Abstract: An interfacial approximation of the streamer stage in the evolution of sparks and lightning can be written as a Laplacian growth model regularized by a ‘kinetic undercooling’ boundary condition. We study the linear stability of uniformly translating circles that solve the problem in two dimensions. In a space of smooth perturbations of the circular shape, the stability operator is found to have a pure point spectrum. Except for the eigenvalue λ 0 = 0 for infinitesimal translations, all eigenvalues are shown to have negative real part. Therefore perturbations decay exponentially in time. We calculate the spectrum through a combination of asymptotic and series evaluation. In the limit of vanishing regularization parameter, all eigenvalues are found to approach zero in a singular fashion, and this asymptotic behavior is worked out in detail. A consideration of the eigenfunctions indicates that a strong intermediate growth may occur for generic initial perturbations. Both the linear and the nonlinear initial value problem are considered in a second paper.

Abstract: Positive streamers are thought to propagate by photo-ionization; the parameters of photo-ionization depend on the nitrogen : oxygen ratio. Therefore we study streamers in nitrogen with 20%, 0.2% and 0.01% oxygen and in pure nitrogen as well as in pure oxygen and argon. Our new experimental set-up guarantees contamination of the pure gases to be well below 1 ppm. Streamers in oxygen are difficult to measure as they emit considerably less light in the sensitivity range of our fast ICCD camera than the other gases. Streamers in pure nitrogen and in all nitrogen–oxygen mixtures look generally similar, but become somewhat thinner and branch more with decreasing oxygen content. In pure nitrogen the streamers can branch so much that they resemble feathers. This feature is even more pronounced in pure argon, with approximately 10 2 hair tips cm −3 in the feathers at 200 mbar; this density can be interpreted as the free electron density creating avalanches towards the streamer stem. It is remarkable that the streamer velocity is essentially the same for similar voltage and pressure in all nitrogen–oxygen mixtures as well as in pure nitrogen, while the oxygen concentration and therefore the photo-ionization lengths vary by more than five orders of magnitude. Streamers in argon have essentially the same velocity as well. The physical similarity of streamers at different pressures is confirmed in all gases; the minimal diameters are smaller than in earlier measurements. S Online supplementary data available from stacks.iop.org/JPhysD/43/145204/mmedia (Some figures in this article are in colour only in the electronic version)

Abstract: Positive streamers are thought to propagate by photo-ionization whose parameters depend on the nitrogen:oxygen ratio. Therefore we study streamers in nitrogen with 20%, 0.2% and 0.01% oxygen and in pure nitrogen, as well as in pure oxygen and argon. Our new experimental set-up guarantees contamination of the pure gases to be well below 1 ppm. Streamers in oxygen are difficult to measure as they emit considerably less light in the sensitivity range of our fast ICCD camera than the other gasses. Streamers in pure nitrogen and in all nitrogen/oxygen mixtures look generally similar, but become somewhat thinner and branch more with decreasing oxygen content. In pure nitrogen the streamers can branch so much that they resemble feathers. This feature is even more pronounced in pure argon, with approximately 10^2 hair tips/cm^3 in the feathers at 200 mbar; this density could be interpreted as the free electron density creating avalanches towards the streamer stem. It is remarkable that the streamer velocity is essentially the same for similar voltage and pressure in all nitrogen/oxygen mixtures as well as in pure nitrogen, while the oxygen concentration and therefore the photo-ionization lengths vary by more than five orders of magnitude. Streamers in argon have essentially the same velocity as well. The physical similarity of streamers at different pressures is confirmed in all gases; the minimal diameters are smaller than in earlier measurements.

TL;DR: The intrinsic stochastic particle noise triggers branching of positive streamers in air at atmospheric pressure, and it is concluded that the ratio of branching length to streamer diameter agrees within a factor of 2 with experimental measurements.

Abstract: Branching is an essential element of streamer discharge dynamics. We review the current state of theoretical understanding and recall that branching requires a finite perturbation. We argue that, in current laboratory experiments in ambient or artificial air, these perturbations can only be inherited from the initial state, or they can be due to intrinsic electron-density fluctuations owing to the discreteness of electrons. We incorporate these electron-density fluctuations into fully three-dimensional simulations of a positive streamer in air at standard temperature and pressure. We derive a quantitative estimate for the ratio of branching length to streamer diameter that agrees within a factor of 2 with experimental measurements. As branching without this noise would occur considerably later, if at all, we conclude that the intrinsic stochastic particle noise triggers branching of positive streamers in air at atmospheric pressure.

Abstract: Streamer discharges determine the very first stage of sparks or lightning, and they govern the evolution of huge sprite discharges above thunderclouds as well as the operation of corona reactors in plasma technology. Streamers are nonlinear structures with multiple inner scales. After briefly reviewing basic observations, experiments and the microphysics, we start from density models for streamers, i.e. from reaction–drift–diffusion equations for charged-particle densities coupled to the Poisson equation of electrostatics, and focus on derivation and solution of moving boundary approximations for the density models. We recall that so-called negative streamers are linearly stable against branching (and we conjecture this for positive streamers as well), and that streamer groups in two dimensions are well approximated by the classical Saffman–Taylor finger of two fluid flow. We draw conclusions on streamer physics, and we identify open problems in the moving boundary approximations.

55 citations

Cites background from "A moving boundary problem motivated..."

...Details can be found in [83] where the present figure appears as figure 1....

Abstract: Streamers are rapidly extending ionized fingers that can appear in gasses, liquids and solids. They are generated by high electric fields but can penetrate into areas where the background electric field is below the ionization threshold. Streamers occur in nature as a precursor to sparks and lightning, but also independently as sprites (large discharges high above thunderclouds) or St. Elmo’s fire. Their main applications are gas and water cleaning, ozone creation, particle charging and flow control. Streamers are very efficient in creating active chemical species as no energy is lost in heating of the background gas and surrounding materials. Furthermore, as streamers are the first phase of sparks, they are relevant for any application of sparks, e.g., in the ignition process in a combustion engine or a discharge lamp. Finally, streamers can occur in high voltage applications, like switch-gear. In this thesis, a number of aspects of the physics of streamers are investigated experimentally. In our study, we have created streamers by applying a high voltage pulse to a wire or sharp tip that is located 40 to 160 mm above a grounded plate. These experiments were conducted inside a vacuum chamber at various pressures between 25 and 1000 mbar, with various gasses and gas mixtures, most of high purity (up to less than 0.1 ppm contaminations). We create the voltage pulses by two different high voltage pulse sources. The C-supply can give pulses between 5 and 60 kV with a minimum risetime of about 15 ns and an exponential decay of varying duration. The newly built Blumlein pulser creates quasi-rectangular pulses with an amplitude between 20 and 35 kV, a duration of about 130 ns and a risetime of about 10 ns. Both pulse sources can produce pulses of positive and negative polarity but have primarily been used with positive polarity.First, the interaction of individual streamer channels and the streamer branching angles are analysed by stereo-photography. Then insight into the propagation mechanism of positive streamers (i.e., against the electron drift direction) is gained by changing the gas composition and the repetition frequency of voltage pulses. Finally, morphology, channel diameters, propagation velocities and spectra of laboratory streamer discharges in a variety of gasses and gas mixtures are studied. Some of these studies are used as a "simulation" of sprite discharges on earth as well as on other planets. Interaction and branching of streamers Pictures show that streamer or sprite discharge channels emerging from the same electrode sometimes seem to reconnect or merge even 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. We have used stereo-photography to investigate the full three-dimensional structure of such events. We analyse reconnection, possibly an electrostatic effect in which a late thin streamer reconnects to an earlier thick streamer channel, and merging, a suggested photo-ionization effect in which two simultaneously propagating streamer heads merge into one new streamer. We find that reconnections as defined above occur frequently. Merging on the other hand was only observed with a double tip electrode at a pressure of 25 mbar and a tip separation of 2 mm, i.e., for a reduced tip distance of p . d = 50 mmbar. In this case the full width at half maximum of the streamer channel is more than 10 times as large as the tip separation. We have also investigated streamer branching with the stereo-photography method and have found that the average branching angle of streamers under the conditions that were investigated is about 42° with a standard deviation of 12°. The role of photo- and background ionization in streamer propagation Positive streamers in air are thought to propagate against the electron drift direction by photo-ionization whose parameters depend on the nitrogen:oxygen ratio. Therefore we study streamers in nitrogen with 20%, 0.2% and 0.01% oxygen and in pure nitrogen and argon. Our new experimental set-up guarantees contamination to be below 0.1 ppm for our purest nitrogen. Streamers in pure nitrogen and in all nitrogen/oxygen mixtures look generally similar, but become thinner and branch more with decreasing oxygen content. In pure nitrogen the streamers can branch so much that they resemble feathers. This feature is even more pronounced in pure argon, with approximately 102 hair tips/cm3 in the feathers at 200 mbar; this density can be interpreted as the density of free electrons that create avalanches towards the streamer stem. It is remarkable that the streamer velocity is essentially the same for similar voltage and pressure in all nitrogen/oxygen mixtures as well as in pure nitrogen, while the oxygen concentration and therefore the photo-ionization lengths vary by more than five orders of magnitude. This is supported by recent modelling results byWormeester et al. in 2010. To study the effects of background ionization on streamers, we have used two methods: variation of pulse repetition frequency (0.01–10 Hz) and addition of about 9 parts per billion of radioactive 85Kr gas to pure nitrogen. We found that higher background ionization levels lead to smoother and thicker streamers. This is similar to the effect of increased photo-ionization close to the streamer tip, created by increasing the oxygen concentration. Again, we do not see any major effects on streamer properties, except that initiation probabilities go down significantly in pure nitrogen with low (0.01 Hz) repetition frequency. At 200 mbar, the estimated background ionization level from the 85Kr was about 4 ?? 105 cm-3, which corresponds to the theoretical level in non-radioactive gas at a pulse repetition frequency of about 1 Hz under similar conditions. This fits with the observed variations in streamer morphology as function of repetition frequency for both pure nitrogen and the nitrogen-krypton mixture. Furthermore, we have found that streamers do not follow the paths of streamers in preceding discharges for pulse repetition frequencies around 1 Hz. This can be explained by the combination of recombination and diffusion of ionization after a discharge pulse which nearly flattens any leftover ionization trail in about 1 second. Streamers in other gasses and streamer spectra In order to get more insight in positive streamer propagation, we have studied more than just nitrogen-oxygen mixtures. We have studied pure oxygen, argon, helium, hydrogen and carbon dioxide. Each of these gasses has different properties like ionization levels, excitation levels, cross sections and electronegativity. Furthermore, we have studied streamers in binary gas mixtures that simulate the atmospheres of Venus (CO2–N2) and Jupiter (H2–He). Streamers in these gasses, as well as in air are physically similar to large scale sprite discharges on the corresponding planets. Therefore, the results of our measurements can be used to better equip (space) missions that study sprites on earth and other planets and can help in the interpretation of the observations of these missions. For all gasses and mixtures, overall morphology, velocities, diameters and emission spectra have been investigated. We have found that it is possible to create streamers in all gasses. Streamer diameters are more or less the same for all gasses, except for pure helium and the Jupiter atmosphere where minimal streamers are respectively 3 and 5 times thicker than in the other gasses. The physical similarity between streamers at different pressures has been confirmed for all gasses that enabled us to measure streamer diameters; the minimal diameters in air and other nitrogen-oxygen mixtures are smaller than in earlier measurements. Streamer velocities are even more similar; for a given combination of pressure and pulse voltage all propagation velocities are within a factor 2. Streamer brightness on the other hand is very different for the different gas mixtures. Streamers are brightest in nitrogen-oxygen mixtures, nitrogen, argon and helium and dimmest in oxygen, CO2 and the venusian mixture. The difference between the brightest and dimmest gasses is about three to four orders of magnitude in the optical range. Streamer spectra from molecular gasses are characterised by molecular bands. In gasses containing a significant amount of nitrogen (including the venusian mixture), the nitrogen second positive system dominates the emission spectrum. In contrast, spark-like discharges in the same gasses are dominated by radiation from neutral and ionized atoms. Spectra in atomic gasses (argon and helium) are different: the argon spectrum contains mainly atomic argon lines, but the helium spectrum also contains many lines of impurities, while we have no indication that the gas purity is below specification. The reason for the many impurity lines in helium are the high excitation and ionization levels of helium compared to the impurities. These high levels (and low cross sections for electron-atom collisions at low energies) may also explain the large diameter of streamers in pure helium.

34 citations

Cites background from "A moving boundary problem motivated..."

...These concepts are further elaborated in [46, 86, 119, 137, 208] and were discussed in section 2....

TL;DR: The volume now gives a somewhat exhaustive account of the various ramifications of the subject, which are set out in an attractive manner and should become indispensable, not only as a textbook for advanced students, but as a work of reference to those whose aim is to extend the knowledge of analysis.

Abstract: This classic work has been a unique resource for thousands of mathematicians, scientists and engineers since its first appearance in 1902 Never out of print, its continuing value lies in its thorough and exhaustive treatment of special functions of mathematical physics and the analysis of differential equations from which they emerge The book also is of historical value as it was the first book in English to introduce the then modern methods of complex analysis This fifth edition preserves the style and content of the original, but it has been supplemented with more recent results and references where appropriate All the formulas have been checked and many corrections made A complete bibliographical search has been conducted to present the references in modern form for ease of use A new foreword by Professor SJ Patterson sketches the circumstances of the book's genesis and explains the reasons for its longevity A welcome addition to any mathematician's bookshelf, this will allow a whole new generation to experience the beauty contained in this text

8,955 citations

"A moving boundary problem motivated..." refers background in this paper

...When 1 − λ = n is a positive integer, from well-known theory [47] for regular singular points, instead of (31), the...

[...]

...where the general solution near ω = 1 can be written as [47]...

Abstract: 1. Introduction.- 1.1 What Is the Subject of Gas Discharge Physics.- 1.2 Typical Discharges in a Constant Electric Field.- 1.3 Classification of Discharges.- 1.4 Brief History of Electric Discharge Research.- 1.5 Organization of the Book. Bibliography.- 2. Drift, Energy and Diffusion of Charged Particles in Constant Fields.- 2.1 Drift of Electrons in a Weakly Ionized Gas.- 2.2 Conduction of Ionized Gas.- 2.3 Electron Energy.- 2.4 Diffusion of Electrons.- 2.5 Ions.- 2.6 Ambipolar Diffusion.- 2.7 Electric Current in Plasma in the Presence of Longitudinal Gradients of Charge Density.- 2.8 Hydrodynamic Description of Electrons.- 3. Interaction of Electrons in an Ionized Gas with Oscillating Electric Field and Electromagnetic Waves.- 3.1 The Motion of Electrons in Oscillating Fields.- 3.2 Electron Energy.- 3.3 Basic Equations of Electrodynamics of Continuous Media.- 3.4 High-Frequency Conductivity and Dielectric Permittivity of Plasma.- 3.5 Propagation of Electromagnetic, Waves in Plasmas.- 3.6 Total Reflection of Electromagnetic Waves from Plasma and Plasma Oscillations.- 4. Production and Decay of Charged Particles.- 4.1 Electron Impact Ionization in a Constant Field.- 4.2 Other Ionization Mechanisms.- 4.3 Bulk Recombination.- 4.4 Formation and Decay of Negative Ions.- 4.5 Diffusional Loss of Charges.- 4.6 Electron Emission from Solids.- 4.7 Multiplication of Charges in a Gas via Secondary Emission.- 5. Kinetic Equation for Electrons in a Weakly Ionized Gas Placed in an Electric Field.- 5.1 Description of Electron Processes in Terms of the Velocity Distribution Function.- 5.2 Formulation of the Kinetic Equation.- 5.3 Approximation for the Angular Dependence of the Distribution Function.- 5.4 Equation of the Electron Energy Spectrum.- 5.5 Validity Criteria for the Spectrum Equation.- 5.6 Comparison of Some Conclusions Implied by the Kinetic Equation with the Result of Elementary Theory.- 5.7 Stationary Spectrum of Electrons in a Field in the Case of only Elastic Losses.- 5.8 Numerical Results for Nitrogen and Air.- 5.9 Spatially Nonuniform Fields of Arbitrary Strength.- 6. Electric Probes.- 6.1 Introduction. Electric Circuit.- 6.2 Current-Voltage Characteristic of a Single Probe.- 6.3 Theoretical Foundations of Electronic Current Diagnostics of Rarefied Plasmas.- 6.4 Procedure for Measuring the Distribution Function.- 6.5 Ionic Current to a Probe in Rarefied Plasma.- 6.6 Vacuum Diode Current and Space-Charge Layer Close to a Charged Body.- 6.7 Double Probe.- 6.8 Probe in a High-Pressure Plasma.- 7. Breakdown of Gases in Fields of Various Frequency Ranges.- 7.1 Essential Characteristics of the Phenomenon.- 7.2 Breakdown and Triggering of Self-Sustained Discharge in a Constant Homogeneous Field at Moderately Large Product of Pressure and Discharge Gap Width.- 7.3 Breakdown in Microwave Fields and Interpretation of Experimental Data Using the Elementary Theory.- 7.4 Calculation of Ionization Frequencies and Breakdown Thresholds Using the Kinetic Equation.- 7.5 Optical Breakdown.- 7.6 Methods of Exciting an RF Field in a Discharge Volume.- 7.7 Breakdown in RF and Low-Frequency Ranges.- 8. Stable Glow Discharge.- 8.1 General Structure and Observable Features.- 8.2 Current-Voltage Characteristic of Discharge Between Electrodes.- 8.3 Dark Discharge and the Role Played by Space Charge in the Formation of the Cathode Layer.- 8.4 Cathode Layer.- 8.5 Transition Region Between the Cathode Layer and the Homogeneous Positive Column.- 8.6 Positive Column.- 8.7 Heating of the Gas and Its Effect on the Current-Voltage Characteristic.- 8.8 Electronegative Gas Plasma.- 8.9 Discharge in Fast Gas Flow.- 8.10 Anode Layer.- 9. Glow Discharge Instabilities and Their Consequences.- 9.1 Causes and Consequences of Instabilities.- 9.2 Quasisteady Parameters.- 9.3 Field and Electron Temperature Perturbations in the Case of Quasisteady-State Te.- 9.4 Thermal Instability.- 9.5 Attachment Instability.- 9.6 Some Other Frequently Encountered Destabilizing Mechanisms.- 9.7 Striations.- 9.8 Contraction of the Positive Column.- 10. Arc Discharge.- 10.1 Definition and Characteristic Features of Arc Discharge.- 10.2 Arc Types.- 10.3 Arc Initiation.- 10.4 Carbon Arc in Free Air.- 10.5 Hot Cathode Arc: Processes near the Cathode.- 10.6 Cathode Spots and Vacuum Arc.- 10.7 Anode Region.- 10.8 Low-Pressure Arc with Externally Heated Cathode.- 10.9 Positive Column of High-Pressure Arc (Experimental Data).- 10.10 Plasma Temperature and V - i Characteristic of High-Pressure Arc Columns.- 10.11 The Gap Between Electron and Gas Temperatures in "Equilibrium" Plasma.- 11. Suslainment and Production of Equilibrium Plasma by Fields in Various Frequency Ranges.- 11.1 Introduction. Energy Balance in Plasma.- 11.2 Arc Column in a Constant Field.- 11.3 Inductively Coupled Radio-Frequency Discharge.- 11.4 Discharge in Microwave Fields.- 11.5 Continuous Optical Discharges.- 11.6 Plasmatrons: Generators of Dense Low-Temperature Plasma.- 12. Spark and Corona Discharges.- 12.1 General Concepts.- 12.2 Individual Electron Avalanche.- 12.3 Concept of Streamers.- 12.4 Breakdown and Streamers in Electronegative Gases (Air) in Moderately Wide Gaps with a Uniform Field.- 12.5 Spark Channel.- 12.6 Corona Discharge.- 12.7 Models of Streamer Propagation.- 12.8 Breakdown in Long Air Gaps with Strongly Nonuniform Fields (Experimental Data).- 12.9 Leader Mechanism of Breakdown of Long Gaps.- 12.10 Return Wave (Return Stroke).- 12.11 Lightning.- 12.12 Negative Stepped Leader.- 13. Capacitively Coupled Radio-Frequency Discharge.- 13.1 Drift Oscillations of Electron Gas.- 13.2 Idealized Model of the Passage of High-Frequency Current Through a Long Plane Gap at Elevated Pressures.- 13.3 V - i Characteristic of Homogeneous Positive Columns.- 13.4 Two Forms of CCRF Discharge Realization and Constant Positive Potential of Space: Experiment.- 13.5 Electrical Processes in a Nonconducting Electrode Layer and the Mechanism of Closing the Circuit Current.- 13.6 Constant Positive Potential of the Weak-Current Discharge Plasma.- 13.7 High-Current Mode.- 13.8 The Structure of a Medium-Pressure Discharge: Results of Numerical Modeling.- 13.9 Normal Current Density in Weak-Current Mode and Limits on the Existence of this Mode.- 14. Discharges in High-Power CW CO2 Lasers.- 14.1 Principles of Operation of Electric-Discharge CO2 Lasers.- 14.2 Two Methods of Heat Removal from Lasers.- 14.3 Methods of Suppressing Instabilities.- 14.4 Organization of Large-Volume Discharges Involving Gas Pumping.- References.

Abstract: A classic reference, intended for graduate students mathematicians, physicists, and engineers, this book can be used both as the basis for instructional courses and as a reference tool.

3,934 citations

"A moving boundary problem motivated..." refers background in this paper

...(102) As is well known in asymptotics [48] the coefficients in the asymptotic relation (99) may jump when χ crosses a Stokes line emerging from a turning point....

Abstract: A variety of non-equilibrium growth processes are characterized by phase boundaries consisting of moving fingers, often with interesting secondary structures such as sidebranches. Familiar examples are dendrites, as seen in snowflake growth, and fluid fingers often formed in immiscible displacement. Such processes are characterized by a morphological instability which renders planar or circular shapes unstable, and by the competing stabilizing effect of surface tension. We survey recent theoretical work which elucidates how such systems arrive at their observed patterns. Emphasis is placed upon dendritic solidification, simple local models thereof, and the Saffman-Taylor finger in two-dimensional fluid flow, and relate these systems to their more complicated variants. We review the arguments that a general procedure for the analysis of such problems is to first find finger solutions of the governing equations without surface tension, then to incorporate surface tension in a non-perturbative manne...

612 citations

"A moving boundary problem motivated..." refers background or result in this paper

...Similar results follow for a needle crystal [18] though in the latter case, convective instability of wave packets caused by significant normal speed along the parabolic front is believed to cause dendritic structures [1]....

[...]

...Themotion of interfaces in a Laplacian field is of general interest and has been a subject of intense study over many years (see for instance the reviews [1,2])....

Abstract: This review is an expository treatment of the displacement of one fluid by another in a two-dimensional geometry (a Hele-Shaw cell). The Saffman-Taylor equations modeling this system are discussed. They are simulated by random-walk techniques and studied by methods from complex analysis. The stability of the generated patterns (fingers) is studied by a WKB approximation and by complex analytic techniques. The primary conclusions reached are that (a) the fingers are linearly stable even at the highest velocities, (b) they are nonlinearly unstable against noise or an external perturbation, the critical amplitude for the noise being an exponential function of a power of the velocity for high velocities, (c) such exponentials seem to dominate high-velocity behavior, as can be seen from a WKB analysis, and (d) the results of the Saffman-Taylor equations disagree with experiments, apparently because they leave out film-flow phenomena.