JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. A5, 10.1029/2001JA900117, 2002

Exponential relaxation of optical emissions in sprites

Christopher P. Barrington-Leigh

Space Sciences Laboratory, University of California, Berkeley, CA

Victor P. PaskoCommunications and Space Sciences Laboratory, Pennsylvania State University, University Park,PA

Umran S. Inan

Space, Telecommunications, and Radioscience Laboratory, Stanford University, CA

Abstract. The optical emissions in a large number of bright sprites observedover one storm in 1998 exhibit a relaxation that is closely exponential intime. This feature was unexpected but might be explained by the presenceof quasi-constant electric fields over times of several milliseconds, in whichcase the optical relaxation would be a direct indication of the exponentiallychanging electron density. The relaxation rates for sprites appear to havean upper bound that is consistent with the dissociative electron attachmentrates expected at sprite altitudes. The experimental results are consistentwith existing large scale electrodynamic models of sprites as well as with thestreamer mechanism as the underlying physical process for sprite ionization.

Introduction

A photometric array named the Fly’s Eye was previ-ously used to detect a predicted signature of “elves”,the lower ionospheric (80 to 95 km altitude) opticalflash due to heating by an impinging electromagneticpulse launched by intense lightning currents [Inan et al.,1997; Barrington-Leigh and Inan, 1999; Barrington-Leigh et al., 2001]. Nine narrow individual photome-ter fields-of-view of (2.2◦×1.1◦) are horizontally ar-rayed to provide a spatial resolution of ∼20 km ata range of 500 km. The Fly’s Eye array, operatedwith a high-speed triggered data acquisition system,bore-sighted image-intensified CCD video camera, anda VLF radio receiver, was also used to investigate sprites[Barrington-Leigh et al., 1999, 2001]. Sprites are diffuseto highly structured discharges lasting 5 to 100 ms andextending from 40 to 85 km altitude which result fromintense electric fields following a major redistributionof electric charge in the troposphere — usually in as-sociation with a positive cloud-to-ground return stroke.Sprites may involve or consist of a brief diffuse lumi-nous region above ∼70 km altitude [Pasko et al., 1998a]

named the sprite “halo” [Barrington-Leigh et al., 2001]but typically include columnar or filamentary opticalstructures thought to correspond to the propagationpath of corona streamers [Pasko et al., 1998a; Stanleyet al., 1999].

Several criteria used for the identification of elvesin narrow field-of-view photometers are described byBarrington-Leigh and Inan [1999]. One additional cri-terion not mentioned there is the fast relaxation timescale (<100 µs) which is often a characteristic of opti-cal pulses due to elves. Such fast relaxation, observed innarrow field of view photometers, is not typical for scat-tered light from lightning [Thomason and Krider , 1982;Guo and Krider , 1982], and observations outlined belowshow that it is also generally not observed for sprites.

On the night of 19 July 1998 a large mesoscale con-vective system over northwestern Mexico produced ex-ceptionally bright sprites. Measurements were madefrom the Langmuir Laboratory for Atmospheric Re-search (107.19◦W × 33.98◦N × 3200 m) in New Mexicousing the Fly’s Eye camera, optical array, and VLF re-ceiver. In addition, many sprites were bright enough tobe visible to the unaided and unadapted eye.

1

BARRINGTON-LEIGH ET AL.: Exponential relaxation in sprites 2

Determination of total sprite luminosity lifetimes hasgenerally been challenging [Rairden and Mende, 1995;Winckler et al., 1996]. Video recordings give generallypoor (>1 ms) time resolution and some systems, suchas the image intensifier of the Fly’s Eye video, exhibit aphosphor persistence following intensely bright signals.This afterglow may last for several frames, making ourinstrument unreliable for quantifying long sprite dura-tions. On the other hand, photometers designed withhigh time resolution exhibit no such “ghosting” but,because they rely on the photoelectric effect, are notoptimized for measurement in the near infrared regionof the spectrum (as compared with band-gap materials)[Winckler et al., 1996]. Moreover, because a photome-ter field of view is typically large compared with animager pixel or with variation in sprite structure, thespatially-integrated background can make difficult thetask of measuring slowly-varying, weak signals. In theFly’s Eye photometers, the slow glow of typical spritesoften appears to decay gradually into the backgroundphotometer signal level, implying the existence of evenmore persistent, unresolved luminosity. Extra brightsprites facilitate the measurement of these longer timescales using the Fly’s Eye.

Sprites are known sometimes to occur well after (upto tens of milliseconds) an associated lightning returnstroke [Bell et al., 1998]. It has been proposed thatthis property may be due to slowly-varying currents,possibly undetectable by remote ELF radio measure-ments, which may be flowing along the ionized returnstroke channel or possibly horizontally within the thun-dercloud [Bell et al., 1998; Cummer et al., 1998; Cum-mer and Stanley , 1999; Johnson and Inan, 2000]. Insome cases, a series of (positive) cloud-to-ground dis-charges may occur sequentially over a large horizontaldistance within a fraction of a second (spider lightning),suggesting the existence of an expansive travelling net-work of intracloud currents [Lyons, 1996]. These eventsare typically accompanied by a series of sprites mirror-ing the propagation of the lightning below (“dancingsprites”). In such cases several sprites can occur withcontinuous luminosity over a large fraction of a secondand may appear to be associated with several lightningstrokes. This paradigm was typical for the sprites ob-served on 19 July 1998.

In the following sections, several notable features ofsprites are discussed in the context of the observationscarried out on 19 July 1998. The extra signal avail-able on this night may have highlighted some hard-to-observe (with instrumentation used so far) but com-mon features of sprites, or the observations may corre-

spond only to the special case of unusually intense ion-ization and emissions. Several multi-color photometryand imaging studies have suggested that the degree ofionization in sprites can vary greatly [Armstrong et al.,1998b, 2000], and that brighter sprites do not necessar-ily exhibit a higher fraction of emissions from ionizedstates [Heavner et al., 1998].

Time scales in sprite photometry

Figure 1 shows a sample photometric record of abright sprite cluster, and illustrates the existence ofmore than one time scale in sprites. The inset imageshows an intense sprite halo [Barrington-Leigh et al.,2001; Wescott et al., 2001] and bright patches nearits lower boundary at ∼75 km which appear to haveinitiated downward streamers in a manner similar tothat described by Stanley et al. [1999]. This eventis accompanied by the photometric signature of elves[Inan et al., 1997] in the full array of photometers (notshown). The record from a single narrow field-of-viewphotometer shows an optical pulse, initially due to elves,which becomes very bright and is protracted for >2 ms.This brightness is likely to be due to the sprite halo ev-ident in the video image. Approximately 6 ms after theevent onset a pulse with characteristic rise and fall timesboth of ∼2 ms appears and then relaxes into >50 ms ofless intense glow.

This example highlights several features of spriteswhich were frequently observed on 19 July 1998 and inthe course of the annual sprite campaigns conducted bythe authors. Many events exhibit a bright peak which isoften only a few ms in duration and tends to grow anddecay with similar time scales. In addition, overall pho-tometric durations much larger than 10 ms were foundto be normal on this day, in contrast to the observationsof Winckler et al. [1996].

Cummer and Stanley [1999] found that the peak inoptical intensity of sprites occurred after the propaga-tion of streamers to their lowest altitudes was complete.The same phenomenology is observed for the halo eventdetailed by Barrington-Leigh et al. [2001] and may beanalogous to the luminous return stroke of lightningfollowing the connection of a leader channel to ground.In this analogy, the slower sprite glow evident in Fig-ure 1 may correspond to lesser excitation of the channelduring the analogue of the continuing current phase inlightning.

Measurements from the Fly’s Eye’s video camera,ELF/VLF sferic receiver, and three photometers, aswell as from an ultra low frequency (ULF, <30 Hz)

BARRINGTON-LEIGH ET AL.: Exponential relaxation in sprites 3

-10 0 10 20 30 40 50 60 70

1

10

elves

sprite halo

symmetrical pulse (bright streamers)

slow sprite glow

Time (ms) after 04:38:27 UT

Bri

ghtn

ess

(MR

)

50

90 km

Figure 1. Photometric features of a bright sprite. The signal from one narrow field-of-view photometer is shownfor part of an event at 04:38:27 UT on 19 July 1998. This sprite occurred 0.5 s after a series of intense spritesassociated with a series of positive cloud-to-ground lightning discharges. The inset shows a video field of the spritefrom the Fly’s Eye camera, superposed by a box showing the approximate field-of-view of the photometer and ascale showing altitudes overlying a causative CG.

search coil (operated by Universitat Frankfurt), areshown in Figures 2 and 3.

Several notable features are apparent in the eventshown in Figure 2. Two cloud-to-ground dischargescause sprites exhibiting both short, bright features anda longer dimmer luminosity which is not well resolved bythe photometers. Based on the video images, it is likelythat this sprite sustained some luminosity during theentire time between the two lightning strokes. The ULFmagnetic field persists at a low level during the intervalbetween the lightning strokes (compare with the mag-netic field magnitude before the first stroke), indicatingthe existence of a vertical current flowing continuouslyfor >140 ms after the first stroke. Figure 3 shows theunusual fact that the brightening of the sprite in pho-tometer 11 appears to anticipate the onset of the secondcloud-to-ground discharge. Given the 31 km proximityof the two lightning strokes as reported by the NationalLightning Detection Network (NLDN), it is likely thatthey were closely coupled electrically through intracloudcurrents. The time scale for stepped leader breakdownis >10 ms altogether, and ∼1 ms for the final leaderpulse preceding the return stroke [Uman, 1987, p. 14-16], implying that the channel taken by the second re-turn stroke in Figure 2 must have been developing wellbefore the onset of the bright optical signature. Whilerecent results of Cummer and Fullekrug [2001] suggestthat inferred vertical current moments may be sufficientto account for sprite breakdown even in long-delayed

(>30 ms) sprites, the occurrence of a sprite just pre-ceding the return stroke in Figure 3 suggests that alarge (horizontal) charge motion within the cloud mayhave both led to a sprite and been involved in the ini-tiation of the return stroke, which would indicate thathorizontal currents might themselves sometimes excitesprites.

For reasons discussed elsewhere [Cummer et al., 1998;Johnson and Inan, 2000], it is difficult to determine ex-perimentally the relative contributions of vertical andhorizontal lightning currents in the production of meso-spheric electric fields. Discussion of sustained lightningsource currents throughout the rest of this paper couldin principle apply to intracloud charge redistribution orto cloud-to-ground currents.

Observations of exponential photometricdecay

While the relative timing of the sprite and lightningin Figure 3 is highly unusual, another remarkable fea-ture of this event is one that was commonly observed inphotometry of bright sprites on this day. The dashedlines superposed on the photometer traces in Figure 3show curves of the form

y(t) = C +Ae−t/τO (1)

fit to the data by choosing values of C, A, and τO. Thedecay of bright optical pulses due to sprites is found in

BARRINGTON-LEIGH ET AL.: Exponential relaxation in sprites 4

1

2

3

10

20

50Photometer 11

25 MR

-80 -60 -40 -20 0 20 40 60Time (ms) after 07:51:24 UT

Saturated at 13 MR

Photometer 5

Photometer 6

(kR

) (M

R)

(MR

)

10

1

10

1

ULF

(nT

) |Bh|

VLF/ELF

+CG+CG

P5 P6

P11

Figure 2. Slow sprite development and ULF currents on 6 August 1998.. The ULF magnetic field is obtainedby integrating in time data (provided by Martin Fullekrug) from a magnetic loop receiver. Photometers 5 and 6have narrow (2.2◦×1.1◦) fields-of-view, while the dimensions of Photometer 11 are three times as large. The videoimages are shown time-aligned to the plots. According to NLDN, the two lightning discharges were 31 km apart.

BARRINGTON-LEIGH ET AL.: Exponential relaxation in sprites 5

τ = 0.61 ms

τ = 0.83 ms

τ = 0.27 ms

15 20 25 30Time (ms) after 07:51:24 UT

τ = 0.37 ms

τ = 0.39 ms

1

2

3

10

20

50

(kR

) (M

R)

(MR

)

10

1

10

1

VLF/ELF

P 11

P 5

P 6

|Bh |ULF

(nT

)

P5 P6

P11

Figure 3. Sprite preceding cloud-to-ground lightning. A close-up of Figure 2.

BARRINGTON-LEIGH ET AL.: Exponential relaxation in sprites 6

a large number of cases to closely follow the exponentialform in (1) over several time constants τO, although thevalue of τO varies considerably between different opticalpeaks. This rather remarkable feature of sprite opticaldecay has only been mentioned for one case [Suszcynskyet al., 1998] but may be discernable in photometric dataof Winckler et al. [1996] and Fukunishi et al. [1996].

An electric field imposed on a uniform, weakly con-ductive medium by a rapid rearrangement of charges isexpected to decay exponentially in time [Pasko et al.,1997, 1999]. Indeed, the typical time scales τO for theobserved decay are comparable to the expected ambientelectric relaxation time constant τE = ε0/σ at the ob-served altitudes (Figure 4). Here, ε0 is the permittivityof free space and σ is the local conductivity. However,the optical emissions should not relax exponentially insuch a case because of their highly nonlinear depen-dence on the electric field strength. Figure 5 shows thiselectric field dependence of the excitation rates νk re-sponsible for the dominant optical emissions in sprites.The electric fields shown are normalized to the localconventional breakdown field Ek, which is proportionalto the local air density. Also shown are the coefficientsof ionization (νi) and electron dissociative attachment(νa), defined such that the electron density follows

dnedt

=(νi(E)− νa(E)

)ne. (2)

The optical emissions (i.e., number of photons emittedper unit time from unit volume) due to excited state kare then [e.g., Pasko et al., 1997]

Aknk ' νk(E)ne, (3)

where nk is the density of the excited state and Ak isthe radiation transition rate. This expression neglectsthe effect of quenching, which is negligible above 50 kmfor the dominant optical bands in sprites, and the ef-fect of cascading, which does not affect the argumentsthat follow. The approximation in (3) is valid becausethe radiative decay times for the relevant states are fast(<6 µs) in comparison with any changes in the elec-tron density. Thus, considering the form of νk shown inFigure 5, it may be seen that the optical emission ratedoes not vary in a simple way in an electric field whichis relaxing exponentially with time.

On the other hand, the observed photometric expo-nential relaxation would be obtained if we adopt thead hoc assumption that the electric field remains ap-proximately constant in time during the lifetime of thesprite. For E=E0=constant, Aknk ' νk(E0)ne(t) and

0.001 0.1 1 10 10030

40

50

60

70

80

90

Electric relaxation time (ms)

Alti

tude

(km

)

τ E

τamin

(E=0)

(E=E )k

Figure 4. Electric relaxation and attachment timescales as a function of altitude. τmin

a corresponds to thetwo-body dissociative attachment rate at E'0.8Ek; seeFigure 5. Values of τE = ε0/σ are calculated using anambient nighttime electron density.

we have from equation (2),

ne = ne

∣∣∣t=0

e[νi(E0)−νa(E0)]t, (4)

where the absolute value of νi− νa is shown in Figure 5and is also constant in time for E = E0. For E0 < Ekin a sprite, this condition would produce an exponentialrelaxation in the rate of optical emissions.

It may be cautioned that a wide variety of physi-cal systems may be well approximated by exponentialbehavior, sometimes due to statistical or geometric rea-sons rather than those relating to local physics. Forinstance, in the case of elves the temporal structureof optical emissions is locally determined by tempo-ral properties of the causative lightning pulse, and ata ground observer site is determined largely by geo-metrical considerations [Veronis et al., 1999]. Thesegeometrical effects can lead to an apparently closely-exponential relaxation of luminosity from the “back”part of elves both in theory and observations for thecase of a photometer with a field-of-view as large asthat of photometer 11 (6.6◦×3.3◦) in the Fly’s Eye.

Nevertheless, the exponential decay feature is foundin a majority of the bright sprites observed between04:00 and 06:00 UT on 19 July 1998 and often with amore exact fit than the cases shown in Figure 3. Fig-ure 6 shows values of the relaxation time constant τOdetermined via curve fitting to 255 measured photo-

BARRINGTON-LEIGH ET AL.: Exponential relaxation in sprites 7

νC3Πu

νi

νa

|νi − νa |

ionization dominates

attachment dominates

νB3Πg

0.2 10

1

102

103

104

105

106

ν(s-1

)

0.4 0.6 0.8 1 1.2 1.4 1.6 E/E

k

maximum in net ionization loss rate

Figure 5. Model ionization and dissociative attachment rates at 70 km altitude. Also shown are the electronimpact excitation coefficients νk for the states B3Πg and C3Πu producing optical emissions in the N2(1P) andN2(2P) bands, respectively. Each rate shown is proportional to atmospheric density, so values at 60 km are ∼3.5times higher than those shown (for 70 km). Data are based on the compilations and calculations of Pasko et al.[1999].

BARRINGTON-LEIGH ET AL.: Exponential relaxation in sprites 8

metric pulses exhibiting good to excellent closeness offit with equation (1). The photometric recordings weremade during 27 sprite sequences which triggered theFly’s Eye to record ∼1 second of data from the nine nar-row field-of-view, red-filtered photometers and a blue-filtered photometer with a larger field-of-view. The al-titudes corresponding to the narrow fields-of-view forthese observations were primarily in the range 60 km to85 km, with considerable uncertainty (±12 km) basedon the possible distance between the sprites and theobserver.

The curve fitting is done by finding a least squaresfit for periods chosen by hand to correspond well to adecaying exponential form. In some cases the initialperiod following a bright peak relaxes faster than theexponential fit, and not all of it is included. Instead,whenever possible, the fit period is chosen to includemany times the duration of τO so as to appropriatelyfix the value of C in equation (1) to the backgroundluminosity. The quality, or closeness, of fit is then as-sessed by comparing the values of log(y − C) from dataand fit using the linear correlation parameter R givenby Bevington and Robinson [1992, p. 199].

Also shown for reference are some time constants de-termined with the same algorithm and associated withoptical pulses from the same storm but which were de-termined to be due to elves, based on the criteria de-scribed by Barrington-Leigh and Inan [1999]. The ap-parent close fit in these cases, however, is less significantsince the parameter τO is barely resolved by the sam-ple period of the photometer. Nevertheless, the valuesof τO given for elves in Figure 6 do give an indicationof the time scales for the optical signals due to elvesviewed with a narrow field-of-view. The sample periodof the data is shown by a dashed vertical line.

It is apparent that while the instrument and methodare capable of resolving decay constants well below100 µs, and while the measured variation in τO extendsover nearly two orders of magnitude for sprites, a lowerlimit of ∼200 µs exists among the observed sprite cases.

Two more dashed vertical lines show the fastest rateconstant n−1

e (∂ne/∂t) = νa − νi expected at two dif-ferent altitudes in the regime where dissociative at-tachment dominates over ionization. As shown in Fig-ure 5, this maximum rate (νa − νi)max is reached atE ' 0.8Ek and is also the fastest optical relaxationthat is predicted for a constant electric field, accord-ing to equation (3). Figure 4 shows the variation ofτmina = 1/(νa − νi)max with altitude. The suggestion

that the observed optical relaxation time scale τO maybe bounded by τmin

a supports the assumption of an es-

sentially constant electric field during these times.

Model and Interpretation

It is remarkable that a constant electric field shouldarise so repeatably in dynamically driven sprite regions.Below we provide a simple physical explanation of thisphenomenon using a large scale electrodynamic modelof sprites [Pasko et al., 1998b] based on the interpre-tation of current and charge systems in a conductingatmosphere following a lightning discharge, originallyproposed by Greifinger and Greifinger [1976]. This dis-cussion will be followed by a consideration of the micro-scopic time dynamics of optical emissions arising fromindividual streamer channels constituting sprites, allow-ing direct comparisons with the observations reportedin the preceding section.

Steady electric fields in sprites

We adopt here a “moving capacitor plate model”of large scale charge and current systems associatedwith sprites, as used by Pasko et al. [1998b] and de-picted in Figure 7. The model defines a downward mov-ing boundary hi which separates regions of atmospheredominated by the conduction (above) and displacement(below) currents. In the case of atmospheric ionizationassociated with sprites, hi is closely related to the lowerextent of sprite ionization. It is generally believed thatthe sprite body is highly ionized. The spatially averagedsprite conductivities derived from ELF mesurements areon the order of 10−7 mho/m [e.g., Pasko et al., 1998b].We note that localized conductivity enhancements as-sociated with individual streamer channels are muchhigher (e.g., '5×10−5 mho/m at 70 km altitude [Paskoet al., 1998a]). The time dynamics of the electric fieldin this kind of highly conducting medium are defined bythe continuity equation ~∇ · ~J=0, where ~J is a currentflowing through the conductor, which is related to theelectric field ~E through Ohms law ~J = σ ~E. Assumingfor simplicity that only the vertical field component ispresent in sprites and that a sprite has an effective hor-izontal cross section area S one can rewrite the abovecontinuity equation in the form σES = Isprite, fromwhich it is clear that the nonzero electric field can per-sist only if there is a net current flow through the spritebody Isprite 6= 0.

The current Isprite flowing through the sprite bodycan be readily evaluated using the model depicted inFigure 7. We assume that the system is driven by thelightning current I(t) = dQ(t)/dt which effectively de-posits thundercloud charge Q(t) at the altitude hQ (see

BARRINGTON-LEIGH ET AL.: Exponential relaxation in sprites 9

1

0.70

0.75

0.80

0.85

0.90

0.95

Optical relaxation time constant10 ms1 ms100 µs10 µs

Fly’s Eye resolution (1998)

Maximum (ν -ν ) at 80 km

ia

Maximum (ν -ν ) at 70 km

ia

sprites (red) sprites (blue) elves

Qua

lity

of fi

t (R

)

Figure 6. Exponential decay times in sprites. Values shown as ×’s and ◦’s are from fits giving R > 0.7 using datafrom one or more red or blue filtered photometers in each of 27 events; each point corresponds to one photometerduring one optical pulse of a sprite. The ∗’s show the result of fits to photometric signatures of elves.

Alti

tude

0

hQ

hi(t)

−Q(t)

+Q(t)(1−hQ/hi(t))

+Q(t) hQ/hi(t)

-- ---- --- --- ---+ + ++ + + ++ + ++ +

+ + +

Isprite

Figure 7. Diagram of charge systems in sprites, afterPasko et al. [1998b].

Figure 7). The total charge which accumulates at thelower end of the sprite is

Qi(t) = Q(t)hQ/hi(t)

The current flowing through the sprite is therefore

Isprite(t) =dQidt

= I(t)hQhi(t)

− Q(t)hQh2i

dhidt

(5)

The first term here has to do with the portion of spritecurrent due to the source (lightning) current, and thesecond term is due to the descending lower edge ofthe sprite. The derivative dhi/dt is generally negativeand for estimates can be set to the velocity of positivestreamers (v=200–2000 km/s [e.g., Grange et al., 1995;Dhali and Williams, 1987; Babaeva and Naidis, 1997]).Given that the apparent vertical extent of sprites is typ-ically several tens of kilometers (see for example Fig-ure 1 and Stanley et al. [1999]; Gerken et al. [2000];Barrington-Leigh et al. [2001]) at this speed streamerscan propagate for several milliseconds. As will be illus-trated below, during these several milliseconds the elec-tric field remains quasi-constant inside the conductingmedium of the sprite, thus supporting the main argu-ment proposed earlier in this paper for the explanationof observed exponential optical relaxation. Substitutingspecific numbers (I ' 5 kA, Q '100 C, hQ = 10 km,hi '50 km) in the above formula indicates that both

BARRINGTON-LEIGH ET AL.: Exponential relaxation in sprites 10

terms can make comparable contributions to the spritecurrent, which appears to be ∼1 kA in this case.

Let us now separately consider two cases to illustratetwo physical means by which the electric field can beheld quasi-constant in the highly conducting mediumof a sprite. The first case corresponds to the situationwhen a steady current continues to flow between cloudand ground and the sprite is static in space, and thesecond case to a situation when the source current iszero, but the sprite extends downward with some nonzero velocity.

Case 1: A sprite is modeled as a cylinder with ra-dius Rs=50 km and conductivity σs=10−7 mho/m asderived from ELF measurements [Pasko et al., 1998b].The cylinder is assumed to be static and its lowerboundary is placed at hi=50 km altitude. The ambi-ent ionospheric/conductivity parameters are the sameas in Pasko et al. [1998b]. A steady current I=5kA flows between the cloud (at hQ=10 km) and theground, starting at time equal zero and continuing un-til the end of the simulation (5 ms). This circum-stance closely resembles lightning continuing currentsassociated with sprites recently reported by Cummerand Fullekrug [2001]. Since the sprite is assumedto be static, hi=const in (5) and only the first termcontributes to the sprite current Isprite = const =IhQ/hi=1 kA. We can easily evaluate the static elec-tric field which we expect inside the sprite in this caseas E = Isprite/(πR2

sσs)=1.3 V/m. This one dimensionalestimate is in excellent agreement with what we observein our two-dimensional numerical experiment for Case1 (see Figures 8 and 9). Figure 8 shows electric fields atthe end of the simulation (time =5 ms), and Figure 9shows the time dynamics of the electric field at altitude70 km in the center of the sprite (r=0).

Case 2: In this case we impose an impulse of cur-rent lasting ∼1 ms (rise time 0 sec, fall time 0.5 ms,peak current 225 kA) which deposits Q=100 C athQ=10 km. Then the charge Q remains constant andthe current is zero (I=0) until the end of the sim-ulation. Only the second term in (5) contributes tothe sprite current in this case. It is assumed that thesprite lower boundary begins at hi(0)=50 km and de-scends at a slowly varying rate. We choose the formhi(t) = hi(0)/(1 + Isprite

QhQhi(0)t), with which the sprite

current (in 1-D) defined by the second term in (5) willhave exactly the same constant value Isprite=1 kA as inthe previous case 1. In 5 ms the sprite lower boundaryextends down to 40 km, so the effective average spritevelocity is 2000 km/sec. The velocity of the sprite infact is time dependent, being 2500 km/s at t=0 and

0 40 80 120 160 200 r (km)

0

1

2

3

E (V

/m)

10-2 100 102 104 106

E (V/m)

z=70 km

Case 1

Case 2

020

40

60

80

Alti

tude

(km

)

Case 1

Case 2

r=0

020

40

60

80

Alti

tude

(km

)

Case 1 Case 2

E (V/m)10 10 10

r (km) 0 100 200200 100

r (km) 0 100 200200 100

-2 0 2

Figure 8. Spatial variation of the model electric fieldshown at the moment of time 5 ms.

1600 km/s at t=5 ms. There are some differences be-tween cases 1 and 2 observed in our numerical experi-ments (see Figures 8 and 9). These differences are ex-plained by the fact that our two-dimensional numericalmodel treats the sprite as a conducting cylinder withfinite lateral extent and thus contains the horizontalcomponent and the horizontal variation of the electricfield (as apparent in Figure 8). This horizontal vari-ation is not present in the simplified one-dimensionalanalytical model used to derive equation (5). The 1-Dmodel (5) nevertheless provides a good physical insightand serves for better understanding of more general 2-Dsimulation results presented in Figures 8 and 9.

These cases illustrate two different means by whichthe electric field can remain constant in the highly con-ducting medium of a sprite. Note that the dielectric re-laxation constant εo/σs is 88 µs inside our model spritebut the field can stay constant for many milliseconds.

In real sprites the total sprite current Isprite(t) iseffectively carried through the sprite by a number ofstreamer channels with relatively small cross section[e.g., Gerken et al., 2000]. In order to produce opti-cal emissions, electric fields inside these channels mustbe on the order of the conventional breakdown thresh-old field Ek (see Figure 5 and Pasko et al. [1998a]).The above modeling, which assumes that sprites havehomogeneous conductivity, certainly is not able to re-produce this aspect of sprites. Nevertheless, the physi-

BARRINGTON-LEIGH ET AL.: Exponential relaxation in sprites 11

0 1 2 3 4 510-1

100

101

Time (ms)

Ele

ctri

c fi

eld

(V/m

)

Case 1

Case 2

r=0, z=70 km

Figure 9. Temporal variation of the model electricfield.

cal reasons for the persistence of electric fields in highlyconducting streamer channels constituting real spritesare similar to those described for the two model casesconsidered above and have to do with either the exten-sion of streamers in space or the temporal growth ofthe ambient electric field, or a combination of these twoeffects.

Time dynamics of electron density and opticalemissions in streamer channels

We now use a simple microscopic model of an in-dividual streamer channel and demonstrate that theexponential optical relaxation is a robust feature ofthe streamer physics, given that the quasi-steady cur-rent is flowing through the streamer channel. In thismodel the streamer current is assumed to be a givenfunction of time which is dictated by the large scaleelectrodynamics of sprites as discussed in the previ-ous section. The streamer is a physical object whichcreates large electric field enhancements (an order ofmagnitude above the conventional breakdown thresh-old) leading to very high ionization rates around itstip, allowing the streamer to propagate long distancesin the form of a filamentary plasma. The physics ofstreamers has been extensively studied and documented[e.g., Dhali and Williams, 1987; Vitello et al., 1994;Pasko et al., 1998a]. For short streamers (i.e., ∼1 cmat ground pressure) the electric field value inside thestreamer channel behind the streamer tip depends onthe applied external field. For streamers which propa-gate in external fields above the breakdown threshold

this value is usually close to the breakdown field (i.e.,∼30 kV/cm for air at ground pressure) with a generaltendency to increase with increased external field [e.g.,Dhali and Williams, 1987; Vitello et al., 1994; Babaevaand Naidis, 1997; Kulikovsky , 1997]. Streamers can alsopropagate in very low external fields, a factor of 6 be-low the breakdown threshold [Allen and Ghaffar , 1995,and references therein]. In this case the electric fieldvalues inside the streamer channel are also very low, onthe order of 5 kV/cm at ground pressure [e.g., Grangeet al., 1995; Morrow and Lowke, 1997], and are not suf-ficient to produce any significant optical emissions (seefor example Figure 5). We assume that bright streamerchannels observed in the upper portions of sprites andpresumably responsible for the observed optical signa-tures reported in this paper are produced by strong ex-ternal electric fields above the conventional breakdownthreshold which therefore create large electric fields in-side streamer channels.

It is usually assumed that the loss of electrons insidea streamer channel due to dissociative attachment is notan important process for propagation of short stream-ers in air (∼1 cm at ground pressure) [e.g., Babaeva andNaidis, 1997]. The distance 1 cm scales to only '40meters at 60 km altitude in accordance with similaritylaws derived by Pasko et al. [1998a]. Assuming that thestreamer propagates with the highest speed measuredin sprites, ∼107 m/s [Stanley et al., 1999], it would takeit ∼2 ms to travel from 60 km to 80 km altitude. Thecharacteristic time scale of dissociative attachment ofelectrons at 60 km is '0.03 ms and '0.1 ms at 70 km(see Figure 5). Since the time scale of the dissocia-tive attachment process is much smaller than the timescale of propagation of streamers, we expect that dis-sociative attachment for the long streamers observed insprites is an important process which is able to affectthe streamer internal ionization balance and to controlits dynamics. It is expected in particular that the elec-tric field inside the streamer channel stays very close tothe conventional breakdown field Ek, at which electronattachment is equalized by ionization (see Figure 5).If the field deviates to lower values the attachment ofelectrons is enhanced, reducing the conductivity of thechannel. Since the current should be continuous alongthe streamer channel the conductivity reduction leadsto a self-consistent increase in the electric field. If thefield deviates to higher values, additional ionization isproduced and leads to the enhancement of conductiv-ity; thus the field is self-consistently reduced to main-tain continuity of current. The electric field thereforestays at the equilibrium point E = Ek. In our modeling

BARRINGTON-LEIGH ET AL.: Exponential relaxation in sprites 12

below we therefore assume that immediately after thepassage of the streamer head we have equal amountsof electrons and positive ions and that E = Ek in thestreamer channel. The time duration of our modelingwill be limited to several ms and therefore we accountonly for the relatively fast processes of electron impactionization and electron dissociative attachment, ignor-ing effects of electron and ion recombination and threebody attachment processes [e.g., Glukhov et al., 1992,and references therein].

We solve the following set of dynamic equations fornumber densities of electrons ne, positive n+ and neg-ative n− ions:

dnedt

=(νi(E)− νa(E)

)ne (6)

dn+

dt= νi(E)ne (7)

n− = n+ − ne (8)

At the initial moment of time t=0 it is assumedthat ne=n+=ne,0, and n−=0, where ne,0 is the electronnumber density of stable streamers derived from simi-larity laws (ne,0=5.5×106 cm−3 at 60 km and ne,0=4.6×105cm−3

at 70 km [Pasko et al., 1998a]). The electric field E inthe above equations is assumed to be a function of timedetermined by current continuity as

E =J(t)σ

=J(t)

eneµe + e(n+ + n−)µi(9)

where µe and µi are mobilities of electrons and ionswhich in the present model are assumed to be con-stants (µe=656 m2/V/s, µi=3.4 m2/V/s at 70 km, andµe=189 m2/V/s, µi=0.98 m2/V/s at 60 km [Paskoet al., 1997]). The time dependent streamer currentdensity J(t) is defined at time t = 0 as J(0) = Jo =(ene,0µe + ene,0µi)Ek (Ek=761 V/m and 219 V/m ataltitudes 60 and 70 km, respectively), and then assumedto decay as J(t) = JoτI/(t+τI). The characteristic timescale τI of current decay in the sprite is considered asthe only input parameter in our model, with all the restcalculated self-consistently. The functional dependenceof νi and νa on electric field depicted in Figure 5 is usedin our calculations.

An interesting aspect of the system which we con-sider is that even under conditions when E = Ek andne is expected to be constant (νi(Ek) = νa(Ek)), bothpositive and negative ions accumulate in the streamerbody with their densities linearly increasing in time (asis obvious from equations (7) and (8)). The mobility ofions is only∼200 times lower than that of electrons so as

10-4 10-3 10-2 10-1 1000

2

4

6

8

10

τI(s)

60 km

70 km

Opt

ical

rela

xatio

n tim

e (m

s)Figure 11. Modeled dependence of τO, the opticalrelaxation time constant, on τI , the time scale of changein the sprite current.

soon as the concentrations of ions reach ∼100 times thatof electrons (assuming equal concentrations of negativeand positive ions — see further discussion and resultspresented in Figure 10a) they become the dominant car-riers of the current in the streamer.

Before considering the results of numerical calcula-tions let us qualitatively inspect a case when J(t)=Jo=const.The above equation (9) for the electric field can be re-arranged as(

eneµe + e(n+ + n−)µi)E = Jo (10)

Even a small increase in ion concentration would leadto an increase in the ion conductivity contribution to Joand therefore to a drop in the E field. However, even asmall reduction in the electric field leads to exponentialrelaxation of the electron density, which quickly com-pensates for the increase in the ion conductivity. Aremarkable feature of this system is that the electrondensity relaxes exponentially in time, while the elec-tric field is reduced by a very small amount (specificnumbers will be specified below) and essentially remainsconstant in time with high accuracy.

The current J(t) is not expected to be constant insprites. In Figure 10 we show results of numerical so-lutions of the above dynamic system for different τI .Figure 10a shows an example of the time dynamics ofelectrons and ions at 70 km altitude corresponding to acase τI=1 ms. The electron dynamics are represented

BARRINGTON-LEIGH ET AL.: Exponential relaxation in sprites 13

0 1 2 3 4 5104

105

106

107

Time (ms)

Num

ber d

ensi

ty (c

m

)−3

(a)

ne

ni

ni

+

−

0 1 2 3 4 510-7

10-6

10-5

10-4

Con

duct

ivity

(mho

/m)

Time (ms)

(b)

σi σe

σtot

0 1 2 3 4 50

2

4

6

8

10

Opt

ical

inte

nsity

(A

rb. u

nits

)

Time (ms)

model result

exponential fit (1.53 ms)

(d)0 1 2 3 4 5200

210

220

230

(c)

Ele

ctri

c fi

eld

(V/m

)

Ek

E

Time (ms)

Figure 10. Numerical model of optical relaxation. (a) Electron and ion densities for τI=1 ms; (b) the correspondingconductivities; (c) the corresponding electric field; and (d) resulting optical emissions.

BARRINGTON-LEIGH ET AL.: Exponential relaxation in sprites 14

by a straight line on a log-linear plot, thus showing ex-ponential relaxation. Densities of positive and negativeions increase linearly with time and the ion conductivitydominates the total conductivity in the streamer chan-nel after t ∼3 ms (Figure 10b). The maximum variationin the magnitude of the electric field in this case (Fig-ure 10c) does not exceed 2.1%. The electron densityexhibits exponential relaxation with time scale 1.53 ms,as does the optical intensity, shown in Figure 10d. Fig-ure 11 provides a summary of all calculated optical re-laxation time scales obtained with the model for alti-tudes 60 and 70 km for τI changing between 0.1 msand 1 second. No changes were observed for τI >1 sec,which is simply a limiting case of constant J(t)=Jo. Itis truly remarkable that model relaxation times stay inthe range 0.2–10 ms for so wide a range (4 decades) ofvariation in time scales of sprite current. Our modelingindicates that the electric field tends to stay within sev-eral percent of Ek so that optical relaxation does notnecessarily proceed at each altitude with the maximumpossible rate, which is realized at E = 0.8Ek as shownin Figure 5.

We believe that these results provide a simple andstraightforward explanation for the experimentally ob-served exponential optical relaxation in sprites reportedin this paper.

Discussion

The vertical and possibly horizontal lightning cur-rents evidenced in Figure 3 might each serve as exam-ples of the source I(t) of equation (5). However, theexponential relaxation observed in sprites is explainedhere to be a sign of a slowly varying sprite conductiv-ity coupled with any combination of the two terms onthe right hand side of equation (5), relating to eithera sustained lightning source term or to the descent ofstreamers in the sprite.

The exponential relaxation of optical signatures re-ported here was also observed to be a feature of brightsprites in Fly’s Eye data on other days in 1997 and1998. Determining the degree of ionization in spriteshas been a recurring theme and point of debate overthe last decade, largely due to the difficulty of measur-ing optical output from emission-lines of ionized molec-ular states. However, based on recent telescopic imag-ing of sprite fine structure [Gerken et al., 2000] andon the modern understanding of the streamer break-down process [Pasko et al., 1998a, 2000; Raizer et al.,1998] it seems incontrovertible that intense ionizationoccurs when filamentary structure is seen in sprites,

although this ionization may occur only briefly [e.g.,Armstrong et al., 1998a, 2000]. Recently, Barrington-Leigh et al. [2001] showed that with the help of model-ing, upwardly-concave curvature in sprite halos can beinterpreted as a sign of intense ionization augmentingconductivity gradients and preferentially expelling theelectric field. The exponential relaxation of sprite lumi-nosity described in this work also appears to manifestelectron density changes, typically over more than oneorder of magnitude.

In addition, the persistence of electric fields nearEk gives reason to predict a characteristic post-onsetmean electron energy for luminous sprite regions of<3 eV [Pasko et al., 1997, 1999], after possibly moreenergetic initial processes. This is roughly consistentwith spectrally-determined time-averaged electron en-ergies of 1–2 eV measured in sprites [Green et al., 1996;Morrill et al., 1998; Green et al., 1998]. In principle,knowledge of the electric field or mean electron energyderived from the exponential relaxation time of opticalemissions at a very precisely known altitude could becombined with calibrated photometric intensity to de-termine the line-integrated absolute electron density inthe sprite, by making use of equation (3) and Figure 5.Such a measurement would, however, require tight con-straints on the vertical field of view and on knowledgeof the location of the optical source.

Summary

Quasi-steady currents in sprites are described interms of two large scale physical processes – the slowvariation of either the lightning source current or thedescent of the sprite’s lower edge. In turn, sprite cur-rents which vary slowly compared with the time scaleof electron dissociative attachment to O2 lead to quasi-constant total conductivity, even while the electron con-tribution to conductivity is decreasing. This results in aproperty of streamers driven by strong external electricfields, namely, that the local electric field is maintainedself-consistently near the breakdown electric field, Ek.

A quasi-constant electric field just below Ek leadsstraightforwardly to an exponential relaxation of theelectron density and the optical emission rate in sprites.According to our model total ion densities become sig-nificant and may dominate the conductivity. Expo-nential decay of photometric features associated withsprites has been observed in a number of bright spritesand appears to be a form of non-spectroscopic evidencefor large ionization changes in sprites. The observedphotometric behavior is consistent with the streamer

BARRINGTON-LEIGH ET AL.: Exponential relaxation in sprites 15

mechanism as the underlying physical process for spriteionization. In future sprite measurements, more atten-tion could be paid to this phenomenon, in particular inthe context of multicolor photometry and photometrywith fine altitude discrimination.

Acknowledgments. We thank Ken Cummins of GlobalAtmospherics Inc. for provision of NLDN data, Rick Rairdenfor the use of an intensified video camera, Langmuir Lab-oratory for the use of their facilities, Martin Fullekrug forsupplying ULF data, and a referee for useful suggestions.This work was supported by the Office of Naval Researchunder grant N00014-94-1-0100 and by the National ScienceFoundation under grant NSF-ATM-9731170. Participationof Victor Pasko was supported by the Electrical Engineeringdepartment of Penn State University.

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C. P. Barrington-Leigh, Space Sciences Laboratory,University of California, Berkeley, CA 94720 (e-mail:cpbl@ssl.berkeley.edu)

V. P. Pasko, Communications and Space SciencesLaboratory, Pennsylvania State University, UniversityPark, PA 16802 (email: vpasko@psu.edu)

U. S. Inan, STAR Laboratory, Stanford University,Stanford, CA 94305. (e-mail: inan@nova.stanford.edu)Received 27 March 2001; revised 9 July 2001; accepted 14 July2001.

This preprint was prepared with AGU’s LATEX macros v5.01,

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from 1999/08/19.