on a dielectric surface when a negative streamer propagates along it, which can lead to a high electric field inside the dielectric. If we initially put negative surface charge on the dielectric, the growth of negative surface discharges is delayed or inhibited. Positive surface charge has the opposite effect., Comment: 8 pages

We use numerical simulations to study the dynamics of surface discharges, which are common in high-voltage engineering. We simulate positive streamer discharges that propagate towards a dielectric surface, attach to it, and then propagate over the surface. The simulations are performed in air with a two-dimensional plasma fluid model, in which a flat dielectric is placed between two plate electrodes. Electrostatic attraction is the main mechanism that causes streamers to grow towards the dielectric. Due to the net charge in the streamer head, the dielectric gets polarized, and the electric field between the streamer and the dielectric is increased. Compared to streamers in bulk gas, surface streamers have a smaller radius, a higher electric field, a higher electron density, and higher propagation velocity. A higher applied voltage leads to faster inception and faster propagation of the surface discharge. A higher dielectric permittivity leads to more rapid attachment of the streamer to the surface and a thinner surface streamer. Secondary emission coefficients are shown to play a modest role, which is due to relatively strong photoionization in air. In the simulations, a high electric field is present between the positive streamers and the dielectric surface. We show that the magnitude and decay of this field are affected by the positive ion mobility., Comment: 13 pages, 18 figures, 47 references

Comparative studies of capacitively coupled radio-frequency discharges in helium and argon at pressures between 10 and 80 Pa are presented applying two different fluid modelling approaches as well as two independently developed particle-in-cell/Monte Carlo collision (PIC/MCC) codes. The focus is on the analysis of the range of applicability of a recently proposed fluid model including an improved drift-diffusion approximation for the electron component as well as its comparison with fluid modelling results using the classical drift-diffusion approximation and benchmark results obtained by PIC/MCC simulations. Main features of this time- and space-dependent fluid model are given. It is found that the novel approach shows generally quite good agreement with the macroscopic properties derived by the kinetic simulations and is largely able to characterize qualitatively and quantitatively the discharge behaviour even at conditions when the classical fluid modelling approach fails. Furthermore, the excellent agreement between the two PIC/MCC simulation codes using the velocity Verlet method for the integration of the equations of motion verifies their accuracy and applicability.

We investigate discharge inception in air, in uniform background electric fields above and below the breakdown threshold. We perform 3D particle simulations that include a natural level of background ionization in the form of positive and O$_{2}^-$ ions. When the electric field rises above the breakdown and the detachment threshold, which are similar in air, electrons can detach from O$_{2}^-$ and start ionization avalanches. These avalanches together create one large discharge, in contrast to the `double-headed' streamers found in many fluid simulations. On the other hand, in background fields below breakdown, something must enhance the field sufficiently for a streamer to form. We use a strongly ionized seed of electrons and positive ions for this, with which we observe the growth of positive streamers. Negative streamers were not observed. Below breakdown, the inclusion of electron detachment does not change the results much, and we observe similar discharge development as in fluid simulations.

The Maxwell time is a typical time scale for the screening of an electric field in a medium with a given conductivity. We introduce a generalization of the Maxwell time that is valid for gas discharges: the \emph{ionization screening time}, that takes the growth of the conductivity due to impact ionization into account. We present an analytic estimate for this time scale, assuming a planar geometry, and evaluate its accuracy by comparing with numerical simulations in 1D and 3D. We investigate the minimum plasma density required to prevent the growth of streamers with local field enhancement, and we discuss the effects of photoionization and electron detachment on ionization screening. Our results can help to understand the development of pulsed discharges, for example nanosecond pulsed discharges at atmospheric pressure or halo discharges in the lower ionosphere.