The dynamics of an anode-directed streamer can be described by advection-diffusion equations for the charged particles, including a local field-dependent impact ionization term, and coupled to the Poisson equation for the electric field. We present the results of new simulations that use a local uniform grid refinement strategy. Even on very fine grids, provided the electric field is high enough, the streamer appears to branch spontaneously. These results are supported by new analytical solutions based on a moving boundary approximatio

Ebert, U. (Ute)

Hundsdorfer, W. (Willem)

Meulenbroek, B.J. (Bernard)

Montijn, C. (Carolynne-Sireeh)

English

CWI's Institutional Repository

C e n t r u m  v o o r  W i s k u n d e  e n  I n f o r m a t i c aMASModelling, Analysis and Simulation Modelling, Analysis and SimulationNumerical simulations and conformal analysis of growing and branching negative discharge streamersC. Montijn, B.J. Meulenbroek, U.M. Ebert,W.H. HundsdorferREPORT MAS-E0409 AUGUST 2004CWI is the National Research Institute for Mathematics and Computer Science. It is sponsored by the Netherlands Organization for Scientific Research (NWO).CWI is a founding member of ERCIM, the European Research Consortium for Informatics and Mathematics.CWI's research has a theme-oriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms.Probability, Networks and Algorithms (PNA)Software Engineering (SEN)Modelling, Analysis and Simulation (MAS)Information Systems (INS)Copyright © 2004, Stichting Centrum voor Wiskunde en InformaticaP.O. Box 94079, 1090 GB Amsterdam (NL)Kruislaan 413, 1098 SJ Amsterdam (NL)Telephone +31 20 592 9333Telefax +31 20 592 4199ISSN 1386-3703Numerical simulations and conformal analysis ofgrowing and branching negative discharge streamersABSTRACTThe dynamics of an anode-directed streamer can be described by advection-diffusion equationsfor the charged particles, including a local field-dependent impact ionization term, and coupledto the Poisson equation for the electric field. We present the results of new simulations that usea local uniform grid refinement strategy. Even on very fine grids, provided the electric field ishigh enough, the streamer appears to branch spontaneously. These results are supported bynew analytical solutions based on a moving boundary approximation.2000 Mathematics Subject Classification:  65M12 ; 65M20 ; 65M50Keywords and Phrases: Negative streamers; minimal streamer model; local grid refinement; moving boundaryapproximationNote: Financial support for C. Montijn is provided by NWO in the Computational Science program. This research wasfurther supported by the Dutch government through the national program BSIK: knowledge and research capacity, in theICT project BRICKS, theme MSV1.1Numerical simulations and conformal analysis ofgrowing and branching negative discharge streamersCarolynne Montijn, Bernard Meulenbroek, Ute Ebert and Willem HundsdorferAbstract— The dynamics of an anode-directed streamer canbe described by advection-diffusion equations for the chargedparticles, including a local field-dependent impact ionizationterm, and coupled to the Poisson equation for the electric field.We present the results of new simulations that use a local uniformgrid refinement strategy. Even on very fine grids, provided theelectric field is high enough, the streamer appears to branchspontaneously. These results are supported by new analyticalsolutions based on a moving boundary approximation.Index Terms— Negative streamers, minimal streamer model,local grid refinement, moving boundary approximationDischarge streamers can emerge when a strong potential isapplied to a sufficiently large sample of nonconducting matterlike a gas. They consist of channels of nonequilibrium plasmathat is generated at the channel tip. The rapid propagation ofthe tip is characterized by local self-induced enhancement ofthe electric field.For anode directed streamers, essential properties of theprocess can be analyzed within a minimal model whichcontains only two charged species (electrons and positiveions), an impact ionization reaction and the Poisson equationof electrostatics. The model can be applied to non-attachinggases like nitrogen or argon. In dimensionless units, it is [1]:∂tσ = ∇ · (σE) + D∇2σ + σ|E|e−1/|E| , (1)∂tρ = σ|E|e−1/|E| , (2)∇ · E = −∇2φ = ρ− σ , (3)where σ and ρ are the electron or positive ion density,respectively, D is the diffusion coefficient of the electrons,and E and φ the electric field and potential, respectively. Thepositive ions can be assumed to be immobile on the shorttime scales considered here because their mobility is at leasttwo orders of magnitude smaller than that of the electrons [2].The dimensionless quantities have been defined by scaling thecorresponding dimensional quantities with the most naturalscales for the length l0, time t0, electric field E0 and chargeq0. For N2 under normal conditions, they are:l0 = 2.3 µm , t0 = 3 · 10−12 s ,E0 = 200 kV/cm , q0 = 4.7 · 1014 e/cm3 ,(4)where e is the elementary charge.Up to now, all simulations [2]–[6] performed on this modelhave been carried out on uniform grids, and some of them[4]–[6] show that, provided the background electric field isManuscript received July 1, 2004. This work was supported by the Dutchfunding agency NWO and by CWI Amsterdam. The authors are with theCenter for Mathematics and Computer Science (CWI), P.O.Box 94079,1090GB Amsterdam, The Netherlands. U.E. has also a part time appointmentwith the physics faculty of Eindhoven Univ. Techn., The Netherlands.high enough, the streamer tends to grow into an unstable state,leading to spontaneous branching. The simulations show verysteep ionization fronts around the propagating channel, nextto wide inert space where only the Poisson equation in theabsence of space charges has to be solved.We therefore have implemented a simulation code with localuniform grid refinement in order to be able to use finer grids[7], to investigate the nature of the instablities in detail andto be able to deal with larger system sizes. The computationaldomain is three dimensional with radial symmetry, where rand z are the radial and axial coordinates, respectively. Thecathode is planar, situated at z = 0, and the distance betweenthe electrodes is L = 2000, corresponding to 4.6µm. The finestgrid size is 1/2. The initial condition is a Gaussian ionizationseed at the cathode,σ(r, z, t = 0) = ρ(r, z, t = 0) = σ0 e−(r2+z2)/R20 (5)where σ0 corresponds to a maximum density of 1014cm−3,and R0 is the 1/e radius, corresponding to 25µm. Thebackground electric field is set to Ebg = −0.4zˆ, zˆ beingthe unit vector in the axial direction, which corresponds to80kV/cm. Figure 1 shows the temporal evolution of the initialionization seed under these conditions. We used a Neumannboundary condition for the electrons at the cathode, whichcorresponds to a net flux of electrons into the computationaldomain. Therefore the particle densities at the cathode willalways grow, and in the figures the densities at the cathodehave been cut off in such a way that the front of the channelis well reproduced. The plot of the total charge density att = 100 show already space charge effects. A charged layerappears, strongly affecting the electric field, which becomesrelatively low in the body of the streamer and very high at theoutside of the streamer front. The ratio of layer thickness overchannel radius decreases and approaches an interfacial limit,after which the streamer becomes unstable and branches, ascan be seen at t = 500.The physical nature of the instability is illustrated in Fig. 2.Here a simplification of the streamer problem is consideredas first suggested by Lozansky and Firsov: the interior of thestreamer is assumed to be equipotential, the ionization frontto be infinitely thin and the interface velocity to be the localelectron drift velocity. New analytical solutions of this problemcan be found in [8]. We here illustrate that this model indeedcan show the same evolution as the numerical solutions ofEqs. (1)–(3): the propagating tip becomes flatter and eventuallybranches.2Electron density Ion density Total charge densityand equipotential lines Electric field strengtht = 100−40 −20 0 20 4020406080100120rz00.050.10.150.20.250.3−40 −20 0 20 4020406080100120rz00.050.10.150.20.250.3−40 −20 0 20 4020406080100120rz00.0050.010.0150.020.0250.030.0350.040.0450.05−40 −20 0 20 4020406080100120rz0.30.350.40.450.50.550.60.65t = 300t = 500Fig. 1. Temporal evolution of the electron density σ (first column), positive ion density ρ (second column), total charge density σ−ρ and equipotential linesφ (third column) and the electric field strength |E| (last column). The background electric field always corresponds to the white level. The rows correspondto t = 100, 300, 500, respectively.REFERENCES[1] U. Ebert and W. van Saarloos, “Propagation and structure of planarstreamer fronts”, Phys. Rev. E, vol. 55(2), pp. 1530-1549, Feb. 1997.[2] S.K. Dhali and P.F. Williams,“Two-dimensional studies of streamers ingases”, J. Appl. Phys., vol. 62(12), pp.4696-4707, Dec. 1987.[3] P.A. Vitello, B.M. Penetrante and J.N. Bardsley, “Simulation ofnegative-streamer dynamics in nitrogen”, Phys. Rev. E, vol. 49(6), pp.5574-5598, June 1994.[4] M. Arraya´s, U. Ebert, W. Hundsdorfer, “Spontaneous branching ofanode-directed streamers between planar electrodes”, Phys. Rev. Lett.vol. 88, pp. 174502 [4 pages], Apr. 2002.[5] A. Rocco, U. Ebert, and W. Hundsdorfer, “Branching of negativestreamers in free flight”, Phys. Rev. E vol. 66, pp. 035102(R) [4 pages],Sep. 2002.[6] N. Liu and V.P. Pasko, “Effects of photoionization on propagationand branching of positive and negative streamers in sprites”, Jour. ofGeophys. Res. vol. 109, pp. A04301 [17 pages], Apr. 2004.[7] C. Montijn, W.Hundsdorfer, U. Ebert and J. Wackers, “Numericalsimulations of growing and branching ionization channels using localgrid refinements”, in preparation.[8] B. Meulenbroek, A. Rocco, U. Ebert, “Streamer branching rationalizedby conformal mapping techniques”, Phys. Rev. E vol. 69, pp. 067402,June 2004.Fig. 2. Temporal evolution of a streamer in the moving boundary approxi-mation. The convex streamer becomes flatter and eventually concave.

Numerical simulations and conformal analysis of growing and branching negative discharge streamers

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