computational studies of filamentary pattern formation in a high power microwave breakdown generated...

IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 38, NO. 9, SEPTEMBER 2010 2281

Computational Studies of Filamentary PatternFormation in a High Power Microwave

Breakdown Generated Air PlasmaBhaskar Chaudhury and Jean-Pierre Boeuf

Abstract—Simulations of the dynamics of high power mi-crowave breakdown of air at atmospheric pressure and 110 GHzare presented. The model reproduces well the formation and mo-tion of filamentary plasma arrays observed experimentally withfast camera imaging. The numerical model is based on finite-difference time domain solutions of Maxwell equations coupledwith a simple fluid description of the plasma growth and diffusion.The computational procedure is discussed in details along withnumerical experiments, to show the sensitivity of the results todifferent numerical parameters.

Index Terms—Air plasma, finite-difference time domain(FDTD), fluid model, high power microwave (HPM), microwavebreakdown, plasma array, plasma filaments, plasma modeling,plasma propagation, streamers.

I. INTRODUCTION

A IR breakdown using high power microwave (HPM)source in a large range of pressures and frequencies has

been extensively investigated and is relatively well understoodtheoretically, as well as experimentally [1]. Early studies of airbreakdown were focused on the determination of the break-down field as a function of several parameters such as pressure,frequency, and pulse duration, but it is only relatively recentlythat detailed observations of the plasma dynamics during break-down have been possible with the use of fast-speed camerasfor microwave [2]–[7] or other discharge conditions [8]–[10].Microwave breakdown at high frequency (around 100 GHz)has been investigated experimentally very recently by Hidakaet al. [2]–[4], thanks to the development of Gyrotrons capableof producing high power pulses of long duration [11], [12].The experiments of Hidaka et al. show very clearly the forma-tion of regular filamentary plasma arrays propagating towardthe microwave source. Note that the existence of small-scalestructures and filaments in high pressure microwave breakdownhas been known for a long time (see, e.g., [13] and references

Manuscript received December 1, 2009; revised April 21, 2010; acceptedJune 18, 2010. Date of publication July 26, 2010; date of current versionSeptember 10, 2010.

B. Chaudhury is with the UPS, INPT, Laboratoire Plasma et Conversiond’Energie (LAPLACE), Universite de Toulouse, 31062 Toulouse Cedex 9,France (e-mail: [email protected]).

J. P. Boeuf is with the UPS, INPT, Laboratoire Plasma et Conversiond’Energie (LAPLACE), CNRS and Universite de Toulouse, 31062 ToulouseCedex 9, France (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPS.2010.2055893

therein), but the detailed dynamics of the self-organized struc-tures has been studied only recently.

Self-organization in nonequilibrium discharges takes placein a variety of conditions (for example, in dielectric barrierdischarges, [14], [15]) and several interesting papers showingplasma filaments and streamers in different conditions can befound in a special issue of this journal (Fifth Triennial SpecialIssue on Images in Plasma Science, Vol 36, Number 4, 2008).

The experimental conditions of [2] (high power microwavebreakdown in the millimeter range at atmospheric pressure)offer a new and spectacular example of self-organized dynamicstructure in a nonequilibrium plasma. The experiments of [2]–[4] show that this unique filamentary plasma structure existsonly at high pressures and that the structure changes, withdecreasing pressure, into layers of curved plasma sheets andinto a more familiar diffuse plasma. Images of the array cap-tured in the E-plane of the incident electromagnetic wave showthe multiple plasma columns elongated along the electric fieldpolarization. Images in the B-plane of the incident wave revealthat these filaments are regularly arranged and form a patternthat moves toward the microwave source. The average axialdistance between adjacent filaments on the beam axis is about0.76 mm, which is slightly larger than a quarter wavelength(λ/4 = 0.68 mm) at 110 GHz.

Similar discrete structures have been observed at lower mi-crowave frequencies and lower power; however, very few theo-retical and modeling attempts have been made to understandthese phenomena [16]–[18]. The complex coupling betweenthe wave and the plasma cannot be described analyticallyand numerical simulations that can cope with a continuouslychanging plasma with sharp density gradients must be de-veloped. Recently, a one-dimensional (1-D) model [18] hasbeen proposed to investigate the experimental observations of[2]. The pattern description, however, needs at least a two-dimensional (2-D) approach. We have performed such study[19], coupling Maxwell equations with plasma fluid equations,to describe the formation of patterns under conditions similarto the experiments [2].

In this paper, the same physical model as in [19] is usedand we focus on some numerical aspects of the model. In[19], we showed how two 2-D simulations performed for atransverse electric and transverse magnetic wave can reproducethe experimental three-dimensional patterns observed in twodifferent planes. In this paper, we focus on the case where theincident electric field is perpendicular to the simulation domain,

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2282 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 38, NO. 9, SEPTEMBER 2010

which corresponds to the experimental images of [2] taken inthe B-plane. We specifically study and discuss the parametersthat control the numerical accuracy of the simulation keepingthe physics as simple as possible.

II. PRINCIPLES OF THE FILAMENTARY ARRAY FORMATION

Hidaka et al. [2] hypothesized the observed development offilamentary arrays as a result of sequential process of elec-tromagnetic (EM) wave diffraction and reflection from theinitial filaments formed early in the breakdown process. Thishypothesis was verified to some extent by using a simulationtool with metal posts in place of high density plasma fila-ments [2] and later by a 1-D model involving fluid and waveequations [18].

The basic physics of the phenomena observed in [2]–[4] canbe summarized as follows. If the breakdown criterion is sat-isfied (i.e., if ionization overcomes attachment, diffusion, andrecombination, [1]), the plasmoid initiated by avalanches froma free electron grows until the plasma angular frequency of theplasmoid (ωp = (nee

2/ε0me)1/2) reaches a value close to theincident wave angular frequency. Because of field enhancementat its poles [13], [20], the plasmoid stretches in the directionof the electric field and forms a filament. The plasma startsshielding the incident microwave and reflection of the EM wavecomes into play, leading to the formation of maxima of thetotal field (incident + scattered) ahead of the initial filament, inthe direction of the microwave source. Diffusion and enhancedelectron multiplication at these maxima lead to the formation ofnew filaments ahead of the previous ones, which in turn modifythe field pattern. The filamentary array therefore propagatestoward the source.

The stretching of the filaments in the direction of the E fieldhas some similarities with the formation and propagation ofstreamers and the filaments that form during microwave break-down are sometimes called microwave streamers [13], [20].The streamer mechanism was initially proposed by Raether,and later by Loeb and Meek [21], to explain the electricalbreakdown at near atmospheric pressure under a dc field. Aconsiderable amount of theoretical, numerical, and experimen-tal work has been performed to understand the developmentof an electron avalanche, its transition into streamers and thepropagation of streamer fronts under a dc field (on numer-ical simulations of streamers, see, e.g., [22] and referencestherein).

The development of microwave streamers has not been so ex-tensively described. In this paper, we focus on the experimentsof Hidaka et al. [2], and we limit our investigations to the 2-Dsimulations of the pattern formation in a plane perpendicular tothe E field of the incident wave. The microwave filaments orstreamers are therefore developing in the direction perpendicu-lar to the simulation domain. We only show calculation resultsin the context of an incident electric field perpendicular tothe simulation domain which corresponds to the experimentalimages in the B-plane. We will therefore not describe theelongation of the filaments along the E field (experimentalimages in the E-plane), when the electric field is parallel to thesimulation domain.

III. PHYSICAL AND COMPUTATIONAL MODEL

Maxwell equations are coupled with the air plasma transportequations through the electron current density

∇× �E = − μ0∂ �H

∂t(1)

∇× �H = ε0∂ �E

∂t+ �J (2)

�J = − ene�ve. (3)

Here, �E and �H represent the electric and magnetic field ofthe EM wave; the details of the incident wave are given inSection IV. �J is the plasma current density induced by theincident waves. ne is the electron density, e represents theelectron charge, �ve is the electron mean velocity, μ0 and ε0represent the magnetic permeability and electric permittivityof vacuum, respectively. The ion contribution to the currentdensity is neglected.

A usual approximation to the mean electron velocity in thecurrent density equation is obtained from the simplified electronmomentum transfer equation

∂�ve

∂t= − e �E

me− νm�ve (4)

where me represents the electron mass and the electron-neutralcollision frequency in air νm is estimated by [1]

νm(s−1) = 5.3 × 109p (5)

where p is the pressure in torr (ambient temperature). In thiscase, p = 760 Torr.

We choose to describe the time evolution of the plasma den-sity by a simple fluid equation accounting for plasma diffusionand growth or decay associated with ionization, attachment, andrecombination.

∂ne

∂t− div(Deff∇ne) = S (6)

where

Deff = (ξDe + Da)/(ξ + 1) (7)

with ξ = νiτm, is an effective diffusion coefficient; the detailsof which are discussed in [19]. νi is the ionization frequencyand τm = ε0/[ene(μe + μi)] is the dielectric or Maxwell re-laxation time, (μe, μi) represents the electron and ion mobilityrespectively, and (De,Da) are free and ambipolar diffusioncoefficients. Comparison between numerical solutions of thedrift-diffusion Poisson system for a given ionization frequency,with solutions of continuity equation (6) with effective diffusivecoefficient, shows excellent agreement and will be presented ina forthcoming paper.

S = ne(νi − νa) − rein2e (8)

is the net electron production rate and includes ionization (ion-ization frequency νi), attachment (attachment frequency νa),

CHAUDHURY AND BOEUF: COMPUTATIONAL STUDIES OF FILAMENTARY PATTERN FORMATION IN A HIGH POWER MICROWAVE 2283

and recombination (recombination coefficient rei, assumingne = ni).

We have assumed the classical effective field approximationwhere the electron transport properties are supposed to dependonly on the local effective field

Eeff =

√E2

rms

(1 + ω2/ν2m)

(9)

where Erms is the total rms field at the considered location andω is the angular frequency of the incident field. The dependenceof the ionization and attachment frequency in air on Eeff/p isdeduced from solutions of the Boltzmann equation [23] under auniform dc reduced electric field equal to Eeff/p.

The effective diffusion coefficient is equal to the ambipolardiffusion coefficient in the plasma bulk where νiτm � 1 andto the free electron diffusion coefficient on the edge of theplasma (plasma front) where the electron density goes to zeroand νiτm � 1. We find that describing properly the transitionfrom ambipolar diffusion in the plasma to free diffusion at theplasma edge is essential since the propagation velocity of thefilamentary pattern only depends on the diffusion coefficientand ionization frequency in the plasma front (see Section III-Abelow).

The continuity equation with effective diffusion equationabove allows to use the simplifying assumption of a quasi-neutral plasma bulk while ensuring a correct description ofthe plasma edge propagation velocity (having to solve electronand ion transport equations coupled with Poisson’s equationin this context would lead to a much more complicated andmore time consuming numerical method). The use of the ef-fective diffusion coefficient above is discussed and justified in[19]. Note that the 1-D model of [18] assumed that diffusionwas purely ambipolar (in the plasma front as well as in theplasma bulk) and we think that this assumption led to erroneousresults.

The electron mobility is obtained from the electron-neutralmomentum collision frequency by μe = e/(meνm), the elec-tron diffusion coefficient is given by De = μekTe/e andthe ambipolar diffusion coefficient by Da = (μi/μe)De, withμe/μi = 100. The electron-ion recombination coefficient issupposed to be constant and equal to rei = 0 in this case.Solution of the electron Boltzmann equation and experimentsshows that the electron temperature Te in air varies between1 and 2 eV for the range of electric fields considered here.In this paper, we have taken Te = 2 eV, which correspondsto an electric field of 6 MV/m. The accuracy of the modelcan certainly be improved by taking into account the electrontemperature dependence with the rms field and by including amore detailed description of the plasma chemistry. However,this is beyond the scope of this paper. We have shown in [19]that this simple model can well reproduce some experimentalobservations, and our goal here is only to focus on somenumerical aspects of the model.

The characteristic timescale for plasma evolution is muchlonger than that of the wave considered here (110 GHz) andit is therefore not necessary to solve the Maxwell and fluidequations with the same time steps. The same applies to scale

lengths, which determine the grid spacing for the solutionsof the two sets of equations. In the next two subsections,we describe the numerical procedure for the solution of theseequations on two different grids with different time steps.

A. Maxwell’s Equation Using FDTD

Finite-difference time domain (FDTD) is an explicit second-order accurate time-domain method using centered finite differ-ences on a uniform Cartesian grid, yielding the spatio-temporalvariation of the electric and magnetic fields, and has beenapplied to a wide variety of applications of electromagneticscattering problems [24]–[26]. The discrete �E and �H are stag-gered in both time and space which means they are shifted bya half time and space step. The velocity equation is discretizedby the direct integration scheme [27]. A scattered-field FDTDformulation has been used. Contrary to the total-field FDTDcodes [25], which propagate the incident wave through the grid,scattered-field codes [26] accurately generate the incident wavevia an exact analytical function at each field vector location.In this way, we can get rid of the progressive accumulatingerrors of the incident wave due to numerical dispersion andanisotropy. The electric and magnetic field update equations areobtained by decomposing the field components into incidentand scattered terms so that the total fields are Et = Ei + Es

and Ht = Hi + Hs, where subscripts i, s, and t stand forincident, scattered, and total fields, respectively. Using a directintegration approximation for electron momentum equation (4),we can write

vn+1e − vn

e

Δt+ νm

vn+1e + vn

e

2= − e

me

En+1t + En

t

2(10)

and using this with leapfrog approximations of (2), we get

En+1s = En

s

1 − β

1 + β+

eneΔt

2ε0

1 + α

1 + βvn

e

− β

1 + β

(En+1

i + Eni

)+

Δt

(1 + β)ε0∇× H (11)

vn+1e = αvn

e − eΔt

2meγ

(En+1

t + Ent

)(12)

α =1 − a

1 + a, β =

ω2pΔt2

4γ, γ = 1 + a, a =

νmΔt

2.

(13)

Here, the temporal index ‘n’ denote the time t = nΔt. Theseequations for electric field and velocity, along with the con-ventional equation for the magnetic field are stepped in time,alternately updating the electric and magnetic field componentsat each grid point.

For our 2-D simulation, the system of Maxwell equations canbe solved for an electric field E perpendicular to the simulationdomain, z-polarized x-directed (say TE case, the only nonzerocomponents of the EM fields are Ez , Hx, and Hy) or in thesimulation domain, y-polarized x-directed (say TM case, Ex,Ey , and Hz are nonzero). The 2-D grids for the two cases are


Fig. 1. 2-D FDTD grid showing the location of electric and magnetic fields.Densities are located at the corners whereas velocities and current densities arecomputed at the locations of E. Density, velocity, and electric field are definedat the same locations in the TE case. (a) TM. (b) TE.

illustrated in Fig. 1. The spatial grid indices (i, j) correspondto the physical coordinates (iΔx, jΔy). The grid configurationfor the current density �J is to place Jx, Jy , and Jz at thelocations of Ex, Ey , and Ez , respectively. Hence, the velocitiesare calculated at the locations of electric fields. The densities,on the other hand, are considered at the corners of the cells.When electric field �E is in the simulation domain, we need to doa spatial averaging to estimate the density at the same locationas the electric field, while density, velocity, and electric fieldEz are defined at the same location in the other case when �E isperpendicular to the simulation domain.

We use the same grid spacing, noted ΔsM in the x- andy-directions, and the time step of the FDTD scheme is ΔtM =0.5ΔsM/c, where c is the velocity of light.

Since we are using a scattered field formulation, only scat-tered fields need to be absorbed at the boundaries of thesimulation domain, and these can be more easily absorbed thana total field by an outer radiation boundary condition. We areusing Mur’s boundary condition [28] for the absorption of thewaves at the computational boundaries.

B. Solution of the Continuity Equation

The continuity equation (6) is solved using a simple explicitscheme for the diffusion and ionization term, whereas theloss terms are treated implicitly or semi-implicitly to imposepositivity of the solution

nn+1e(k,l) =

11 + ΔtF (νa + reinn

(k,l))

×{

nne(k,l)[1 + ΔtF νi] +

DeffΔtFΔs2

F

×[nn

e(k+1,l) + nne(k−1,l) + nn

e(k,l+1)

+nne(k,l−1) − 4nn

e(k,l)

] }. (14)

In this problem of microwave breakdown at high pressure,the density gradients can be very large, and we will see inSection IV below, that the grid spacing for the density equation(6) must be, in some cases, much smaller than the FDTDgrid for the Maxwell equations. The density gradient can beestimated by considering the asymptotic solution of equation(6), assuming only ionization and in the case of a uniform and

steady state field (constant ionization frequency). In that case,the density can be written as

ne(r, t) = At−3/2 exp[νit − r2/4Deff t]. (15)

The density equation (15) exhibits a front that propagatesat a speed v = 2(Deffνi)1/2, where the value of Deff has tobe taken at the front (i.e., Deff = De), and the characteristiclength of the front, defined as |∇ne/ne|−1 in a reference framemoving at speed v, is L = (Deff/νi)1/2. We see that thecharacteristic length of the front decreases when the ioniza-tion frequency increases, i.e., when the applied electric fieldincreases. In our conditions, Deff is on the order of 10−3 m2/sin the front and νi may be on the order of a few 109 s−1, so thatL is in the micrometer range, i.e., very small with respect to thewave length (2.7 mm).

An efficient way to deal with the requirement of a fine gridto accurately describe the sharp density gradients would be touse an adaptive mesh refinement (AMR) scheme which adaptsthe distribution of grids according to the density gradients. Forthe cases considered here (simulations of the conditions of theexperiments of [2]), we found that using a fixed grid fine enoughto resolve the density gradients led to reasonable computationtimes. It also appeared that accurate results could be obtainedusing a FDTD grid much coarser than the density grid (seeSection IV). In the following, we use the same grid spacingin the x- and y-directions, and we call ΔsM the grid spacingfor the Maxwell equations (FDTD scheme) and ΔsF the gridspacing for the fluid equation for the density. The grid size ratiom is defined by m = ΔsM/ΔsF .

Solutions of the continuity equation needs the transportcoefficients which are functions of the electric field. Since theelectric field is available only at the coarser FDTD grid points,an interpolation is needed to obtain the effective field on the finegrid in order to estimate the ionization and attachment frequen-cies in the density equation. Once the new density is known onthe fine grid, proper interpolation must be used to update thedensity on the coarser grid, which is used in the FDTD scheme.We have employed a simple 2-D linear interpolation schemefor this purpose, which is briefly described below. The largedots in Fig. 2, A, B, C and D, are coarse grid points where rmselectric fields are available after solving the Maxwell equations.Proper averaging must be done for the TM case to get the rmsvalue of the E field at the corners of the coarse cells, wherethe density is defined. Finally, rms values at all the fine meshlocations represented by the small circles are interpolated usingthe formula

ek,l =(m − nl)

m

(m − nk)m

Ei,j +(m − nl)

m

nk

mEi+1,j

+nl

m

(m − nk)m

Ei,j+1 +nl

m

nk

mEi+1,j+1 (16)

where nk, nl varies from 0, 1, 2, . . . ,m in the x- andy-directions, respectively. For example, in Fig. 2, m = 5, andwe have 36 points in the fine grid.

Using the interpolated values of the field, the new density atthe fine grid locations is obtained from the discretized continu-ity equation (14). The density on the coarse grid is then obtained


Fig. 2. Overlapping coarse FDTD and fine fluid grid. Interpolation is used tofind the electric fields at the fine mesh points.

by a proper weighting (similar to the one used for interpolationabove) of the density of the fine grid.

The time step (ΔtF ) for the continuity equation is calculatedusing the CFL condition ΔtF < (ΔsF )2/(2Dξmax

), whereΔsF is the fluid mesh size and Dξmax

corresponds to maximumeffective diffusion.

C. Algorithm

Maxwell equations and the density equation are solvedsuccessively in time (this is possible because the time scaleof the density variations is much longer than that of the110-GHz EM field). Using the density calculated at time t,Maxwell equations, together with the electron momentumequation [eqs. (1)–(4)], are solved with the FDTD scheme (eqs.(11), (12)) for one cycle of the EM wave, TM . The rms field isthen calculated on the coarse grid and interpolated on the finegrid, and the density equation is solved (using the same rmsfield) for a time duration equal to TM (it is possible, under someconditions and in order to save computation time, to solve thedensity over a time duration equal to several TM ; this will notbe discussed here). Once the density at time t + TM has beencalculated on the fine grid, the density is updated on the coarsegird, and the Maxwell equations are solved over the next cycleusing the updated density.

IV. RESULTS AND DISCUSSION

In this section, we show the 2-D simulation results for break-down in air at a pressure of 760 torr with 110 GHz, 6 MV/mEM field for the case of E field perpendicular to the simulationdomain. The simulation domain is 3λ × 3λ. Breakdown isinitiated around an initial charged particle density profile of1013 m−3 with a Gaussian shape and a standard deviation of50 μm, centered on the central x-axis at a distance of 2.25λfrom the left boundary (x = 0). The incident plane wave (of theform E0cos(ωt − kx)) is injected at x = 0 with a wave vectorin the x-direction.

Fig. 3. Evolution of plasma filaments with time. Plasma densities at (a) 15 ns,(b) 50 ns, (c) 75 ns, and (d) 143 ns. Distances are in terms of λ and densitiesare measured in m−3.

Fig. 3 shows the plasma density distribution at four differ-ent times of the simulation for a 6 MV/m 110-GHz incidentfield. We see that breakdown takes place at the initial locationof the seed electrons. Once the density of the plasmoid issufficient, the incident field is scattered and standing waveswith electric field maxima appear ahead of the plasmoid, asdescribed in Section II. The plasmoids are actually filamentsthat stretch along the E field, i.e., in the direction perpendicularto the simulation domain. New filaments grow due to diffusionand enhanced ionization at the field maxima ahead from theprevious filaments. The filamentary pattern propagates in thedirection of the microwave source and at a velocity controlledby ionization and diffusion coefficient on the plasma edge. Thedensity distributions of Fig. 3 show that the distance betweenthe on-axis filaments are of the order of λ/4, which matchesthe experimental results.

We have seen above that in order to resolve properly thedensity gradients, the grid spacing for the density equation maybe quite small. In order to minimize the computation time, it isimportant to optimize the sizes of the computational grids andthe purpose of the discussion below is to study the influence ofthe grid spacing for the FDTD (coarse grid) and density (finegrid) on the accuracy of the results. In Fig. 3, the coarse gridsize was λ/50, while the fine grid size was λ/1200.

In all the figures below, the distances are in terms of incidentwave wavelength λ. Generally, the cell size (ΔsM ) for theFDTD grid is determined by the restrictive dispersion require-ments, which require it to be smaller than roughly 1/10 or1/20 of the incident wavelength λ or smallest electromagneticfeature of interest. In this case, the important length scale whichmust be properly resolved is the minimum skin depth whichdetermines the penetration of the EM field in the plasma.

For an incident wave of 6 MV/m, the electron density reachesvalues on the order of 4 × 1021 m−3, corresponding to a skindepth on the order of 3 × 10−4 m, i.e., about λ/10.

As a first test (see Table I), calculations with four differ-ent FDTD grid sizes (25, 50, 75, and 100 grid spacings perwavelength) less than this skin depth have been performed,the density grid size being kept constant. We have compared


TABLE INUMERICAL PARAMETERS USEDFOR DIFFERENT CASES ALONG WITH

THE APPROXIMATE PROPAGATION TIME OFTHE FILAMENTS

Fig. 4. Filamentary pattern after the front propagates one λ from its startingpoint toward the source, for a constant size of the density grid (λ/1000) anddifferent FDTD grid sizes (a)λ/25, (b)λ/50, (c)λ/75, and (d)λ/100. Thebeam enters from x = 0, and electric field is perpendicular to the plane of paper.Densities (ne) are in m−3.

the shape and time of propagation of the filaments toward thesource, until the pattern travels a distance of one wavelengthλ from the initial breakdown point. Note that the time ofpropagation over one wavelength may be slightly different fordifferent grid sizes (last column of Table I) and convergestoward 102 ns.

Fig. 4 shows the pattern obtained in the four different casesafter propagation over one wavelength and Fig. 5 shows thetime evolution of the location of the plasma front. The littlemismatch in Fig. 5 is due to front detection procedure, accuracyof which depends on the FDTD grid size. The front position isdetermined by following a particular density level on the axisof symmetry. In this case, the level is taken as 10−4 times themaximum density in the filaments (typically, in the order of1017 m−3). It should be noted that the speed of propagationis not constant and the observed decrease in speed is associatedwith the formation of new filaments. Newly formed filamentstake some time to attain a significant density level before itstarts reflecting the incident wave and propagates forward. Thisdelay leads to the change in speed. The results of Figs. 4 and 5show that the speed, as well as the structure of the filaments, areinsensitive to the FDTD grid size if it is less than λ/25, whichis good enough to resolve the associated skin depth. However,higher electric fields lead to higher densities and hence a lower

Fig. 5. Position of the front (edge of the 1st filament on the left in Fig. 4)as a function of time for different FDTD grid and with a constant grid size(λ/1000) for the density equation, in the cases shown in 4.

Fig. 6. Filamentary pattern after the front propagates over one wavelength, λfrom its starting point, with a constant FDTD grid size (λ/50) and for differentgrid sizes of discretized density equation (a)λ/100, (b)λ/300, (c)λ/500,(d)λ/1200. The beam enters from x = 0, and electric field is perpendicularto the plane of paper. Densities (ne) are in m−3

skin depth, therefore the grid sizes must be adjusted with it.The speed of propagation can be easily found from the slope ofFig. 5 and it is of the order of 30 km/s. We have performed aparametric study to find the speed of propagation as a functionof incident power, as well as at different pressures with constantpower. We get similar trends as reported in the experiment [4],and for an incident power of 3 Mw/cm2 ( 4.75 MV/m), weobtain a similar speed of 14 km/s, which is in good quantitativeagreement with the experimental results of [4].

A second set of tests has been performed for different sizesof the fluid grid using a constant FDTD grid size equal to λ/50.As shown in Table I, the ratio between the Maxwell grid sizeand the density grid size has been varied as 2, 6, 10, 20, 24,corresponding to density grid sizes from λ/100 to λ/1200.Figs. 6 and 7 show that the results are very sensitive to thedensity grid size. Grids coarser than λ/500 (i.e., about 5.4 μm)give unacceptable errors for both speed of propagation anddensity distribution. The results start improving with decreasinggrid sizes, below λ/500. This confirms that the grid spacingof the density equation must resolve the characteristic length


Fig. 7. Position of the front as a function of time for different grid sizes of thediscretized density equation and with a constant FDTD grid size λ/50.

of the front, defined (see the discussion in Section III-B) as(De/νi)1/2).

Fig. 6 clearly shows the effect of numerical diffusion fordensity grids coarser than λ/500. Grid sizes smaller than thislength scale can reproduce the experimental results quite well;however, the simulation becomes computationally expensive.It must be noted that the ionization rate of air scales approx-imately as the electric field strength to the fifth power [4],which means that larger incident electric fields will requireeven finer grids. Adaptative refinement schemes based on thediffusion/ionization scale length criteria for the density gridcan lead to much less computational demands which will beparticularly useful for a 3-D study.

Limitation: The computational scheme described here,based on the use of a fine grid for the plasma equation anda coarse grid for Maxwell equations, works perfectly whenthe electric field E is perpendicular to the simulation domain.However, it is less accurate when the electric field is in thesimulation plane because the simple field interpolation betweenthe coarse grid and the fine grid that we have used in thispaper is no longer adequate. Solutions of the problem with theE-field in the simulation domain and with a dual mesh havebeen compared with “exact” solutions obtained with a uniquefine mesh (using a parallelized code). The comparisons showthat using a dual mesh in that case still provides a good prop-agation velocity of the filaments, but that the detailed structureof the filaments may not be accurately described.

V. CONCLUSION

The formation of plasma patterns during microwave break-down in atmospheric air at 110 GHz has been simulated usingan FDTD method coupled with a diffusion-ionization descrip-tion of the plasma growth. The results show a good qualitativeagreement with the experiments and good quantitative match isobtained for the propagation speed of the filaments. The com-putational procedure and the convergence of the simulationshave been studied as a function of grid size. The results showthat two different grid sizes can be used for the Maxwell anddensity equations. A sufficient accuracy is obtained when theFDTD grid size is on the order or less than the plasma skin

depth, i.e., 25 grid spacing per wavelength (about 100 μm)can be used for air breakdown at 6 MV/m, 110 GHz, while agrid spacing in the μm range (more than 1000 grid spacing perwavelength) is necessary to resolve the sharp density front. Thiscombination leads to reasonable computation times, of about12 CPU hours on a PC workstation to follow the propagationof the filamentary plasma pattern for 100 ns in a 3λ × 3λsimulation domain for the maximum accuracy of Table I. Wecan conclude that 3-D simulations are feasible under theseconditions for a fixed FDTD grid, but an adaptative meshrefinement method for the density equation would be probablynecessary.

In this paper, we have considered, as in [19], a very simpli-fied physical model (constant electron temperature, no plasmachemistry, etc.) of the dynamics of microwave breakdown. Theresults show [19] that this model is sufficient to reproduce someimportant experimental features, provided that an effectivediffusion coefficient is taken into account. Also, the purposeof this paper was to focus on some numerical aspects of themodel, and it was therefore not necessary to add any complexityto the model of [19]. A more detailed analysis of the effectsof the assumptions of the discharge model (field dependenceof the electron temperature, recombination coefficients, role ofnegative ions . . .) will be presented in forthcoming papers.

ACKNOWLEDGMENT

This work has been performed in the frame of the RTRASTAE PLASMAX project. The authors would like to thankmembers of this project for fruitful discussions.

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Bhaskar Chaudhury was born at Giridih,Jharkhand, India, in 1978. He received the B.Sc.degree in physics from the Vinoba Bhave University,Hazaribagh, India, in 1999, the M.Sc. degree inapplied physics from the Sikkim Manipal Instituteof Technology, Gangtok, India, in 2002, and thePh.D. degree at the Institute for Plasma Research,Gandhinagar, India, in 2008.

He is currently working as a Post-Doctoral Re-searcher at LAPLCAE, Paul Sabatier University,Toulouse, France. His research interests include

computational electromagnetics, computational plasma physics, millimeter/microwave interaction with plasma, high pressure/atmospheric pressureplasmas and its applications, fluid models, streamer dynamics, antenna theory,radar cross section studies, FDTD method, PIC simulations, wave propagationin plasmas, high-performance computing, etc.

Jean Pierre Boeuf received the Master degree fromthe Ecole Superieure dElectricité in Gif sur Yvette,France, in June 1977. He received the Ph.D. degreein plasma physics from the Université de Paris XIOrsay, France, in 1981 and is Docteur ès Sciences ofUniversité de Paris XI, in 1985.

In 1983, he joined the National Center forScientific Research (CNRS) at the Laboratoirede Physique des Décharges, Ecole Supérieured¿Electricité, Gif sur Yvette, France. He moved toUniversité Paul Sabatier, Toulouse, France at Centre

de Physique des Plasmas et Applications de Toulouse (CPAT) in 1986. He iscurrently the Directeur de Recherche CNRS at Laboratoire Plasma et Conver-sion d¿Energie (LAPLACE), a joint laboratory between CNRS, Université PaulSabatier, Toulouse, France, and Institut National Polytechnique de Toulouse,Toulouse, France. He is currently in charge of the LAPLACE Research Groupin Energetic and Non-equilibrium Plasmas (GREPHE). His recent researchprograms include projects on plasma thrusters for satellite propulsion, plasmasfor aerodynamic applications, negative ion source for the ITER neutral beaminjection, microwave breakdown, micro-discharges and applications, in collab-oration with academic and industrial partners.

computational studies of filamentary pattern formation in a high power microwave breakdown generated...

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