# Model of red sprites due to intracloud fractal lightning discharges

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Model of red sprites due to intracloud fractal lightning discharges

Radio Science,Volume 33, Number 6, Pages1655-1668,November-December1998 Model of red sprites due to intracloud fractal lightning discharges J. A. Valdivia NRC NASA Goddard SpaceFlight Center,Greenbelt, MD G. M .Milikh Departinent of Astronomy, University of Maryland, College Park K. Papa,dopoulos Departments of Physicsand Astronomy,University of Maryland, CollegePark Abstract. A new and improved model of red sprites is presented. Emphasisis placed on accountingfor the puzzlingobservationof the spatial structurein the exnissions. The model relies on the electromagneticpulse (EMP) fields createdby a horizontallightning dischargeand includesthe observedfractal structureof suchdischarges in the computation of the EMP powerdensity.It, is shownthat the modelcan accountfor the observedspatial structureof the red spriteswhile reducingthe typical chargerequiredto approximately 100 C. 1. Introduction seemto touch the cloud tops. Among the most puzzling aspects of the observationsis the presenceof spatial structure in the elnissionsreported by Winck- High-altitude optical flasheswere detected more than 100 yearsago [Kerr, 1994]. Interest in the subject was renewed recently following the observations let et al. [1996]. Vertical striations with horizonof such flashes by the University of Minnesota group [Franz ct al., 1990]. Sincethen, observations of the optical emissionsa.t altitudes between 50 and 90 km associated with thunderstorms have been the focus of many groundand aircraft campaigns[Boecket al., 1992; Vaughan ½t al., 1992; ['l/•nckler ½t al., 1993; tal size of 1 km or smaller, often limited by the instrumental resolution, are apparent in the red sprite elnissions. The first published theoretical model of red sprites [Milikh ½tal., 1995]associatedtheir generationwith transient electric fields induced by large intracloud Sentman et al., 1995; Lyons, 1994]. Sentman and lightning discharges. The intracloud dischargewas Wescott[1993]and Winckleret al. [1996]described lnodeled as a horizontal electric dipole at an altitude the phenomenon,called "red sprites", as a luminous colulnn that stretches between 50 and 90 kin, with peak lmninosity in the vicinity of 70-80 km. The flasheshave an averagelifetime of a few milliseconds and an optical intensity of about 100 kR. Red sprites are associatedwith the presenceof massivethunderstorm clouds,although the luminouscolumnsdo not of 10 kin, and the field calculation included quasistatic, intermediate, and far-field components. Mi- likh et al. [1995]demonstratedthat energizationof the ionosphericelectrons by the transient fields could account for several of the observed features. Two subsequentpapers reached the same conclusionsusing similar methodology as above but emphasizing different sourcesfor the lightning-generatedelectric fields. The first [Paskoet al., 1995]modeledthe electric fields energizing the electrons as due to a point (?opyright1998by the AmericanGeophysicalUnion. charge Q (monopole)at an altitude of 10 km. The Paper number 98RS02201. laxation of the field. The second[Rowlandet al., 1995] assumedthat the fields were due to the far 0048-6604/98/98RS-02201511.00 1655 analysis emphasized the i•nportance of dielectric re- 1656 •LiLDIVIA ET AL.: MODEL OF RED field of a vertical electric dipole generated by cloudto-ground discharges.All of the above models, while successtiffin explaining some observed characteristics of red sprites, such as the color and the gener- SPRITES ,EMISSIONS ,INTERACTION OF ELECT1;•ON$ WITH NEUTtALS ation altitude of the emissions, suffer froin two im- port.ant drawbacks. First,, dipole or monopole distributions generate electric fields smoothly distributed at, ionosphericheights, thereby failing to accountfor the persistent spatial structure of the red sprites. Second,the threshold charge and dipole moment requirements for all three models have been criticized ,ELECTRON ENERGIZATION FOKKE• ?LANCKCODE as unrealistically large (based on the lightning paralneterscited by Uma• [1987]). ,?ROFAGATIONOF EM FIELDS INCLUDINGSELFABSOI•?TION The objective of the present paper is to extend the resultsof Milikh ½tal. [199,5]and it Valdivia et al. [1997]by incorporatingin the calculationof the transient field the internal fine structure of the intra- cloud discharge.It will be shownthat incorporating in the calculation of the lightning-inducedfields the dendritic fine structure of the lightning channel, as ,SOURCEIS LIGHTNING ,HORZObFAL DENDRITIC , •ACTAL ANTE•A - DLA ,(STRUCTURE) describedby Williams [1988] and by Lyons [1994] (who termed it spider lightning), results in a natural explanation of the observed spatial structure of the red sprites and in a significant reduction of the required threshold charge. The fractal lightning dis- charge must be of the intracloud type to have a radiation pattern that is preferentially upward (as well as downward). Figure 1 illustratesthe major elementsusedin our model. The first element is a computer model of the intra,clouddischargethat tries to accountfor its dendritic fine structure. This is modeled as a horizon- tal fractal dischargestructure that producesthe spariotemporal distribution of the amplitude and phase of the electromagnetic field in the far zone. As the field propagatesfrom the lower atmosphere, self-absorptionbecomesin•portant, requiring, as the Figure 1. A diagramof the tasksinvolvedin the treatment.. From the fractal struct.urc, wc compute the fields generated and their interaction with the medium in the lower ionosphere. driving the red sprites, which have a total optical emission of about 100 kR, to levels more consistent with observational requirements. In section2 we describea model of lightningas a fractal antenna, estimate its fractal dimension, and presenta few examplesof fractal discharges.In sec- tion 3 we considerthe propagationof the lightninggeneratedfields in the lower ionosphere,including the effect of self-absorption,and compute the elec- tron distribution function in the presenceof the electric fields. In section 4 we discussthe intensity of putation of the propagation of the lightning-induced the optical emissionsdue to lightning and the spafields in the lower ionosphere. The fields interact tial structure of the red sprites generatedby fractal with a.nd energize the ambient electronsgenerating discharges.This is followedin section5 by conclunext element of our model, the self-consistent com- non-Maxwellian distribution fimctions. The colli- sionsof energetic electrons with neutral particles restilt in the observedemissions.The structuringof the elnissionsis attributed to the highly inhomogeneous fields projected into the lower ionosphere,when the internal structure of the dischargeis included in the model. It will be shown that the model not only accounts for the structure but also simultaneously reducesthe required chargefor the lightning discharge sions. 2. Lightning as a Fractal Antenna It is well known that lightning dischargesfollow a tortuouspath [L½Vine and Mcncghini,1978].It was shown [William,s, 1988] that intraclouddischarges resemble the well-known Lichtenberg patterns ob- servedin dielectricbreakdown[seealsoIdone,1995]. VALDIVIA ET AL.' MODEL These patterns have been recently identified as fractal structuresof the diffusionlimited aggregate(DLA) type with a fractal dimensionD • 1.6 [Niemeycret al., 1984; Sander, 1986]. A fractal lightning dischargeradiates as a fractal antenna and, unlike a dipole type of antenna, gener- OF RED SPRITES 1657 to the spatial distribution of the individual elements over the fractal. In fact., for an oscillating current,of the forin ei•t the phasesvary as q•,,• k. x - k.r=, wherek = •/c and x is the positionof the observation point,. In the senseof statistical optics, we can consider ates a spatially nonuniform radiation pattern, with the ensembleaverageof equation (1), using an erregionsof high field intensityand regionsof low field godicprinciple,over the spatialdistributionP(•, intensity. This is equivalent to a two-dimensional 02,03,..., A1,A2, A3.... ) ofthefractalelements [Goodphased array having an effectivegain factor. Such man, 198,5].For simplicitywe assumethat.the disa fractal lightning dischargecan be modeled as a tributions for each of the elements are independent set, of nonuniform distributed sinall current line el- and the same; hence ements[Nie'meycret al., 1984; k•'cchiet al., 1994] {ri,, Li,, In, si,)[n: 0 ..... N}, wherer. and Un are the position and orient,ation of the nth line element, .s'. is the distance traveled by the discharge,and I• is the strength of the current,at, the nth line element. For future reference,the dischargeis placed horizontally at, a height of 5 kin, and an image dischargeis placed 5 kin below the conductiveground. 2.1. Why Is a Fractal Lightning Discharge so relevant? A fractal lightning dischargecan be consideredas a two-dilnel•sional phased array antenna, that will naturally exhibit a spatially dependent radiation pattern with an effective gain factor. Such a gain factor is extremely ilnportant, for it will reduce the lightning energetics(seebelow), coinparedwith the dipole model, required to produced the optical emissionsin the ionosphere. In order to stndy how this gain factor comesabout, we consider a. fractal antenna as a nontlniform dis- tribution of radiating elements as described above. For an oscillating currentof the formei•t, eachof the elements contributes to the total radiated power density at. a given point. with a vectorial amplitude and phase [Alla.i• a•d Cloitre, 1987; Jaggard,1990; Ti;er•e'ron [4'erne'r,1995],i.e., N N r. E*,---, (Z Aneid'")' (E AxneiCm )* n--1 m--1 : E(An' Ai•)ei(½"-½m) (1) where E* denotesthe colnplex conjugate. The vector amplitudes A. represent,the strength and orientation of the field generatedby eachof the individual elements,while the phases4•, are, in general, related •r From this equat.ion the antenna gain G yields [A2/ N- 1 2 This is one of the most important relationships that. ca,l• be used to study fractal antennae. The first, term representsthe incoherent radiation COlnponent, and the second term is the coherent (interference) radiation component. If we further assulne that ([AI e)-I(A)[e- 1,weobtain that theensemble average is . i N- I •)[ (2) If the distribution of the phasesis random (or uni- forin), then < ei0 >- 0 and G - N. On the other hand, if there is perfect coherence, we have < •i½ >_ 1 and G - N •. If the distribution of the vector a.mpli[udes does not satisfy the above relations, for example, if the radiators are oriented in arbit.rary directions, [hen the power density will be less coherent since (IAI2)• 2 [{A}[ (true for any distri- bution). A similar result can be achievedby having a distribution of amplitudes instead of phases. In general,a ffa.ctal antennawill displaypartial coherencein somedirection(s), and the peak radiated power will lie in between these two limits and can showa significant,gain over a random distribution of phases. Thecoherence term N(N- 1)I<A>I depends on the spatial structure of the ffactal antenna. Such spatial structure is usually describedby the fractal dimensionD [Ott, 1993]. We conjecture 1658 VALDIVIA ET AL.' MODEL OF RED SPRITES' shows the dipole, or linear, model of the current mining the radiation pattern of a fractal dischargeis channel between two points; in general, it will genits fractal dimension, which we can estimate as fol- erate a dipole radiation pattern. The middle drawlows. If we partition the volumewhere the discharge ing showsa tortuous model of the dischargebetween the same two points; this tortuous current channel occurs into boxes of side s, the number of boxes that contain at least one of the dischargeelementswill can be considered as a two-dimensional phased arscaleasN(e) ,.•e-D . It is easyto verifythat a point ray, as shownin the bottom drawing. Note that the correspondsto D - 0, a line correspondsto D - 1, tortuous model will have a longer path length that and a compact surfacecorrespondsto D = 2 . The will increaseN or the individual.4• in equation(2) box countingdimension[Ott, 1993]is then defined with respect to the dipole model. Clearly, there will be positions at which the radiation pattern from the by tortuous line elements will add constructively, while at other positionsthey will add destructively. Thereln(•) fore we expect that a tortuous lightning channelwill Note that if e is too small, then the elements of have, besidesthe spatial structure, a radiation patthe dischargewill look like one-dimensional line eletern with spotsof radiated powerdensitylarger than ments. Similarly, if e is too large, then the discharge the more homogeneousdipole pattern. The fractal will appear as a single point. Therefore it is very dimension will be an important parametrization for important to computeD only in the scalingrange, the fractal dischargemodels that will be generated which is hopefully over a few decadesin s. and will play a significant role in the spatial strucWe will now apply these ideas to modelsof the ture and intensity of the radiation pattern, as we will that the most relevant structural parameter deter- D•_InN(s) (3) lightningdischarge. Figure 2 showstwo different modelsof a lightning discharge. The top drawing see later. 2.2. Time Dependent Fields A currentpulse,I(t- s/v), propagateswith speed v (hence.;Y- v/c), alongthe length s of the horizontal fractal discharge. The dischargewill be mainly horizontal so that. it radiates energy upward, as opposed to a vertical discharge,which will radiate its energy mainly in the horizontal direction. The intra,cloud current pulse is taken as a series of train pulsesthat propagate along the arms of the antenna and have a time dependencegiven by I(t) - (e-• - e-vt)(1+ cos(wt))H(t) (4) with w - 2•rc•n.tand H(t) asthe stepfunction.Here ,.r representthe number of oscillationsduring the decaytimescale1/c•. We chosec• - 10a s-• as the inversedurationof the pulseand 7- 2 x 10s s-• as the risetinae[seeUrnart,1987]. The initial strengths Figure 2. Top drawing is a dipole or straight line model of the lightning discharge. Middle drawing representsthe tortuousmodel betweenthe sametwo endpoints as in the dipole model. The tortuous model can be described as a spatially nonuniform distribution of radiators, eachcontributing to the total radiation field with a given phaseand amplitude (bottom drawing). of the current pulse Io get divided as the discharge branches,but.the total chargedischargedis then Q Io/c•, which for Io - 100 kA givesQ • 100 C. 2.3. Example: Tortuous Discharge Model A fractal tortuous path can be constructed in termsof a randomwalk betweentwo endpoints[ Vecchiet al., 1994]. We start with a, straight line of length L, to which the midpoint is displacedusing a Gaussianrandomgeneratorwith zero averageand VALDIVIA ET AL.' MODEL deviation •r (usually •r = 0.5L). The procedureis then repeated to each of the straight segmentsN t,imes. There is a clear repetition in successive halving of the structureaswegoto smallerscales,making it, a. broadband antenna. Figure 3a showsa typical tortuous fractal, where the division has been taken OF RED SPRITES 1659 and y = 0 km at a. height h = 60 km. Starting with the dipole (straight path), for each successive division, N = 1,2, ..., 8, we compute the array factor R as a functionof the tortuouspath length (Figure3b). to the N = 8 level and in which the path length 5' There is a clear increase in the array factor from the tortuous fractal as compared with the single dipole. The straight line fit agreeswith the theoretical result has increased 5 t,imes, i.e., S •, 5L. We can estimate (to be publishedelsewhere) the fractal dimensionby realizingthat the total path length 5' should go as R(As) As Ro • i q-c•At n,/Lo L where As is the increasein the path length due to the tortuosity and At is the time it takes the pulse where(0 is the averagesegmentsize.This equation to propagate along the fractal. Therefore, even for time dependent dischargecur('an be derivedfrom equation(3). The computationof the radiated electric field is rents, the effect of tortuosity can definitely increase describedin the appendixin the far-field approxima.- the radiated power density at certain locations, as comparedwith a single dipole antenna. Such a retion. Note that. this antenna will radiate every time stilt.will becomeextremely important in our lightning there is a changein direction of the discharge. It is expectedthat it will have a peak radiated power studies, in which by going away from single dipole densitylargerthan that of an equivalentdipole (see models we can increase the power radiated from a fractal antenna.or, in the caseof lightning, a fractal descriptionon antenna gain above). The far-fieldarrayfactorR = rtf dtE2 andthe discharge. Such gain in the radiated power density peak power density dependon the path length or, equivalently,on the numberof segmentsof the fractal. Take the tortuouspath of Figure3a with ,! = 5 so that the peak of the array factor is at x = 0 km --T-C) r-T•r'--I R T V-T7 can be traced to the different fractal dimensions Another important concept related to fracta.1antenna,e is the spatial structure of the radiation field. LJ d._• .%IT¾ T-T- c mr--, , I ' T---T--r b 12 -r ...... T•mmm-rT•-r]-r lmrrrmm•T I ...... -It, 1(-_) 1 o 4 cD N i • < .... '10 256 • D •o • --- L> I -- 0 41 198 I ..... --E% I , , , , I , , , , 0 X •=S 10 D as givenby equation(5). 0 10 •C) •CD S 4 C) .=50 6C) (kraco) Figure 3. (a,) The tortuousdischarge.(b) The array factor dependence,normalizedto the dipole, on the path length. 1660 VALDIVIA ET AL' MODEL OF RED SPRITES For large "7 the array factor scalesas R • exp(-a (on a circlefar awayfronsthe discharge)is q•- 1, a At/c)(2 + cos(27rn7 aArlc)) whereAr- max,•d,• choicethat simplifiesthe computationof the electric - max• ][ x- rl• [[, i.e., the maximumvariation field closeto the discharge.The potentialoutsidethe over the fractal of the distance between the eledischarge structurecanbe computedby iteratingthe ments of the fractal r,, and the field position x (see discretetwo-dimensionalLaplace'sequation the appendix). Consequently,the radiation pattern will have spatial structurewhen a nf Ar/c • 27r, which translateinto nf • 50. A more comprehensive and natural model of fractal dischargesthat includes branching and tortuosity is developedin the next section. 2.4. Two-Dimensional Fractal Discharge Femia 6t al. [1993] found experimentallythat a dielectric discharge pattern is approximately an equipotential. On the basis of this idea, we can construct a fractal dischargeby adapting for our purposes the two-dimensional stochastic dielectric dis- chargemodel proposedby Niemeycr et al. [1984]. This model naturalIv leads to fractal structures where until it, converges.The dischargepattern growsin singlestepsby the additionof an adjacentgrid point (open circlesin Figtire 4) to the dischargepattern, generating a new bond. We assume here that a.n adjacentgrid point, denoted(i,j), has a probability proportional to the •1power of the local electricfield to become part of the pattern. Since the electric fieldfor a point (i,j) adjacentto the discharge(where ½ - 0) is givenby Ei,j • &i,j, the probabilityof addingsucha point t,othe dischargecanbe expressed as the fractal dimension can be easily parametrized by a parameter We start with a charge Q at the center of our computational box. A dischargewill start propagating from this initial point. C•onsidera dischargepattern (line connectingthe solid circlesin Figure 4) at a later time t, which we assumeto be an equipotential with value e5- 0. The boundary condition at infinity ! where the normalization sunsis over all the points adjacent to the discharge. Therefore we apply the Monte (?arlo method to add one of the adjacent points t,o the discharge, i.e., throw the dice and chooseone of the adjacent,points. The new grid point,, being part, of the discharge structure, will havethe samepotential a,sthe dischargepattern, i.e., o - 0. Therefore we lnust resolveLaplace'sequation for the potential every time we add a new bond. The structure generatedfor •/- 1 correspondsto a Lichtenberg pattern. The dimension of this fractal structure is D • 1.5, as follows froin Figure 5, which revealsthe relation betweenzl and D (defined above and COlnputedwith the standard inethods [Ott, We expect that when •/ - 0 the dischargewill have the same probability of propagating in any direction' the discharge will be a compact structure with a, dimension D - 2. On the other extreme, for 0 , •?--> ,:x:the dischargewill become one dimensional, and hence D - Figure 4. Diagram of the discretizedfractal discharge and its adjacent grid points. The solid circles correspondto the discharge,and the open circlescorrespond to the adjacent points that can be added to the discharge. 1. Therefore this model, and also the dimensionof the discharge,is parametrizedby q (Figure 6). In order t,o compute the radiated fields, we must describethe current,along each of the segmentsof the ffactal discharge.We start with a chargeQ at the center of the discharge. The current,is then discharged along each of the dendritic arms. At each VALDIVIA ET AL.- MODEL O0 m OF RED SPRITES 1661 branching a.rm, we assume that the current on the ith a.rm will be proportional to L•. Thereforewecan satisfy charge(or current.)conservationif the current along the ith arm, which branchesat, the branching point,, is .. (6) The calculation of the electric field in the ionosphere fi'om the fractal current structure is described in the appendix for the far field of the small line elements. It turns out that the radiated power den- sity,_•(W/m•")= ceoE•"• I•,32f(D)g(O,D, fi), where #(0, D,/•) describesthe angular distribution of the radiated pa.ttern and f('D) describesthe dimension -10 -5 0 5 10 comesnarrowerandgoesfroma dipoleradiationp•!,- X (kin) Figure 5. Fractal dischargegenerated with the stochastic model for r/ = 1. The gray shading correspondsto the electric potential that generated the discharge. branching point we chooseto ensureconservationof current, but we expect,a. high current to propagate longer than a small current before being stopped. Hence a.teach branching point,a larger fraction of the current will propagate along the longest arm. Suppose that a. current Io arrives a.t,a. given branching point, and if Li is the longest.distance along the ith •'-. Once we have the fra.ctal dischargestructure, we must considerthe propagation of the lightning fields indited in the lower ionosphere. The fields energize the electrons,generating highly non-Maxwellian electron distribution nu•nber of electron-neutral 3.1. .. 1.0 2 , , 4 6 '.........,..............• 8 10 Figure 6. The dimensionof the stochasticdischarge model as a fimction of r/ with the estimated error bars. 3. Electromagnetic Pulse Absorption and Electron Energization functions which inelastic increase the collisions, re- with the help of a Fokker-Planckcode [Tsat•g½tal., 1991]. 1.6 '•. 0 tern that peaks in the direction perpendicular to L,. (contributes to the field in ionosphere)to a radiation pattern that peaks preferentially in the same direction a.sL,. (does not contribute to the field in the ionosphere). Therefore there is also a.n interplay between the electric field pattern #(0) generatedfrom the fractal discharge,and hence between its dimension and speed of propagation ,3. sponsible for the emissions,and inducing field selfabsorption. The electron energization is computed 1.8:'• 1.4 dependence.As we increase•, the angular field distribution from a single horizontal line element be- Self-Absorption As the lightning-induced fields propagate in the lower ionosphere, the field changes t,he properties of the medium by heating the electronswhile experiencing absorption. The solution to Ma.xwell'sequations for the propagation of the electric field is a nonlinear wave equation 4•'0 V-"E-V(V.E) e3(ag) 1 02 ot(m)- 0, (7) 1662 VALDIVIA ET AL.: MODEL where the medium is incorporated in the conductiv- OF RED SPRITES ically beneficial case. It is satisfiedat any height of ity • and in the dielectric•' tensors[Gurevich,1978]. the lower ionospherein the equatorial region or at For the heights and frequenciesof interest, • and • reach a steady state much faster than the timescale the height below 80 km at lniddle latitude. Notice of the field variation, i.e., 1/w. The solutionin the lnore energy from the electric field, increasing the absorption and reducing the field amplitude at the higher altitudes. ray approximation is therefore tE'"(?s,t) ""z(os2 ' _ - csc(x) f: •(zE2)dz c (8) wheresin(x)= z/v/x "•+ yU+ z• istheelevation angle of the point r = •s = {x,y,z}, H(t) is the 2 . + •,;)is the step function, n(z ,E 2) - •oe%/c(g• • _ 4zrn•e •/ m absorptioncoefficientof the wave, we is the plasma frequency,and f• = eB/mc is the electron gyrofrequency, The nonlinearity is incor- poratedself-consistently through•/e= •,•(z, I•l), the that,for (•8/y)u cos• 0o< 1, the electrons cangain The averagedquiver energy •'(E, %), which depends nonlinea.rlyon the steady state averagedcollisiona.1frequency %, is the critical parameter that controls the behavior of the distribution function f(v) under an electric field E at a given height. For valuesof • < 0.02 eV, most,of the energyabsorbedby the electrons excites the low-lying vibrational levels of nitrogen, and emissionsin the visiblerangecannot be excited. For 0.02 eV< F < 0.1, eV the electron energy results in excitation of optical emissionsand electron-neutral collision frequency, due to the non- molecular dissociation. For F > 0.1 eV, ionization is Maxwellian initiated. For simplicity we are going to considerthe nature of the electron distribution func- tion that the fields generate. 3.2. Electron Distribution Fokker-Planck case in which the electric and threshold, i.e., F < 0.1 eV; otherwise a self-consistent equation for the electron density in space must. be included in the analysis. the Approach The electron distribution function in the presence of an electric field is strongly non-Maxwellian, requiring a. kinetic treatment. The kinetic treat.ment will provide •,• = •'e(z, I•l) •,nd the excitation rates of the different electronic levels. We use an existing Fokker-Planck code, which has been developed for the descriptionof ionosphericR.F breakdown[Tsang ½tal., 1991;Papadopoulos et al., 1993]. For givenvalues of E and % (the ambient collisionalfrequency), the electron distribution function f(v) is found by solving numerically the Fokker-Planck equation Of 1 0 field is below the ionization Of at 3mv2 Ov(V•"•,(v)F(E,•,(v))•v ) - œ(f),(9) where 4. Intensity and Structure of Sprites Given the spatioten•poral field profile including self-absorption, we consider the optical emissions,or red sprites, from N2. We discussonly the emissions of the N,•(1P) band causedby the excitation of the Bal-Ivelectronic levelof molecular nitrogenby electron impact. This band dominates in the spectrum of red sprites[Mende et al., 199,5].Here we compute the excitation rate %•, • of the N_,(BaII•)electronic level, which has an excitation energy of 7.35 eV and a lifetime of 8 p.sec,by using the Fokker-Planckcode for a given field strength. The excitation rate per electron is then given by v is the quiver energy or the kinetic electron energy where•r•. is the excitationcrosssectionand .Nk•2 in an oscillatingelectric field, u(v) is the electron- is the number density of N2 [Milikh et al., 1997]. neutral effective collisional frequency, œ is the operator which describes the effect of the inelastic col- lisions[Tsang et al., 1991], and 0o is the angle be- The excitations are then followed by optical emissions, and the number of photons emitted per sec- ond per cubiccentimeteris givenby •:•.n• for an tween the electric and magnetic fields. We have electron density n,•. In order to compare with obserassumed that the frequency of the electromagnetic vations, it is convenientto averagein time the numfields satisfies w << •, %. We have also assumed ber of photons over the duration of the discharge that (•/•,)= cos• 0o< 1, whichis the lessenerget- T (approximately milliseconds), i.e., < •,fl•,,• >= VALDIVIA ET AL.' MODEL .• lie.Br l•,edt/T. Theintensity oftheradiative transition in Rayleighsis given by 10-6 / ( vc,•n• B )dl ' (12) I=4•' where the integral is carried alongthe visual path of the detector (column integrated). In order to apply our model, we must specifythe current amplitude Io, along the discharge,the velocity of propagationof the discharge/3 = v/c and the electron density profile in the lower ionosphere. From here on we shall assume a midlatitude nighttime electrondensityprofile[Gurcvich,1978]. The fractal model is especiallysuitable for understanding the dependenceon the dimensionof the discharge[Valdivia et al., 1997],sinceD(q) can be easily parametrized, as is plotted in Figure 6. We proceednext, to determine t,he spatial structure of the optical emissionsas a function of the dimen- OF RED SPRITES 1663 sion D. We consider the four fractal dischargesq = l, 2, 3, and cx:•shown in Figure 7 with dimensions D •_ 1.5, 1.35, 1.25, and 1.0, respectively, where the thickness of the line correspondsto the strength of the current. In general, the electric field and emissionpattern induced by the fractal dischargesin the ionosphere are three-dimensional(3-D), but for the purposesof illustration we take a two-dimensional (e.g., the line y = l0 kin from the center of the discharge) of this 3-D profile in the ionosphere. The emissionpatterns along the crosssection y = 10 kin, averaged over the duration of the discharge using Equation (12), are shown in Figure 8 for the fractal dischargecorrespondingto q = 3. The velocity of the dischargewas taken as • = 0.025 and the amount of current was taken as Io = 200 kA. The emission rate, i.e., number of photons per cubic centimeter per second, is computed in decibels with respect to the averagedemissionrate over the image area. The ,-f-/= r---• lO -•- i ß ß cross section i 2.0 D= 1 ..5 lO -lO lO 10 - 5 0 X ,c/---- 5 o 10 5 ,5.o o X (km) D= 5 lO 5 lO (km) F)• I .2 1 .0 lO lO o --._% --10 , -lO , • --5 o X (km) , , • 5 .... --10 lO -- o --5 o X (km) Figure 7. The fractal structuresfor dimensionsD= 1.5, 1.35, 1.25 and 1.0 (q - 1, 2, 3, and •.). The thicknessof the linescorrespondsto the current.strength, and current conservation has been satisfied at each branchingpoint. 1664 VALDIVIA ET AL.' MODEL OF RED SPRITES peak emissionintensity for an optimal column integration is about 100 kR. The emissionintensity dependson the type of dis- lOO charge. The maximum intensit, y in kilorayleighsfor a optima,1column integration along the x axis is shown in Figure 9 as a function of t,he dimensionof the discha,rgeD(q). Since the optical emissionintensity is ext,remely sensitive to the power density, a fa,ctor of 2 on the electric field strength can have profound effectson the emissionpattern of a given fractal dis- 9o 8o <1> 70 - 116.o charge. kR As seen in Figure 9, different fracta,ls require different current peaks (or propagationspeed) to pro- mclx I /• - 2.0 x100kA - 0.025 60 , 7 -100 -50 o. 1o. 2o. 3o. 40. n• ""(dB) 3.0 0 50 100 x Figure 8. The time-averagedemissionpattern for q: 3. The temporal emissionpattern has been time averagedfor about a millisecond(dura,tionof sprite). The column-integra,tedemissionintensity was about 100 kR for an optimal optical path. duce similar emissions intensities. For the four frac- ta,ls of Figure 7 we find the necessary currentpeak Io neededto produce an emissionintensity of about 100 kR. The correspondingemissionpatterns are shown in Figure 10 with their peak current Io, which is related to the total charge dischargeas seen above. We note that the emissionpattern correspondingt,o i,he fra,ctal q - 3 has considera, ble spatial structure as comparedwith the or,her casesin the figure. We see that by having a spa, tia,lly structured radiation pa,ttern,the fracta,lscan increasethe powerdensity 120 • = o.o2_• ---- 200 _ 100 200 kA 80 6O 40 -! 2O 0 I L 10 1 20 I ..30 1 .40 , 1 50 1 •,0 D Figure 9. The ma,ximun•intensityin kilorayleighs as a fimctionof the dilnension. The gra,ph hasbeeninterpolated by a cubicspline,but theactualpointsareshownby asterisks. VALDIVIA ß 100 ET AL.: MODEL OF RED SPRITES 1665 100 . = 10.0 = x100kA 90 90 80 80 10.0 x100kA . 70 70 D=1.3• D= 1.50• 60 60 o 5o oo . -50 -lOO i 100 100 2.0 0 ß ß -- x 100kA . i 50 ß 25.0 oo i ß ß ß x100kA o 90 90 80 80 70 70 D= 1.00• D=1.2• 60 - 1 oo - 5o o 5o 1 oo - 1 oo - so X (KM) /3- 0.0 2 5 n- o 5o • oo X (KM) 2 0 0 o. 10. '"'"'"""" '"' 20. N7 (dB) 30. 40. Figure 10. The emissions patternsat the liney-10 km corresponding to the fourfractal structuresof Figure7 whichhavedimensions D "• 1.5, 1.35, 1.25,and 1.0. The currentis chosenso that the peak emissionintensityis about,I(kR)_•100 kR, and/3- 0.025. locally in specificregionsof the ionosphereand generate considerableoptical emissionswith relatively low (lnore realistic) lightning dischargeparameters. 5. Conclusions and Discussions the lightning dischargepropagatesalong the spider channel. Evidence for these types of models has been obtaiued from a set of measurements of the proper- ties of the dischargesin correlation with the sprites. Red spritesseemto be uniquelycorrelatedwith +CG discharges,but only someof the +CG dischargesactually generate sprites. Time correlation studiesof the time delay between the +CG events and the associated sprite have shown that it can reach more A novel model (Figure 1) for the formation of red sprites, which incorporates the fracta.1nature of the horizontal lightning dischargesor of a spider lightning type, was presented. The fractal structure of than tens, and sometimes hundreds, of milliseconds [Lyons,1996],suggestiveof the time delay required the discharge is reflected in the subsequent optical elnission pattern. Such a model fits the qua.1- to developthe horizontal intracloud fractal discharge. ita.t.ive model for the generation of red sprites by As mentionedby Lyons [1996]the sprite-generating L:_qon. s [1996], which is basedon the fact that hor- storms seem to have dimensions in excess of 100 km. izontal dischargesof the order of 100 km have been observedin connection with positive cloud-to-groud (+C.G) events. The model starts with the initial spider lightning followed by the positive leader toward the ground, which, in turn, is followed by the positive return stroke. The latter acts as a charge put, in the center of a Lichtenberg-like figure, i.e., Such large sizesare also required for the generation of the long horizontal discharges.The similarity between lightning dischargesand dielectric breakdown helps to understand the role played by +CG dischargesin red sprite generation. Surface dielectric breakdown develops a much more intense structure if caused by an immerse positive needle rather than 1666 VALDIVIA ET AL.' MODEL by a negativeone, as revealedby the Lichtenbergfiguresgivenby A tten and Saker [1993]. A simplephysical explanation of this effect is that the positively chargedneedlepulls electronsfroin the inedia, which is more energeticallyefficientthan pushingelectrons ejected by the negativelychargedneedle. Similarly, the intracloud lightning dischargewill acquirea bett.er developedfractal structure if causedby a positive rather than a negativereturn stroke. For an optimal configuration,so that, the fieldsget projected upward, the lightning dischargemust be horizontal, i.e., the so-called intracloud lightning or OF RED SPRITES for Io > 200 kA the equation for the evolution of the electrondensity n½shouldbe included). It can be observedfrom our model of sprites that the main body of the sprites is constrained between 80 and 90 km in height. The latest observations of sprites reveal filaments that can be described as streamers[CuremetandInan, 1997]propagating down from the main body of the sprite with a cross-sectional diameter of 100 m or less. Given the nucleated spatial structure in the conductivity produced by the fractal lightning discharge,the streamerswould start naturally in the presenceof a laminar field. The lain"spiderlightning"[L,qous, 1994].Accordingto L.qons inar field includesthe field induced in the ionosphere [1996],lightningdischarges havinga peakcurrentof during the cloud charging process. Also, some efup to 200 kA are associatedwith sprites. We assume fect might come froin the near-zone field described that a fraction of this current will be deflected from in equa, tion (14). Therefore a comprehensivemodel the cloud back into the lightning channel, while ap- of sprites,which includesthe main body producedby proximately ha.lf of the peak current,will go to the inthe fractal lightning and the subsequentstreamerdetracloud discharge.Thus the intracloudcurrent peak velopment, has to be developed. It, involvesboth the could reach almost 100 kA. Statistics of the speed of laminar and electromagneticlightning-inducedfields propagationfor intracloud lightning are incomplete. and their effects in the lower ionospherecan be deThe propagation speedduring a cloud-to-groundre- veloped from a model that solvesthe nonlinear wave turn stroke can reach speeds of about • • 0.1-0.5 equation, equation (7), and includesionization and [•man, 1987], while the propagationspeedof intr- charge separation. Such streamer concept.naturally acloud dischargesis at least an order of magnitude allowsfor the expansionof the heated region (of the lower, and hence we take ,d • 0.01-0.05. sprite) to a wider range in heights. Certain fractals can radiate more effectively than others, but, in general, this problem is very com- Appendix' Electric Fields From plicated. The power density, and thus the emis- the Fractal sion pattern and intensity,scalesas S(W/m•) • 32I• f(D)g(0) in the far field of the smallline el- ements. For a specificdimension,i.e., D(•/ = 3) _• 1.25, we obtained a.nemissionintensity of about 100 kR, with d = 0.025 and Io = 200 kA. However, using the above scalingfor the radiated field, we can generate a similar radiation pattern with /• = 0.05 and Io = 100 kA. The optical emission pattern depends on the structure of the discharge,but we conjecture that the most relevant structural parameter in deter- mining the spatial structure of the emissionsis the dimension of the self-similar fractal. Even though we live in a world where dielectric dischargesseem to have D • 1.6 [Sander, 1986; Nicmcyer ½t al., 1984], lightning dischargesseemto show lower di- A current pulse propagateswith speed/3- v/c along a fractal structure. At. the nth line element with orientation L• and length L,,, which is paramet- rized by 1 ½[0, L,,], the currentis givenby J, (s,,l, t) = I,, I(t-s,, +l/v)(the time dependence I(t)is given by equation4), where.s• is the path length alongthe fractal (or if you prefer, a phase shift.). The radiation field is the superposition, with the respective phases, of the small line current. elements that form the fractal. For a.set.{rn, Ln, I,,, SnI n - 0, ..., N} of line elements, the Fourier transformed hertz vec.tor is given by II(x,w) y•, if..jfo L" J.(s. 1w)eikllx-•"- mensions,a fact that lnight becomerelevant due to -• '' ^ dl, the sensitivity of the emissionstrength on the ffactal {n} [[x- rn-1Ln II dimension of the discharge. The optimal emissions intensity is obtained for dimensionsD ___1.25 for the a.sa solution to Maxwell's equations. Here I,• is the fractal model used above. In this case, the ionization strength of the current at, the nth line element, s• is starts occurring for Io > 200 kA and .3 = 0.02,5(i.e., the path length along the fractal, r,, is the position VALDIVIA ET AL.' MODEL of theelement, J, (s,, l, w) - I, I(w)ei •-(sn+l) is the Fourier-transformedcurrent strength at the nth line element, x is the position at which we measure the fields, w is the frequency,and k - •-. Values with the ½ circumfiexindicateunit vectors,and d,-]] x- r,• ]] means the standa. rd distance from the line element to the field position. To simplify the notation, we will denote d,• = x - rn. Note that, in general, the size of a single element L,• 100 m is much smaller than the distance to the ionosphere d, •, 80 kin, i.e., L, << d, Therefore we can use the far-field approximation of the small elements to carry the above integral. Of course, the nonuniformity in the radiation pattern will be due to the phase coherence,or interference pattern, betweenthe differentelementsthat form the fra.ctal. The electric field is then constructed E(x,w) - X7 x X7 x II(x,w). Fourier transform from We then invert the of the field to real time and rain the spatiotemporal radiation pattern due to the fractal dischargestructure, which is given by OF RED SPRITES Ln E(x, t)- y•.cd•(1 -/3(•7•1•)) 1667 (15) Acknowledgments. The work was supportedby NSF grant ATM 9422594. We expressour gratitude to A. V. Gurevich, A. S. Sharma, and V. A. Rakov for enlightening discussions.We also thank one of the referees,who gave us very helpful comments. References Atten, P., and A. Saker, Streamer propagationover a liquid/solid interface, IEEE Trans. El½ctr. Insul. 28, 230, 1993. Allain, C., and M. Cloitre, Spatial spectrum of a general family of self-similar arrays, Phys. Rev. A, 36, 5751-5757, 1987. Boeck, W. L., O. H. Va.ughanJr., R. 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D. and E. M. Wescott, Observations of upper atmospheric optical flashesrecordedfrom a.n aircraft, Geophys. Res. Lctt., 20, 2857-2860, 1993. Sentman, D. D., E. M. Wescott, D. L. Osborne, D. L. Hampton, and M. J. Hea.vner, Preliminary re- G..M Milikh, Departmentof Astronomy,University of Maryland, CollegePark, MD 20742. K. Papa.dopoulos,Department of Physics,Universit,.x.• of Maryland, CollegePark, MD 20742. J. A. Valdivia, NRCNASAGoddard Space Flight Center,Code692,Greenbelt, MD 20771.(e-mail' alejo•roselott.gsfc.nasa.gov) (Received June 5, 1997; revised June 11, 1998; acceptedJune 30, 1998.)