Non-ionized media subject to strong fields can become locally  ionized by penetration of finger-shaped streamers.  We study negative streamers between planar electrodes in a simple  deterministic continuum approximation. We observe that  for sufficiently large fields, the streamer tip can split.  This happens close to Firsov's limit of ``ideal conductivity''.  Qualitatively the tip splitting is due to a Laplacian instability  quite like in viscous fingering. For future quantitative  analytical progress, our stability analysis of planar fronts  identifies the screening length as a regularization mechanism

Arrayás, M.

Ebert, U. (Ute)

Hundsdorfer, W. (Willem)

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 a Modelling, Analysis and SimulationSpontaneous branching of anode-directed streamers between Planar electrodesM. Arrayás, U.M. Ebert, W.H. HundsdorferREPORT MAS-R0120 DECEMBER 31, 2001MASModelling, Analysis and SimulationCWI 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 © 2001, 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 1386-3703Spontaneous Branching of Anode-Directed Streamersbetween Planar ElectrodesManuel Arrayas1;2, Ute Ebert1and Willem Hundsdorfer11CWI, P.O.Box 94079, 1090 GB Amsterdam, The Netherlands2Instituut{Lorentz, Universiteit Leiden,P.O.Box 9506, 2300 RA Leiden, The NetherlandsABSTRACTNon-ionized media subject to strong elds can become locally ionized by penetration ofnger-shaped streamers. We study negative streamers between planar electrodes in a sim-ple deterministic continuum approximation. We observe that for suÆciently large elds,the streamer tip can split. This happens close to Firsov's limit of \ideal conductivity".Qualitatively the tip splitting is due to a Laplacian instability quite like in viscous n-gering. For future quantitative analytical progress, our stability analysis of planar frontsidenties the screening length as a regularization mechanism.2000 Mathematics Subject Classication: 35B99, 74H60Keywords and Phrases: Tip splitting, branching, discharges, streamers, interface dynam-ics, Stefan problemNote: The paper was submitted to Phys. Rev. Lett. on Nov. 16, 2001. The work wascarried out within project MAS1.4 \Pattern formation and Low Temperature Plasmas",to be continued in the new theme MAS3 \Nonlinear Dynamics and Complex Systems"after 1.1.02. M.A. was supported by the EU-network \Patterns, Noise, and Chaos" andU.E. partially by the Dutch Science Foundation NWO.Streamers commonly appear in dielectric breakdown when a suÆciently high voltage is suddenlyapplied to a medium with low or vanishing conductivity. They consist of extending ngers of ionizedmatter and are ubiquitous in nature and technology [1, 2]. The degree of ionization inside a streamer islow, hence thermal or convection eects are negligible. However, streamers are nonlinear phenomenadue to the space charges inside the ionized body that modify the externally applied electric eld.While in many applications, streamers by a strongly non-uniform background electric eld are forcedto propagate towards the cathode through complex mixtures of gases [2, 3, 4], we here investigate thebasic phenomenon of the primary anode-directed streamer in a simple non-attaching and non-ionizedNew address: Universidad Rey Juan Carlos, Escuela Superior de Ciencias Exp. y Tecnologa, c. Tulipan s/n, 28933Mostoles, Madrid, Spain10 5000200400600800100012001400rzt = 3000.20.40 5000200400600800100012001400rt = 3650.20.40.60 5000200400600800100012001400t = 420r0.20.40.60 5000200400600800100012001400t = 450r0.20.40.6Figure 1: Evolution of spontaneous branching of anode directed streamers in a strong homogeneousbackground eld at times t = 300, 365, 420 and 450. Model, initial and boundary conditions arediscussed in the text. The planar cathode is located at z = 0 and the planar anode at z = 2000(shown is 0  z  1400). The radial coordinate extends from the origin up to r = 2000 (shown is0  r  600). The thin lines denote levels of equal electron density  with increments of 0.1 or 0.2 asindicated by the labels. The thick lines denote the higher electron density levels 1., 2., 3., 4., 5. and6. These high densities appear only at the last time step t = 450 in the core of the new branches.gas and in a uniform background eld as in the pioneering experiments of Raether [5]. In previoustheoretical work, it is implicitly assumed that streamers in a uniform background eld propagate ina stationary manner [6, 7, 8]. This view seems to be supported by previous simulations [9, 10].In this paper we present the rst numerical evidence that anode directed (or negative) streamers dobranch even in a uniform background eld and without initial background ionization in the minimalfully deterministic \uid model" [1, 6, 7, 8, 9, 10], if the eld is suÆciently strong. We argue that thishappens when the streamer approaches Firsov's limit of \ideal conductivity" [6]. The streamer thencan be understood as an interfacial pattern with a Laplacian instability [11], qualitatively similar toother Laplacian growth problems [12]. For future quantitative analytical progress, we identify theelectric screening length as a relevant regularization mechanism.We investigate the minimal streamer model, i.e., a \uid approximation" with local eld-dependentimpact ionization reaction in a non-attaching gas like argon or nitrogen [1, 6, 7, 8, 9, 10, 11]. Indetail, the dynamics is as follows:(i) an impact ionization reaction in local eld approximation: free electrons and positive ions aregenerated by impact of accelerated electrons on neutral molecules @ne+rR je= @ni+rR ji=jeEnej 0(jEj=E0); ne;iand je;iare particle densities or currents of electrons or ions, respectively,and E is the electric eld; in all numerical work, we use the Townsend approximation (jEj=E0) =exp( E0=jEj) for the eective cross-section.(ii) drift and diusion of the charged particles in the local electric eld je=  eEne  DerRne,2where in anode-directed streamers the mobility of the ions actually can be neglected because it ismore than two orders of magnitude smaller than the mobility of the electrons, so ji= 0,(iii) the modication of the externally applied electric eld through the space charges of the particlesaccording to the Poisson equation rR E = e(ni  ne)=0. It is this coupling between space chargesand electric eld which makes the problem nonlinear.The natural units of the model are given by the ionization length R0=  10, the characteristic impactionization eld E0, and the electron mobility edetermining the velocity v0= eE0and the timescale 0= R0=v0. Hence we introduce the dimensionless coordinates [11] r = R=R0and t = =0, thedimensionless eld E = E=E0, the dimensionless electron and ion particle densities  = ne=n0and = ni=n0with n0= "0E0=(eR0), and the dimensionless diusion constant D = De=(R0v0). Afterthis rescaling, the model has the form:@t   r  ( E+D r) =  f(jEj) ; (1)@t =  f(jEj) ; (2)   = r E ; E =  r ; (3)f(jEj) = jEj (jEj)= jEj e 1=jEjin sim.: (4)In the simulations presented here, a planar cathode is located at z = 0 and a planar anode atz = 2000. The stationary potential dierence between the electrodes  = 1000 corresponds to auniform background eld E =  0:5 ezin the z direction. For nitrogen under normal conditions witheective parameters as in [9, 10], this corresponds to an electrode distance of  5 mm and a potentialdierence of  50 kV. The unit of time 0is  3 ps, and the unit of eld E0is  200 kV/cm. We usedD = 0:1 which is appropriate for nitrogen, and we assumed cylindrical symmetry of the streamer.The radial coordinate extends from the origin up to r = 2000 to avoid lateral boundary eects onthe eld conguration. As initial condition, we used an electrically neutral Gaussian ionization seedon the cathode(r; z; t = 0) = (r; z; t = 0) = 10 6e (z2+r2)=1002: (5)The parameters of our numerical experiment are essentially the same as in the earlier simulations ofVitello et al. [10], except that our background electric eld is twice as high; the earlier work had 25kV applied over a gap of 5 mm. This corresponded to a dimensionless background eld of 0.25, andbranching was not observed.In Fig. 1 we show the electron density levels at four time steps of the evolution in the higher back-ground eld of 0.5. We observe that at time t = 420, the streamer develops instabilities at the tip.At time t = 450, these instabilities have grown out into separate ngers. Because of the imposedcylindrical geometry, the further evolution after branching ceases to be physical. On the other hand,the main eect of the unphysical symmetry constraint is to suppress all linear instability modes thatare not cylindrically symmetric. Hence in a fully 3D system, the instability will develop even earlierthan here.Further simulations show: (a) branching does not occur in a system of the same size in the lowerbackground eld of 0.25, in agreement with [10]. (b) Branching is not due to the proximity of theanode, since in a system with twice the electrode distance (with the anode at z = 4000) and with twicethe potential dierence ( = 2000) | so with the same background eld |, the streamer branches30 50 100 150 200 250550600650700750rzt = 3000.10.20.30.40.50 50 100 150 200 2508008509009501000rzt = 3650.30.40.50.6Figure 2: A zoom into the head of the streamer from Fig. 1 at the rst two time steps. The aspectratio is equal and the axis scaling identical at both times. The thin lines are the levels of equalelectron density as in Fig. 1. The thick lines are electrical equipotential lines in steps of  = 12.in about the same way after about the same time and travel distance. (c) However, branching doessomewhat depend on the numerical discretization. In fact, numerical noise due to the discreteness ofthe spatio-temporal grid is the only noise present in our mean eld model (1) { (5). It substitutesthe physical noise caused by the discreteness of the individual charge carriers. A wider numericalmesh leads to a higher eective noise level; and in agreement with this reasoning, in the simulationthe branching then develops somewhat earlier. (d) Occasionally, we observe a dierent tip splittingmode. In Fig. 1 at time t = 450, the nger on the axis develops the strongest with  exceeding 6,while in the ngers o the axis,  stays below 3. In the other branching mode, the rst nger o theaxis outruns the nger on the axis.Before we discuss the physical nature of the instability, we explain our numerical approach: we useduniform space-time grids with a spatial mesh of 1000  1000. The spatial discretization is based onlocal mass balances. The diusive uxes are approximated in standard fashion with second orderaccuracy. For the convective uxes a third order upwind-biased formula was chosen to reduce thenumerical oscillations that are common with second order central uxes. Time stepping is basedon an explicit linear 2-step method, where at each time step the Poisson equation is solved by theFISHPACK routine. References for these procedures can be found in [13].To understand now why and at which stage the streamer becomes susceptible to noise and develops atip splitting instability, in Fig. 2, we zoom into the streamer head. Shown are the rst two time stepsfrom Fig. 1 with the electron density levels again as thin lines, and additionally with the equipotentiallines as thick lines. One observes that during the temporal evolution prior to branching, both thecurvature and the thickness of the ionization front decrease. So the width of the front becomes muchsmaller than its radius of curvature, and an interface approximation becomes increasingly justied.The electric eld inside the streamer head also decreases, so that the ionization front more and morecoincides with an equipotential surface. In summary, the ionization front evolves towards a weakly4curved and almost equipotential moving ionization boundary. At the same time, the electric eldimmediately ahead of the streamer increases.We argue now that a transient stage of an approximately equipotential and weakly curved ionizationboundary leads to tip splitting. Conversely, we argue that tip splitting in the lower background eldof 0.25 is not observed within the presently and previously [10] investigated gap lengths because thetransient stage of Fig. 2 is not reached before the streamer reaches the anode.In fact, the streamer in Fig. 2 approaches the limit of \ideal conductivity": the conducting bodyhas   const:, while in the non-ionized region r2 = 0 due to the absence of space charges. Theboundary between the two regions moves approximately with the drift velocity vf/ r or with thediusion corrected velocity vf/ r1 + 2pD (jrj)=jrj[11]. Our simulations are the rstnumerical evidence that the \ideal conductivity" limit can be approached within our model.This limit of ideally conducting streamers in an electric eld that becomes uniform far ahead of thefront was studied by Firsov [6]. He realized that uniformly propagating paraboloids of arbitraryradius of curvature are solutions of this problem. He did not realize that his paraboloids are mathe-matically equivalent to the Ivantsov paraboloids [12] of dendritic growth found earlier. The uniformlypropagating Ivantsov paraboloids in the early 80'ies were identied as dynamically unstable. This isgenerally the case for such so-called Laplacian growth problems without a regularization mechanism.Since ideally conducting streamers also pose such a Laplacian growth problem [11], the dynamicalinstability of the structure shown in Fig. 2 can be expected, and it actually occurs as can be seen inFig. 1. This explains qualitatively why tip splitting occurs.For a quantitative analysis, a system specic regularization mechanism has to be found [11, 12]. Itsidentication is intricate because negative streamer fronts are so-called pulled fronts whose dynamicsis dominated by the leading edge rather than the nonlinear interior of the front [14]. Thereforestandard methods like the pertubative derivation of a moving boundary approximation for the model(1){(4) does not work [15]. (Pulling also implies that standard numerical methods with adaptivegrids are ineÆcient.) However, the ionization front has two intrinsic length scales, a diusion lengthand an electric screening length. We therefore explore the approximation of D = 0. It is smooth forthe velocity of negative fronts [11] and eliminates the leading edge, and hence suppresses the pullednature of the front. Rather the front becomes a shock front for the electron density, while the intrinsiclength scale of the electric screening layer behind the shock remains.As a rst step to understand the short wave length regularization of perturbations due to this screeninglength, we have investigated the transversal instability modes of a planar ionization front in the limitD = 0 in a eld that approaches the uniform limit E =  E1ezfar ahead of the front. The planarunperturbed front propagates with velocity v = E1, which equals the drift velocity of the electronsprecisely at the shock front. The implicit analytical front solution can be found in [11]. In a comovingframe  = z   vt, we denote it by 0(); 0();0(). The Fourier components ~k; ~k;~kof atransversal linear perturbation are dened through = 0() +Zdk ~k() eikx+st+ : : : etc. (6)For the derivation of the boundary conditions on the shock front, it is more convenient to write asingle Fourier component as  = 0    eikx+st+ k() eikx+st+ : : :. With this ansatz and the5auxiliary eld  k= @k, the Fourier components solve the inhomogeneous equation@0BBBB@kk kk1CCCCA=Ms;k0BBBB@kk kk1CCCCA 0BBBB@s @0=(v + E0)s @0=vE0k201CCCCA; (7)Ms;k=0BBBB@s+20 f(E0) 0v+E0 0v+E0@0 0f0(E0)v+E00 f(E0)=v s=v  0f0(E0)=v 01  1 0 k20 0 1 01CCCCA:The boundary conditions at the shock  = 0 can be obtained from the analytical solution in thenon-ionized area, and from the boundedness of the charge densities:0BBBB@kk kk1CCCCA"0 !0BBBB@f0(v)=(1 + s=f(v))01(vk   s)=(sk)1CCCCA: (8)The other boundary conditions are obtained by imposing that at  !  1 the electric eld decaysand the densities become constant.These equations together with the boundary conditions dene an eigenvalue problem for s = s(k; v)with v = E1. It can be solved numerically by shooting from  = 0 towards  1. In agreement withanalytical limits | details will be given elsewhere |, we nds(k)(jE1j k for k  (jE1j)=2jE1j (jE1j)=2 for k  (jE1j)=2: (9)This means that the electric screening length 1=(jE1j) does regularize the instability of short wavelength perturbations from linear growth in k to the saturation value s(k) = jE1j (E1)=2. Asmall positive growth rate remains, but the analytical derivation of (9) hints to the unconventionalpossibility that suÆciently curved fronts actually are stable to short wave length perturbations. Thisquestion is presently under investigation. If true, it would identify a most unstable wave lengthdetermining the width of the ngers that emerge after tip splitting.In conclusion, we have presented numerical evidence that anode-directed streamers in a suÆcientlystrong, but uniform eld can branch spontaneously even in a fully deterministic uid model. Wehave argued that this happens when the streamer approaches the limit of ideal conductivity. Wehave established a qualitative mathematical analogy with tip splitting of viscous ngers throughthe concept of Laplacian growth, and we have analytically demonstrated that the electric screeninglength leads to an unconventional regularization. This opens the way to future quantitative analyticalprogress.References[1] Y.P. Raizer, Gas Discharge Physics (Springer, Berlin 1991).6[2] E.M. van Veldhuizen (ed.): Electrical discharges for environmental purposes: fundamentals andapplications (NOVA Science Publishers, New York 1999).[3] A.A. Kulikovskii, J. Phys. D: Appl. Phys. 33, 1514 (2000); and Phys. Rev. E 57, 7066 (1998).[4] L. Niemeyer, L. Pietronero, H.J. Wiesman, Phys. Rev. Lett. 52, 1033 (1984); A.D.O. Bawagan,Chem Phys. Lett. 281 325 (1997).[5] H. Raether: Z. Phys. 112, 464 (1939) (in German).[6] E.D. Lozansky and O.B. Firsov, J. Phys. D: Appl. Phys. 6, 976 (1973).[7] M.I. D'yakonov, V.Y. Kachorovskii: Sov. Phys. JETP 67, 1049 (1988); and Sov. Phys. JETP68, 1070 (1989).[8] E.M. Bazelyan, Yu.P. Raizer, Spark Discharges (CRS Press, New York, 1998); Yu.P. Raizer,A.N. Simakov, Plasma Phys. Rep. 24, 700 (1998).[9] S.K. Dhali and P.F. Williams, Phys. Rev. A 31, 1219 (1985) and J. Appl. Phys. 62, 4696 (1987).[10] P.A. Vitello, B.M. Penetrante, and J.N. Bardsley, Phys. Rev. E 49, 5574 (1994).[11] U. Ebert, W. van Saarloos and C. Caroli, Phys. Rev. Lett. 77, 4178 (1996); and Phys. Rev. E55, 1530 (1997).[12] For a review and a collection of original articles, see P. Pelce: Dynamics of Curved Fronts(Academic Press, San Diego 1988).[13] P. Wesseling, Principles of Computational Fluid Dynamics, Springer Series in Comp. Math. 29(Berlin 2001).[14] U. Ebert and W. van Saarloos, Phys. Rev. Lett. 80, 1650 (1998); and Physica D 146, 1{99(2000).[15] U. Ebert, W. van Saarloos, Phys. Rep. 337, 139 (2000).7

Spontaneous branching of anode-directed streamers between planar electrodes

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