Streamer discharges are thin plasma channels that precede lightning and sparks. They usually evolve in bundles as stochastic tree-like structures and are inherently difficult to model due to their multiscale nature. In this paper, we perform a computer investigation of positive streamer discharges in air using contemporary photoionization models. We report on three-dimensional computer simulations under conditions that are available in laboratory spark-gap experiments. The solutions demonstrate a multiscale morphology consisting of streamer fluctuations, branching, and the formation of a discharge tree. Some branches are comparatively thick and noisy, while others are thin, smooth, and carry electric fields exceeding $250\,\textrm{kV/cm}$ at their tips. Our results are consistent with past experiments and clarify the puzzling branching dynamics and stochastic morphology of positive streamer discharges.
In this paper we study the effects of photoionization processes on the propagation of both negative and positive streamer discharges. We show that negative fronts accelerate in the presence of photoionization events. The appearance and propagation of positive streamers travelling with constant velocity is explained as the result of the combined effects of photoionization and electron diffusion. The photoionization range plays an important role for the selection of the velocity of the streamer as we show in this work.
A 2D approach is used to simulate the properties of positive and negative streamers emerging from a high-voltage electrode in a long (14 cm) air gap for standard pressure and temperature. The applied voltage varies from 100 to 500 kV. To reveal the influence of photoionization, the calculations are made for various rates of seed electron generation in front of the streamer head. The difference between the properties of positive and negative streamers is associated with the different directions of the electron drift ahead of the streamer head. As a result, the peak electric field at the streame…
In this paper we study the lateral instability of streamer discharges using a continuum discharge model. We observe similarities to Kelvin-Helmholz instability in fluids. In strong electric fields, lateral long wave length perturbations can grow while small wave length perturbations diminish during the discharge evolution. We perform numerical simulations and carry out asymptotic analysis of the instability. Our results could explain the growth observed in experimental discharges.
In this review we describe a transient type of gas discharge which is commonly called a streamer discharge, as well as a few related phenomena in pulsed discharges. Streamers are propagating ionization fronts with self-organized field enhancement at their tips that can appear in gases at (or close to) atmospheric pressure. They are the precursors of other discharges like sparks and lightning, but they also occur in for example corona reactors or plasma jets which are used for a variety of plasma chemical purposes. When enough space is available, streamers can also form at much lower pressures,…
We introduce the generic structure of a growth model for branched discharge trees that consistently combines a finite channel conductivity with the physical law of charge conservation. It is applicable, e.g., to streamer coronas near tip or wire electrodes and ahead of lightning leaders, to leaders themselves and to the complex breakdown structures of sprite discharges high above thunderclouds. Then we implement and solve the simplest model for positive streamers in ambient air with self-consistent charge transport. We demonstrate that charge conservation contradicts the common assumption of d…
Previous studies of streamer discharge branching mechanisms have mainly been generative other than predictive. To predict or even control branching, a reliable connection between experimental conditions and streamer branching needs to be established. As an important step toward the goal, in this work, a 2D deterministic model of negative streamers in air is numerically solved with the ionization seeds assumed as the superposition of Gaussians. The "indicative profiles approach" developed here can consistently relate the change in a quantitative measure of geometrical irregularity of the seed p…
A major obstacle for the understanding of long electrical discharges is the complex dynamics of streamer coronas, formed by many thin conducting filaments. Building macroscopic models for these filaments is one approach to attain a deeper knowledge of the discharge corona. Here we present a one-dimensional, macroscopic model of a propagating streamer channel. We represent the streamer as an advancing finite-conductivity channel with a surface charge density at its boundary. This charge evolves self-consistently due to the electric current that flows through the streamer body and within a thin …
Branching is an essential element of streamer discharge dynamics but today it is understood only qualitatively. The variability and irregularity observed in branched streamer trees suggest that stochastic terms are relevant for the description of streamer branching. We here consider electron density fluctuations due to the discrete particle number as a source of stochasticity in positive streamers in air at standard temperature and pressure. We derive a quantitative estimate for the branching distance that agrees within a factor of 2 with experimental values. As branching without noise would o…
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 c…
A streamer coupling theory is developed to describe the formation of homogenous emission, and the high moving speed of emission patterns in high pressure discharges. By considering the effects of both electron diffusion and electronic drift in the streamer head, the minimum required preionization level $n_{\rm min}$ for the formation of streamer coupling is found to depend on electric field strength, gas pressure and electron temperature. The homogeneity and moving speed of the emission pattern in streamer coupling head increase with preionization level $n_0$, when $n_0 > n_{\rm min}$. The pre…
We present an open-source plasma fluid code for 2D, cylindrical and 3D simulations of streamer discharges, based on the Afivo framework that features adaptive mesh refinement, geometric multigrid methods for Poisson's equation, and OpenMP parallelism. We describe the numerical implementation of a fluid model of the drift-diffusion-reaction type, combined with the local field approximation. Then we demonstrate its functionality with 3D simulations of long positive streamers in nitrogen in undervolted gaps, using three examples. The first example shows how a stochastic background density affects…
We investigate streamer formation in the troposphere, in electric fields above the breakdown threshold. With fully three-dimensional particle simulations, we study the combined effect of natural background ionization and of photoionization on the discharge morphology. In previous investigations based on deterministic fluid models without background ionization, so-called double-headed streamers emerged. But in our improved model, many electron avalanches start to grow at different locations. Eventually the avalanches collectively screen the electric field in the interior of the discharge. This …
The propagation mechanisms of plasma streamers have been observed and investigated in a surface dielectric barrier discharge (SDBD) using 2D particle in cell simulations. The investigations are carried out under a simulated air mixture, 80\% N$_2$ and 20\% O$_2$, at atmospheric pressure, 100$\,$kPa, under both DC conditions and a pulsed DC waveform that represent AC conditions. The simulated geometry is a simplification of the symmetric and fully exposed SDBD resulting in the simultaneous ignition of both positive and negative streamers on either side of the Al$_2$O$_3$ dielectric barrier. In …
Positive streamer discharges require a source of free electrons ahead of them for their growth. In air, these electrons are typically provided by photoionization. Here we investigate how stochastic fluctuations due to the discreteness of ionizing photons affect positive streamers in air. We simulate positive streamers between two planar electrodes with a 3D plasma fluid model, using both a stochastic and a continuum method for photoionization. With stochastic photoionization, fluctuations are visible in the streamer's direction, maximal electric field, velocity, and electron density. The strea…
Radiation is important for the propagation of streamers in dielectric liquids. Photoionization is a possibility, but the effect is difficult to differentiate from other contributions. In this work, we model radiation from the streamer head, causing photoionization when absorbed in the liquid. We find that photoionization is local in space (μm-scale). The radiation absorption cross section is modeled considering that the ionization potential (IP) is dependent on the electric field. The result is a steep increase in the ionization rate when the electric field reduces the IP below the energy of t…
The numerical simulation of the development of a streamer discharge in a gap with an external longitudinal magnetic field was used to demonstrate the self-focusing of such discharges. Self-focusing is caused by a sharp deceleration of the radial ionization wave due to a change in the electron energy distribution function, a decrease in the average electron energy, the rate of gas ionization and the electron mobility in crossed electric and magnetic fields as compared to the case of the discharge development without a magnetic field. The self-focusing effect of a streamer discharge in an extern…
We discuss spatially and temporally adaptive implicit-explicit (IMEX) methods for parallel simulations of three-dimensional fluid streamer discharges in atmospheric air. We examine strategies for advancing the fluid equations and elliptic transport equations (e.g. Poisson) with different time steps, synchronizing them on a global physical time scale which is taken to be proportional to the dielectric relaxation time. The use of a longer time step for the electric field leads to numerical errors that can be diagnosed, and we quantify the conditions where this simplification is valid. Likewise, …
Streamer discharges are important both in theory and industry applications. This paper proposed a local discontinuous Galerkin method to simulate the convection dominated fluid model of streamer discharges. To simulate the rapid transient streamer discharge process, a method with high resolution and high order accuracy is highly desired. Combining the advantages of finite volume and finite element method, local discontinuous Galerkin method is such a choice. In this paper, a simulation of a double-headed streamer discharge in nitrogen was performed by using 1.5-dimensional fluid model. The pre…
We introduce a 3D hybrid model for streamer discharges that follows the dynamics of single electrons in the region with strong field enhancement at the streamer tip while approximating the many electrons in the streamer interior as densities. We explain the method and present first results for negative streamers in nitrogen. We focus on the high electron energies observed in the simulation.
Streamer discharges play a central role in electric breakdown of matter in pulsed electric fields, both in nature and in technology. Reliable and fast computations of the minimal model for negative streamers in simple gases like nitrogen have recently been developed. However, photoionization was not included; it is important in air and poses a major numerical challenge. We here introduce a fast and reliable method to include photoionization into our numerical scheme with adaptice grids, and we discuss its importance for negative streamers. In particular, we identify different propagation regim…
Streamer electrical discharges are often investigated with computer simulations of density models (also called drift-diffusion-reaction models). We review these models, detailing their physical foundations, their range of validity and the most relevant numerical algorithms employed in solving them. We focus particularly on schemes of adaptative refinement, used to resolve the multiple length scales in a streamer discharge without a high computational cost. We then report recent results from these models, emphasizing developments that go beyond cylindrically symmetrical streamers propagating in…
Streamer discharges pose basic problems in plasma physics, as they are very transient, far from equilibrium and have high ionization density gradients; they appear in diverse areas of science and technology. The present paper focuses on the derivation of a high order fluid model for streamers. Using momentum transfer theory, the fluid equations are obtained as velocity moments of the Boltzmann equation; they are closed in the local mean energy approximation and coupled to the Poisson equation for the space charge generated electric field. The high order tensor in the energy flux equation is ap…
We study the development of a negative discharge driven by a Marx generator of about 1 MV in an air gap of 1 up to 1.5 meter, at standard temperature and pressure. We show the evolution of the discharge with nanosecond-fast photography together with the electrical characteristics. The negative discharge develops through four well-distinguished streamer bursts. The streamers have different velocities and life times in different bursts. The last burst triggers a positive inception cloud on the positive grounded electrode and a burst of positive counter-streamers emerges. The pre-discharge then b…
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 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. …
Positive streamer discharges have been studied and modelled extensively in air. Here we study positive streamers in CO$_2$ with and without oxygen admixtures; they are relevant for current high voltage technology as well as for discharges in the atmosphere of Venus. We discuss that no efficient photoionization mechanism is known for gases with a large CO$_2$ fraction, as photons in the relevant energy range are rapidly absorbed. Hence positive streamers can propagate only due to some other source of free electrons ahead of the ionization front. Therefore we study positive streamer propagation …
To find a viable alternative to SF6 with growing climate change regulations, proper evaluation of alternatives such as compressed air ought to be done. For medium voltage applications, the withstand voltage is used as the dimensioning criteria and this is dependent on the initiation and propagation of streamers which are precursors to electrical breakdown. For design optimization, a thorough understanding of the initiation and propagation mechanisms of such electrical discharges under different stresses, pressure etc. ought to be studied experimentally and numerically also via a predictive mod…
The dynamics of positive and negative streamers is numerically simulated in atmospheric pressure air. It is shown that positive and negative streamers behave in radically different ways when decelerating and stopping. When the head potential drops, the negative streamer transits to the mode in which the propagation is due to the forward electron drift. In this case, the radius of the ionization wave front increases, whereas the electric field at the streamer head decreases further and the streamer stops. The only advancement mechanism for a positive streamer with decreasing head potential is a…
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 fie…
Positive streamers need a source of free electrons ahead of them to propagate. A streamer can supply these electrons by itself through photo-ionization, or the electrons can be present due to external background ionization. Here we investigate the effects of background ionization on streamer propagation and morphology by changing the gas composition and the repetition rate of the voltage pulses, and by adding a small amount of radioactive Krypton 85. We find that the general morphology of a positive streamer discharge in high purity nitrogen depends on background ionization: at lower backgroun…
We investigate structure and evolution of long positive spark breakdown; and we study at which stage pulses of hard X-rays are emitted. Positive high-voltage pulses of standardized lightning impulse wave form of about 1 MV were applied to about 1 meter of ambient air. The discharge evolution was imaged with a resolution of tens of nanoseconds with an intensified CCD camera. LaBr3(Ce+) scintillation detectors recorded the X-rays emitted during the process. The voltage and the currents on both electrodes were measured synchronously. All measurements indicate that first a large and dense corona o…
Starting from the minimal model for the electrically interacting particle densities in negative streamer discharges, we derive a moving boundary approximation for the ionization fronts. The boundary condition on the moving front is found to be of 'kinetic undercooling' type. The boundary approximation, whose first results have been published in [Meulenbroek et al., PRL 95, 195004 (2005)], is then tested against 2-dimensional simulations of the density model. The results suggest that our moving boundary approximation adequately represents the essential dynamics of negative streamer fronts.
Controlling streamer morphology is important for numerous applications. Up to now, the effect of the voltage rise rate was only studied across a wide range. Here we show that even slight variations in the voltage rise can have significant effects. We have studied positive streamer discharges in a 16 cm point-plane gap in high-purity nitrogen 6.0, created by 25 kV pulses with a duration of 130 ns. The voltage rise varies by a rise rate from 1.9 kV/ns to 2.7 kV/ns and by the first peak voltage of 22 to 28 kV. A structural link is found between smaller discharges with a larger inception cloud cau…
We investigate the branching of positive streamers in air and present the first systematic investigation of splitting into more than two branches. We study discharges in 100 mbar artificial air that is exposed to voltage pulses of 10 kV applied to a needle electrode 160 mm above a grounded plate. By imaging the discharge with two cameras from three angles, we establish that about every 200th branching event is a branching into three. Branching into three occurs more frequently for the relatively thicker streamers. In fact, we find that the surface of the total streamer cross-sections before an…
It is by now well understood that large sprite discharges at the low air densities of the mesosphere are physically similar to small streamer discharges in air at standard temperature and pressure. This similarity is based on Townsend scaling with air density. First the theoretical basis of Townsend scaling and a list of six possible corrections to scaling are discussed; then the experimental evidence for the similarity between streamers and sprites is reviewed. We then discuss how far present sprite and streamer theory has been developed, and we show how streamer experiments can be interprete…
The high order fluid model developed in the preceding paper is employed here to study the propagation of negative planar streamer fronts in pure nitrogen. The model consists of the balance equations for electron density, average electron velocity, average electron energy and average electron energy flux. These balance equations have been obtained as velocity moments of Boltzmann's equation and are here coupled to the Poisson equation for the space charge electric field. Here the results of simulations with the high order model, with a PIC/MC (Particle in cell/Monte Carlo) model and with the fi…
Standard photographs of streamer discharges show a two-dimensional projection. We here present stereo-photographic images that resolve their three-dimensional structure. We describe the stereoscopic set-up and evaluation, and we present results for positive streamer discharges in air at 0.2 - 1 bar in a point plane geometry with a gap distance of 14 cm and a voltage pulse of 47 kV. In this case an approximately Gaussian distribution of branching angles of 43 degrees is found; these angles do not significantly depend on the distance from the needle or on the gas pressure.
In this study, we investigate the responsible species and processes involved in repetitive pulsed streamer inception in CO$_{2}$. We applied a 10 kV high-voltage pulse with a repetition frequency of 10 Hz and pulse width of 1 ms to a pin electrode which is placed 160 mm apart from the grounded plane electrode. We measured the inception times (delay between the rising edge of the high-voltage pulse) by a photo-multiplier tube for 600 high voltage cycles. We observed one peak in the histogram of inception times with a median of 1.2$μ$s. To identify the source of this peak, we applied a negative …
Streamers are ionization fronts that occur in gases at atmospheric and sub-atmospheric pressures. Numerical studies of streamers are important for practical applications but are challenging due to the multiscale nature of this discharge type. This paper introduces a spectral element method for modelling streamer discharges. The method is developed for Cartesian grids but can be extended to be used on unstructured meshes. The streamer model is based on the Poisson equation for the electric potential and the electron continuity equation. The Poisson equation is discretized via a spectral method …
Pictures show that streamer or sprite discharge channels emerging from the same electrode sometimes seem to reconnect or merge though their heads carry electric charge of the same polarity; one might therefore suspect that reconnections are an artifact of the two-dimensional projection in the pictures. Here we use stereo-photography to investigate the full three-dimensional structure of such events. We analyze reconnection, possibly an electrostatic effect in which a late thin streamer reconnects to an earlier thick streamer channel, and merging, a suggested photoionization effect in which two…
We use a three-specie fluid model of electric discharge in air to simulate streamer evolution from the avalanche-to-streamer transition to the collision of opposite-polarity streamers. We estimate the upper limit on the production of thermal runaway electrons, which is dominant during the second of these processes. More thermal runaways are produced if the ionization due to natural background and photoionization is reduced, due to possibility of creation of higher electric fields at streamer tips. The test-particle simulation shows, however, that these thermal runaway electrons have insufficie…
This paper focuses on the three-dimensional simulation of the photoionization in streamer discharges, and provides a general framework to efficiently and accurately calculate the photoionization model using the integral form. The simulation is based on the kernel-independent fast multipole method. The accuracy of this method is studied quantitatively for different domains and various pressures in comparison with other existing models based on partial differential equations (PDEs). The comparison indicates the numerical error of the fast multipole method is much smaller than those of other PDE-…
In this paper, we propose an efficient and accurate message-passing interface (MPI)-based parallel simulator for streamer discharges in three dimensions using the fluid model. First, we propose a new second-order semi-implicit scheme for the temporal discretization of the model that relaxes the dielectric relaxation time restriction. Moreover, it requires solving the Poisson-type equation only once at each time step, while the classical second-order explicit scheme typically needs to do twice. Second, we introduce a geometric multigrid preconditioned FGMRES solver that dramatically improves th…
Streamer discharges are often operated in a repetitively pulsed mode and are therefore influenced by species left over from the previous discharge, especially free electrons and ions. We have investigated these effects by applying two consecutive positive high voltage pulses of 200-700 ns duration to a point plane gap in artificial air, pure nitrogen, other nitrogen-oxygen mixtures and pure argon at pressures between 67 and 533 mbar. The pulses had pulse-to-pulse intervals (Δt) between 200 ns and 40 ms. We imaged both discharges with two ICCD cameras and combined this to a compound image. We o…
In this paper we present the complete derivation of the effective contour model for electrical discharges which appears as the asymptotic limit of the minimal streamer model for the propagation of electric discharges, when the electron diffusion is small. It consists of two integro-differential equations defined at the boundary of the plasma region: one for the motion and a second equation for the net charge density at the interface. We have computed explicit solutions with cylindrical symmetry and found the dispersion relation for small symmetry-breaking perturbations in the case of finite re…
We recently have presented first physical predictions of a spatially hybrid model that follows the evolution of a negative streamer discharge in full three spatial dimensions; our spatially hybrid model couples a particle model in the high field region ahead of the streamer with a fluid model in the streamer interior where electron densities are high and fields are low. Therefore the model is computationally efficient, while it also follows the dynamics of single electrons including their possible run-away. Here we describe the technical details of our computations, and present the next step i…
We show that the spatio-temporal ElectroHydroDynamic (EHD) force production in surface AC-Dielectric Barrier Discharge (AC-DBD) actuators is strongly influenced by both the streamer regime during the positive phase and the micro-discharge regime during the negative phase. Focusing on the spatial EHD force profiles, we demonstrate that the ionic wind spatial distribution can only be explained by the positive contribution of the streamer regime. The location of maximum ionic wind is found to be directly linked with the maximum elongation of the streamers at several millimeters from the exposed e…
Streamer discharges can be used as a primary source of reactive species for plasma-assisted combustion. In this research we investigate positive streamers in a stoichiometric air-methane mixture at 1 bar and 300 K with a three-dimensional particle-in-cell model for the electrons. We first discuss suitable electron scattering cross sections and an extension of the photoionization mechanism to air-methane mixtures. We discuss that the addition of 9.5% methane leaves electron transport and reaction coefficients essentially unchanged, but it largely suppresses photoionization and shortens the phot…
