Journal Of Physics D: Applied Physics J. Phys. D: Appl .
Journal of Physics D: Applied PhysicsJ. Phys. D: Appl. Phys. 47 (2014) 435202 (9pp)doi:10.1088/0022-3727/47/43/435202On the propagation of streamers inelectrical dischargesJianqi Qin and Victor P PaskoCommunications and Space Sciences Laboratory, Department of Electrical Engineering, Penn StateUniversity, University Park, Pennsylvania, USAE-mail: jianqiqin@psu.eduReceived 18 June 2014, revised 6 August 2014Accepted for publication 4 September 2014Published 3 October 2014AbstractStreamers are non-thermal filamentary plasmas developing in insulating mediums under theinfluence of strong external electric fields. The present knowledge, based on a widely-acceptedconcept of critical or stability field introduced half a century ago, indicates the existence of aunique minimum electric field in which streamers could propagate stably with constant radiusand velocity. In this work, we present a new understanding indicating that growing, decaying,and stable propagation of streamers is controlled not solely by the external field but also bythe physical dimensions of streamers. Stable propagation is demonstrated to be achievablein a wide range of electric fields, with a lower limit of 5 kV cm 1 for positive streamersand 10 kV cm 1 for negative streamers in air at atmospheric pressure, up to the breakdownfield 28.7 kV cm 1. In these field ranges, the streamer radius required for stable propagationis inversely proportional to the external field, with larger and smaller initial radii, respectively,leading to growing and decaying streamers. The new mechanism suggests possible waysto flexibly control the streamer parameters in applications, such as changing the size andpotential of the point electrode to obtain the required initial streamer dimensions for thedesired propagation pattern.Keywords: streamers, ionization, propagation, electrical discharges(Some figures may appear in colour only in the online journal)1. Introductionby satellites [9]. Those thermal runaway electrons with energies as high as 100 keV can be further accelerated up to several MeVs energies by the electric field of lightning steppedleaders during the stage of negative corona flash, leading to theproduction of TGFs through bremsstrahlung radiation [9]. Asfor applications, streamers have long been used for industrialozone production [5], plasma-assisted combustion [10], pollution control [11], and recently have shown a great medicalpotential [12, 13], for example, in wound healing and treatment of skin diseases [14, 15]. Understanding of streamerpropagation mechanism is of essential importance for thestudies of electrical breakdown phenomena and their relatedapplications. It has been generally believed since the 1970s thatstreamers are propagating in such a way that their dynamicssolely depends on the applied electric field. The external fieldthat is equal to the critical field leads to stable propagation ofstreamers with constant radius and velocity. The fields withThe streamer theory was originally developed by Raether[1], Loeb and Meek [2] in the 1930s as an alternative to theTownsend theory to explain the experimental observationsof spark discharges. Those pioneer works were followed bydecades of extensive research on streamer physics and applications [3, 4]. It is now well-known that streamers are fundamental components in many types of gas discharges, suchas the dielectric-barrier discharges between two electrodes [5],the lightning discharges in thunderstorms [6], and the transientluminous events in the upper atmosphere [7]. Streamers arealso essential for the electrical breakdown phenomena in liquiddielectrics (e.g. transformer oil) and solid insulators [4, 8].Moreover, it has been recently suggested that the thermalrunaway electrons produced by streamers could be responsible for the terrestrial gamma ray flashes (TGFs) observed0022-3727/14/435202 9 33.001 2014 IOP Publishing LtdPrinted in the UK
J Qin and V P Pasko J. Phys. D: Appl. Phys. 47 (2014) 435202propagation of streamers, an external field E0 must be appliedto supply energy for such processes as ionization, excitation,and dissociative attachment. It is found experimentally thatthe critical field Ecr required for the propagation of positivestreamers is close to the value 4.4 kV cm 1 in air at atmospheric pressure [20], and the Ecr for negative streamers isestimated to be a factor of 2–3 higher, about 8–12.5 kV cm 1[18, 21]. It was also believed that these were the electricfields required for stable propagation of streamers, and witha larger or a smaller value, respectively, leading to growing ordecaying streamers.To achieve a better understanding of the streamer dynamics,it should be useful to derive an analytical criterion for streamerpropagation. However, since streamer discharges are highlynonlinear processes, it is difficult to derive a simple criterionby solving analytically the continuity equations of chargedspecies coupled with Poisson’s equation. Hence, we simplifya streamer as a thin charge layer and derive a simple criterionfor streamer propagation using the law of energy conservationon a macroscopic level following approach described on page355 by Raizer [18]. The conclusions derived from the simplecriterion will be demonstrated using plasma fluid modeling inlater sections.As shown in a cross-sectional view in figure 1(b), a streamerhead and its extension in a short time Δt are represented astwo thin cylindrical charge layers with two slightly differentradii Rs and R′s. In this approach, growing, stable and decayingstreamers, respectively, correspond to R′s Rs, R′s Rs, andR′s Rs. As a new section of the streamer head is created inthe time period of Δt, part of the previous streamer head isneutralized by the secondary electrons drifted backward fromthe new section of the positive streamer (or by the electronsdrifted forward from the streamer channel in the case of negative streamers), effectively leading to the propagation of thethin charge layer, namely the streamer head, for a short distance ΔL. The work done by the external field E0 during thisprocess can be written aslarger magnitudes lead to exponentially growing streamers,and those with smaller ones lead to decaying streamers thatcould propagate only for a short distance [16, 17]. In the framework of that theory, some fundamental concepts in streamerphysics such as the critical fields and minimal streamer radiuscould not be explained, and flexible control of the streamerparameters in applications seemed difficult. In this paper, wepresent a new mechanism by deriving analytical criteria fordecaying, stable and growing streamers, and by simulatingstreamer propagation in a wide range of electric fields in air atatmospheric pressure using a plasma fluid model. The resultsdemonstrate that streamer dynamics, more specifically, thegrowing, decaying and stable propagation of streamers are controlled not solely by the external field but also by the physicaldimensions of streamers.2. Analytical criteria for streamer propagationConsider a few free electrons moving in a gas (or a liquiddielectric) immersed in an external electric field E0. Theelectrons are first accelerated by the electric force and thencollide with neutral molecules, losing a part of their kineticenergy during the collision, after which the electron acceleration resumes. This process repeats such that on a macroscopicscale an ensemble of electrons propagates with an averagevelocity, referred to as the electron drift velocity. If theexternal electric field is stronger than the conventional breakdown field Ek (defined by the equality of ionization and dissociative attachment frequencies [18], Ek 28.7 kV cm 1 inair at atmospheric pressure [19]), electrons will gain enoughenergy between collisions to ionize the neutral molecules andcreate secondary electrons, which drift together with the primary ones. As this electron cloud moves forward, the numberof secondary electrons increases exponentially, which isknown as the electron avalanche phenomenon. Once the totalnumber of electrons in the avalanche is so large that theirspace charge field becomes comparable to E0, the avalancheto-streamer transition occurs [18].Streamers are narrow filamentary plasmas driven by highlynonlinear space charge waves [18]. Unlike an electron avalanche that propagates in a drift manner and has negligiblespace charge effect, the dynamics of a streamer is mainly controlled by a highly-enhanced field region in the tip of the filament, known as the streamer head. This head region, depictedas a crescent shape in figure 1(a), contains a large amount ofnet positive or negative space charge, that respectively corresponds to positive and negative streamers. The space chargestrongly enhances the electric field to values about 3–7Ek inthe region just ahead of the streamer, while screening theambient field out of the streamer channel. The intense electron impact ionization in the high field region rapidly raisesthe electron density from an ambient value to the level in thestreamer channel, leading to the extension of the streamer intoa new region. Therefore, streamers are often referred to asspace charge waves, which can penetrate into neutral gas witha velocity much higher than the electron drift velocity, up toa fraction of the speed of light. It is apparent that during the ΔW qeN0 E0 ΔL(1)where qe is the absolute value of electron charge and qeN0is the total amount of charge in the streamer head. Theenergy ΔW is mainly spent on creating the new section ofthe streamer, more specifically, on the chemical processessuch as ionization, attachment, vibrational and electronicexcitation of the neutral molecules in the streamer head.Having assumed that the total average energy expended onthe above-mentioned processes in the discharge during creation of one electron-ion pair is we, the law of energy conservation requires that ΔW Newe(2)where Ne is the total number of secondary electrons created inthe new section of the streamer, so that qeN0 E0 ΔL Newe(3)The electron density in the new section can be approximated as 2n e N0 / (πRs2L s )(4)
J Qin and V P Pasko J. Phys. D: Appl. Phys. 47 (2014) 435202Electron driftLsStreamer extensionin a short time tzNeutralized2RsSimplified as2RsIonization degree: 0.001%LsLs LSpace charge field2Rs’(b)External field2Rs(a)Ls LFigure 1. Simplified model of streamer propagation. (a) Cross-sectional view of a positive streamer propagating in an external field. (b)Schematics of a crescent-shaped streamer head and representation of its extension in a short time interval using two cylinders.and 400Ne n eπR′s 2ΔL(5)300 q E0L sR′s 2w e s2Rswewewe (eV)Hence, we have(6)q E0RsR′s 2 e 2RsC0 weE0Rst we / qe[Morrow and Lowke, 1997][Babaeva et al., 2004]02345Es/Ek6789Figure 2. Effective energy required for producing a secondaryelectron in air at atmospheric pressure. The results shown bythe solid and dashed lines are, respectively, calculated using theionization rate and electron mobility as functions of the reducedelectric field provided in the references [19, 23, 24].Note that this stable condition is identical to that presented onpage 355 by Raizer [18] (with a different interpretation in thepresent work). For negative streamers E0Rst C0 (E0 ) we / qe(9)where Rst represents the stability radius required for the stablepropagation of streamers. Moreover, it can be estimated that we (Es ) q μ Es2Js ⃗ · Es⃗ e eνin eνi(10)where Js ,⃗ Es,⃗ νi and μe are, respectively, the current density,the electric field, the ionization frequency, and the electronmobility in the streamer head. Note that νi and μe are functions of the reduced electric field E/N, where N is the neutraldensity. Figure 2 shows the values of we(E) calculated in air atatmospheric pressure using three different sets of νi(E/N) andμe(E/N) commonly adopted in previous streamer modeling[19, 23, 24]. Although the results are slightly different, they allshow almost constant we for E 4Ek, which are the field valuesusually observed in positive streamer heads. For lower fieldsfrom 4Ek to 2Ek that commonly exist in negative streamerheads, the we value varies up to a factor of 4. This indicatesthat the stability radius Rst required for stable propagation ofpositive streamers is almost linearly proportional to 1/E0, butthe relation between Rst and E0 in negative streamers deviatesfrom such a linear fashion due to the significant variation ofwe. For a streamer propagating in an external field E0 with a(7)where C0 1.0 for positive streamers and C0 is a function ofthe external field for negative streamers in air at atmosphericpressure, as will be demonstrated below. Note that in thepresent work the streamer radius Rs is defined as the radialdistance at which the electron density decreases to half of itsvalue on the axis of symmetry of the streamers. The requirement for stable propagation of positive streamers can be therefore approximated as 200100where ws qeE0Ls is the amount of energy gained by anelectron propagating over a distance of Ls along an electric field of E0. The condition ws we, ws we, and ws werespectively, represents the requirement for the formation ofa growing, a stable, and a decaying streamer. If the energyws can be used to produce exactly one secondary electron,the streamer will create a new streamer head identical to itsprevious one at the next moment of time. Note that identical streamer heads at different moments of time lead toconstant potential drop (i.e. constant electric field) in thestreamer head, which is the most important characteristic ofstable streamers as discussed in previous literature [9, 21].Otherwise with ws we, the extra secondary electrons leadto a radial expansion of the streamer and a slight increase ofthe electron density in the streamer head. On the contrary,for ws we, the streamer has to reduce its radius and theelectron density in the streamer head decreases slightly aswell. The above-derived relation clearly indicates that thestreamer propagation is not only controlled by the strengthof the external field E0, but is also highly dependent on Ls,namely the size of the streamer head.Since the thickness Ls of the streamer head is proportionalto the streamer radius Rs [22], the equation (6) can be rewrittenas [Liu and Pasko, 2004](8)3
J Qin and V P Pasko J. Phys. D: Appl. Phys. 47 (2014) 4352020.50 cm20.51019.510ne (m-3)(c)2.0 cm0. 25 cmE0 10 kV/cmSimulation domainE0Plasma cloudRsph0.625 mmE0 20 kV/cm(b)(a)18.510zrUsph17.510Negative streamersPositive streamersFigure 3. (a) Geometry of the simulation domain. (b) Propagation of growing and decaying positive streamers in an external fieldof 10 kV cm 1. Both positive streamers are initiated from a Gaussian distributed plasma cloud with a peak density of 1020 m 3 and acharacteristic size σ0 of 0.05 mm. The radius of the spherical electrode Rsph is 0.5 mm. The only difference is that in the left panel thespherical electrode has a potential Usph 3.5 kV, whereas in the right panel Usph 3.2 kV. (c) Propagation of negative streamers in anexternal field of 20 kV cm 1. For both negative streamers, the initial plasma cloud has a peak density of 1018 m 3 and a characteristic size of0.10 mm. The electrode radius Rsph 1.0 mm, and in the left panel Usph 4.0 kV, whereas in the right panel Usph 3.4 kV.where ρ [(z Rsph)2 r2]1/2. Note that for z Rsph, UL(r, z) E0z. For computational efficiency, a plasma cloud formed byequal amounts of electrons and positive ions with a Gaussian spatial distribution, i.e. ne np n0exp{ (r/σ0)2 [(z 2σ0)/σ0]2},represented by a black sphere in figure 3(a), was placed near thespherical electrode on the axis of symmetry of the simulationdomain to rapidly initiate a streamer.It was found in those simulations that for a given uniformelectric field E0, lowering the potential Usph of the sphere(that results in a smaller initial streamer radius) leads toslower growth of the streamer in the region of uniform electric field, and with a critically low Usph, the streamer decayseven if it propagates in a uniform electric field that is onlyslightly weaker than Ek. Four representative examples ofsimulations under those conditions are shown in figure 3. Forthe two positive streamers, the only difference in their initiation conditions is that the potential of the charged sphere islowered from 3.5 kV to 3.2 kV, leading to the formation of agrowing streamer with a large radius and the formation of adecaying streamer with a smaller radius in the same uniformfield E0 10 kV cm 1. Similarly, as shown in figure 3(c),lowering the sphere potential from 4.0 kV to 3.4 kV leadsto two negative streamers that are respectively growingand decaying in the external field E0 20 kV cm 1. Thesemodeling results indicate that there should exist a stabilityUsph that initiates a streamer with the stability radius Rst.By intentionally initiating streamers with small initial radiiusing small spherical electrodes, the above finding is confirmed in a wide range of electric fields, with the modelingresults shown in figure 4.In our simulations, stable propagation of streamers is con firmed by constant value of the integral 0 2πrn e (r ) dr in thestreamer head at different moments of time. This criterionis more accurate when compared to the criteria of constantradius or constant velocity, becauseradius Rs Rst, which depends on its initiation conditions, thestreamer will experience an exponential growth if Rs Rst ordecay if Rs Rst. We note that in the above analytical derivation of the simple criterion, we simplified a streamer as purelya thin charge layer and neglected the variations in the streamerchannel, which affect slightly the dynamics of the streamerhead [25]. The effectiveness of the criterion, therefore, needsto be demonstrated using numerical simulations that take intoaccount the contributions of the streamer channel.3. Streamer propagation in numerical simulationsStreamer propagation is simulated using a plasma fluid modelaccounting for the electron impact ionization of N2 and O2,the electron dissociative attachment to O2, and the electrondetachment process O N2 e N2O. Photoionization processes are included using the three-group SP3 model [26]. Theelectron mobility, electron diffusion coefficient, the ionization frequency, and the two-body and three-body attachmentfrequencies are defined as functions of the reduced electricfield E/N using modified formulations of Morrow and Lowke[19]. The motion of charged species is simulated by solvingthe drift-diffusion equations for electrons and ions coupledwith the Poisson’s equation, and open boundary conditionsare used in all simulations [27, 28].We use a technique introduced by Babaeva and Naidis [21]to model point electrode configurations of previous experiments [20]. A small spherical electrode of radius Rsph andpotential Usph was placed in the uniform electric field E0, asshown in figure 3(a). The sphere enhances the electric field inthe region near the sphere but leaves the electric field far awayfrom the sphere almost unaffected. In a cylindrical system ofcoordinates, the potential of Laplacian field (in the absence ofspace charge) at a given location (r, z) is UL (r , z ) Usph3 Rsph E0 1 3 (z Rsph )ρρ Rsph(11) 4 0 2πrn e (r ) dr n eRs2(12)
J Qin and V P Pasko J. Phys. D: Appl. Phys. 47 (2014) 435202(a)Positive streamers 1.25 mm20.5(b)Negative streamers 1.00 cm10E0E00. 5 cmne (m-3)104.0 cm19.518.51017.510Figure 4. (a) Stable positive streamers in external fields of 5, 10, and 20 kV cm 1. The plasma cloud is identical to that used in figure 3(b).The radius and potential of the spherical electrodes are, respectively, 1.0 mm and 6.1 kV, 0.5 mm and 3.40 kV, 0.5 mm and 1.302 kV for thecases from left to right. Note that according to simulations, a smaller streamer needs a more accurate sphere potential to propagate stablyfor a given distance (e.g. 0.5 cm). (b) Stable negative streamers in external fields of 10, 20, and 25 kV cm 1. The plasma cloud is identicalto that used in figure 3(c). The radius and potential of the spherical electrodes are, respectively, 2.0 mm and 25.0 kV, 1.0 mm and 3.75 kV,1.0 mm and 0.23 kV for the cases from left to right.stable negative streamer shown in figures 5(c) and (d). Inaddition, it is known that the electric potential differences inthe heads of growing streamers increase exponentially [9]. Inour simulations, we observed that for decaying and growingstreamers, the potential drop in the streamer heads is, respectively, decreasing and increasing, and the total potential dropin the streamer channels is, respectively, larger and smallerthan E0L.Note that in most simulations except those related to verythin streamers, the actual simulation domains are larger thanzoom in views shown in figures 3 and 4. The spatial resolutionfor positive streamers is 0.5 cm/2000 2.5 µm and for negative streamers is 4 cm/3000 13.3 µm. It is also important toemphasize that in our simulations the transport coefficientsand rate constants of kinetic processes are taken from the workof Morrow and Lowke [19]. Additional tests not included herefor the sake of brevity demonstrate that using coefficients andrate constants taken from different sources in refereed literature, the potential of the spherical electrodes need to beadjusted slightly to produce stable streamers. For example,with the coefficients taken from the work of Babaeva et al[24] or Liu and Pasko [23], the potential applied to the sphereelectrode needs to be, respectively, 2.6 kV and 4.0 kV toproduce stable streamers in an external field of 10 kV cm 1using otherwise identical conditions to those of figure 4(a)(middle panel). Other factors such as the high-order schemeused to calculate the electron flux and the photoionizationmodel might also slightly affect exact conditions for stablestreamers but do not change fundamentally new conclusionsof the present work.where ne on the right hand side represents the average density in the streamer head, such that a slight variation of Rsor ne can lead to a significant variation of the integral. It isalso confirmed in our simulations that all stable streamers,regardless of the external field, propagate with a constantvelocity and have a constant potential drop in the streamerheads, which is consistent with the stable streamer propagation in critical fields Ecr observed in previous literature [21].Constant potential drop in stable streamer heads is due toidentical streamer heads at different moments of time, which,as indicated in our analytical derivation, can be achievedif qeE0Ls we. Moreover, constant potential drop in thestreamer head leads to the fact that the total potential drop inthe stable streamer channel should be approximately equalto E0L, where L is the length of the streamer channel, whichhas also been confirmed in our modeling study with some ofthe results shown in figure 5. Note that stable propagationof streamers are better demonstrated in the case of negativestreamers, which show perfectly identical streamer heads atdifferent moments of time and that the electric field in thestreamer channel is equal to the applied field (see figures 5(c)and (d)). The positive streamer shown in figures 5(a) and(b), which is stable according to the criterion of a constantvalue of the integral 0 2πrn e dr , still exhibits slight variation in the streamer head due to a relatively small simulationdomain used in the positive case as a very high spatial resolution is required for the modeling of thin positive streamers.Nevertheless, we expect that once propagating over a longerdistance the positive streamer shown in figures 5(a) and (b)could propagate in a stable fashion similar to that of the5
J Qin and V P Pasko J. Phys. D: Appl. Phys. 47 (2014) 435202Positive Streamer22105 ns20Negative Streamer2010(a)1015 ns(c)15 ns25 ns35 ns55 ns18ne (m-3)101810161016101414101018E0 10 kV/cmx 106(b)5 ns15 ns14E (V/m)10825 nsE0 25 kV/cmx 106(d)15 ns55 ns35 ns610462200.10.20.3z (cm)0.40.5000.51.0z (cm)1.52.0Figure 5. (a, b) Electron density and electric field on the axis of symmetry of the stable positive streamer shown in the middle panel offigure 4(a). (c, d) Electron density and electric field on the axis of symmetry of the stable negative streamer shown in the right panel offigure 4(b).4. The lower and upper limits of the stability fieldsis related to the process of photoionization produced by thephotons emitted from the streamer head [23, 29]. This is particularly true for positive streamers which propagate in theopposite direction of the electron drift and require ambientseed electrons ahead of them for their spatial advancement[30]. Photoionization is the main process that supplies theseseed electrons [29, 31]. As for negative streamers, they aregenerally more difficult to branch because the electrons in thestreamer channel drift in the same direction as the streamerpropagation and, together with the photoelectrons, serve asthe seed electrons ahead of the negative streamers [23, 32].It should be emphasized that in those previous studies ofstreamer branching, the external fields are assumed to bestronger than the breakdown field Ek. How streamers branchin weak external fields is poorly understood. We find thatstreamers are more easily to reach numerical branching inweaker external fields, most likely because of the relativelyweak space charge fields in the streamer head (see figure 4)that lead to less effective photoionization. However, sincethe plasma fluid model can not fully reproduce the stochasticstreamer branching phenomenon as it is a deterministic model,we could only hypothesize that the maximum streamer radiusRmax should decrease with decreasing external field E0, in contrast to Rst which increases with decreasing E0. The criticalfield Ecr is determined by the relation Rmax(E0) Rst(E0). If thehypothesis is correct, it is expected that in the external fieldEcr, streamers could only propagate in a stable manner withthe radius Rst, because a smaller radius leads to a decayingstreamer, and a larger radius leads to streamer branching. ItIt is necessary to clarify the concepts of critical field Ecr, stability field Est, and stability radius Rst. Identical to its previousdefinition, the critical field Ecr is defined as the minimum electric field required for the propagation of streamers in a givenmedium. However, we emphasize that the stability field Estis not identical to Ecr, but represents a wide range of electricfields, with its lower limit equal to the critical field Ecr andits upper limit confined by the breakdown field Ek. Each stability field Est is related to an (almost) unique streamer radius,referred to as the stability radius Rst, with which streamers canpropagate stably.The lower limits of the stability fields, namely the critical fields Ecr 4.4 kV cm 1 for positive streamers andEcr 8.0 12.5 kV cm 1 for negative streamers in air atatmospheric pressure, have been measured and simulatedextensively in the literature [16, 17, 20, 21]. However, physical explanations for the existence of such specific values werenot given: why streamers cannot propagate in a sub-criticalfield? The answer naturally follows from the present analysis,as it shows that the propagation of streamers in a weaker fieldrequires a larger radius. The streamer branching puts a limiton how large streamer radii could be realized. The critical fieldEcr is determined by the maximum radius Rmax that a streamercould possess in weak external fields. Evaluation of the Rmaxrequires knowledge of the streamer branching mechanismwhich is not well understood yet. Nevertheless, it has beengenerally accepted that the streamer branching phenomenon6
J Qin and V P Pasko J. Phys. D: Appl. Phys. 47 (2014) 435202is also expected that in the case of lower gas pressure (or athigher altitudes with lower air density N in the Earth’s atmosphere), the critical field Ecr required for streamer propagationis lower than Ecr0N/N0, where Ecr0 and N0, respectively, represent the critical field and air density at atmospheric pressure. This is because photoionization is more efficient at lowerpressure due to weaker quenching of the excited states responsible for the photoionizing radiation [23], such that streamerscan have larger radii than those expected from a linearscaling of N0/N.The upper limit of the stability fields is naturally confinedby the conventional breakdown field Ek 28.7 kV cm 1, as ina stronger external field a single seed electron can lead to anexponentially growing electron avalanche and then transforminto a growing streamer. This has been confirmed by numericalsimulations of negative streamers in which stable propagationcan occur in external fields up to 28 kV cm 1. It should benoted that in such a strong external field, the peak electric fieldEs in a (quasi) stable streamer head is as weak as 1.2Ek, withonly 0.2Ek contributed from the space charge in the streamerhead, and the streamer velocity is 1.50 105 m s 1, whichis almost equal to the electron drift velocity 1.45 105 m/sin an electric field of 1.2Ek. In other words, in the upper limitof the stability field, stable negative streamers are in fact electron avalanches. For positive streamers, stable propagationhas been verified by practical calculations in external fieldsup to 20 kV cm 1. In stronger external fields, streamers withsmall radii can easily develop into a stage when the peak electric field Es in the streamer head is higher than 9Ek, that leadsto extremely slow computational advancement.negative streamers, which are essentially electron avalanchesin strong stability fields.6. Differences between positive and negativestreamersThe intrinsic difference in the radii of positive and negativestreamers observed in previous simulations and experimentscan be con
J Qin and V P PaskoJ. Phys. D: Appl. Phys. 3 and N ee nR s 2 L (5) Hence, we have R R qE L w w w s 2 s 2 e 0s e s e (6) where w s q eE 0L s is the amount of energy gained by an electron propagating over a distance o
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