LETTERSPUBLISHED ONLINE: 13 NOVEMBER 2011 | DOI: 10.1038/NGEO1314

Mesospheric electric breakdown and delayedsprite ignition caused by electron detachmentA. Luque* and F. J. Gordillo-Vázquez

The electric discharge of a thundercloud in the troposphere isoften accompanied by upper-atmospheric electric dischargessuch as sprites1,2 or halos3. Present models assume that the netchange in the density of free electrons during these dischargesis determined by the competition between electron impactionization and electron dissociative attachment to oxygenmolecules4–7, and that balance is achieved in an electric fieldtermed the conventional breakdown field. According to thesemodels, free electrons are removed in an electric field whosestrength is below the breakdown field, but multiply in a fieldthat is stronger than the breakdown field. Here we use a simplemodel of the electric response of the mesosphere at timescalesof tens of milliseconds to show that in the upper atmosphere,electrons multiply also under field strengths significantlybelow that of the conventional breakdown field because, at lowpressure, electron associative detachment from atomic oxygenions8,9 counteracts the effect of dissociative attachment.Our model couples chemical kinetics to dielectric relaxation,and our simulations indicate that associative detachmentcould explain recent measurements of the time and altitudeof the inception of delayed sprites10. We conclude thatassociative detachment is a fundamental process in upper-atmospheric electrodynamics.

First predicted in 1924 by Wilson11, electric discharges abovethunderstorms were unambiguously observed in 1989 (ref. 1). Al-though the mechanism of dielectric breakdown in these dischargeswas initially controversial, with some researchers proposing rela-tivistic runaway breakdown12, theoretical models based on conven-tional breakdown by Pasko and coauthors4,13,14 and Raizer et al.15compared more succesfully with the observations7,16. Conventionalbreakdown has been studied in laboratories since the 1920s andinforms our understanding of ground-level electric discharges17, in-cluding lightning18. Conventional breakdown assumes the existenceof a critical or breakdown field Ek, proportional to the air density,with Ek/N ≈ 120 Td, where N is the air number density and theTownsend (Td) is the conventional unit for reduced electric fieldsE/N (1 Td= 10−17 Vcm2). In dry air at standard temperature andpressure (STP), Ek ≈ 3.2× 104 Vcm−1. Our purpose in this articleis to show that at low pressures (high altitude in the atmosphere)one cannot define a fixed breakdown field and the conventionalbreakdown model has to be modified. We base this assertion oncalculations from cross-sections and reaction rates accepted in thepresent literature and in recent observations of the inception timesand altitudes of long-delayed sprites10.

Conventional breakdownIn the conventional breakdown model4,6,17,19 the dominant sourceof free electrons is electron impact ionization ofN2 andO2:

e+N2→ 2e+N2+ (1a)

IAA-CSIC, PO Box 3004, 18080 Granada, Spain. *e-mail: aluque@iaa.es.

e+O2→ 2e+O2+ (1b)

whereas electron losses come from dissociative attachment tomolecular oxygen:

e+O2→O+O− (2)

The reaction rates of ((1a), (1b)) and (2) depend stronglyon the electron energy distribution and hence are usuallywritten as functions of the local reduced electric field E/N . Thethreshold field Ek is the field that balances the weighted rates of((1a), (1b)) and (2).

Electrons attached to a heavy species (such as O−) are unableto ionize further molecules; therefore conventional breakdownmodels usually ignore O− ions created by the attachmentreaction (2), which is considered merely as a loss process for theelectrons. But in air O− ions are short-lived. On one hand, theyare transformed into much more stable ozone ions by a three-bodyclusterization reaction19

O−+O2+M→O−3 +M (3)

with a kinetic reaction rate k ≈ 10−30 cm6 s−1 and where Mrepresents any species (usually an abundant one such as N2 orO2). On the other hand, electrons can be released by associativedetachment to N2:

O−+N2→N2O+e (4)

Note that the stabilizing reactions (3) are three-body reactions andtherefore their importance relative to detachment (4) diminishes asthe air density decreases. The conventional breakdown assumptionthat electrons are inert after they form O− is valid only at highpressures, close to 1 atm, where three-body stabilization of O−dominates. The relevance of associative detachment was knownin the low-pressure plasma community already in the 1970s,when studies of low-pressure electric discharges8,9 showed thatthere is no apparent electron attachment in air below ∼0.1 atm(corresponding to altitudes above ∼15 km in the atmosphere),but it has apparently escaped the attention of researchers workingon atmospheric electricity. Only very recently, we pointed outthe relevance of associative detachment in sprites using a fullchemical kinetic model20,21.

Breakdown initiated by associative detachmentWe propose to extend the conventional breakdown model into amodel that also includes the associative detachment reaction (4).We call this the CB + AD model. To analyse how associativedetachment modifies the conventional breakdown model, let usassume that the air density is low enough to neglect reaction (3);this is certainly the case in the upper atmosphere, where sprites

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NATURE GEOSCIENCE DOI: 10.1038/NGEO1314 LETTERS E

/N (

Td)

Ratio O¬/electrons, η

120

100

80

60

40

20

0

Growth rate, (103 s¬1)

100

80

60

40

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0

120

0 0.5 1.0 1.5 2.0 2.5 3.0

¬2 ¬1 0 1 2

dne/dt<0

dne/dt>0

a b c

a b c

Figure 1 |At low pressures the net growth rate of the electron densitydepends not only on the reduced electric field but also on the ratio of O−

ions to free electrons. Here we plot the regions where electrons multiply(red) or are depleted (blue). The inset shows the net growth rate ofelectrons at 75 km as function of the electric field for three different valuesof η= nO−/ne. Note that, for η larger than about 2.6, the electron densityincreases for any value of the reduced field E/N.

and halos occur. One can then model the local net growth rate ofelectrons by combining the reaction rates of (1), (2) and (4):

1ne

dnedt= νion(E/N )−νattach(E/N )+ηνdetach(E/N ) (5)

where ne is the electron density, η= nO−/ne is the ratio of O− ionsto electrons, νion = [N2]k (1a) + [O2]k (1b) is the ionization rate,νattach=[O2]k (2) is the attachment rate and νdetach=[N2]k (4) is thedetachment rate. Here k represents the kinetic rate of the indicatedreaction. Note that, because we consider only two-body reactions,the left-hand side of (5) is proportional to the air density.

Figure 1 shows the areas in (η,E/N )-space where there is a netincrease or decrease of free electrons according to (5). These areasdo not depend on the air density. We also selected three values of η(indicated by a,b and c) to show, in the inset, the left-hand side of(5) at 75 km altitude as a function of E/N . For these calculationswe used reaction rates obtained from the BOLSIG+ Boltzmannsolver22,23 and ref. 9 for the rate of associative detachment (seethe Methods section).

As the O− density is increased relative to the density of free elec-trons, the range of reduced electric fields where electron attachmentdominates is reduced. When η is higher than about 2.6 the releaseof electrons dominates and electric breakdown occurs for any valueof the reduced electric field. We will see that this value is reachedwithin tens ofmilliseconds in upper-atmospheric discharges.

Inception of delayed spritesThe relevance of associative detachment is exemplified in theinception of delayed sprites. In most cases the delay betweena cloud-to-ground return stroke and its associated sprite isshorter than 10ms, with 2–3ms being typical. However, thereare consistent observations of sprites emerging with significantlylonger delays, up to about 150ms (refs 16,24,25). Remote magneticfield measurements by Cummer and his group7,16,25 have linkedthe inception of delayed sprites to a long-lasting cloud-to-ground,positive continuing current with an intensity of a few kiloamperes(assuming a∼10 km altitude of the discharging cloud).

Recently, Gamerota et al.10 used triangulation and high-speed video imaging to determine inception times and altitudes

for delayed sprites. They also estimated the lightning sourcecurrent moment waveform from broadband magnetic fieldmeasurements26 and, using a 2D finite-difference time-domainmodel, they obtained the lightning-induced mesospheric electricfield. They determined that delayed sprites start within a ±5 kmrange of the instantaneous maximum of reduced electric fieldE/N . However, their model, which did not include associativedetachment, could not explain the inception times or why spritesare sometimes initiated at significantly different values of E/N . Theauthors of that work claim that sprites are initiated at randomlylocated inhomogeneities.

Although inhomogeneities exist and seem to be relevant tostreamer inception and propagation27, we show here that the longdelays between the return stroke and sprite initiation are explainedby the build-up of the density of O−.

A simplemodel gives a first approximation to the response of themesosphere to a lightning-generated electric field. At high altitudes,far from the cloud charge, the electric field and species densities canbe assumed to depend only on altitude. In this case, the total currentis only a function of time and the electric field satisfies

ε0dEdt=−σE+ JT(t ) (6)

where σ is the local conductivity, ε0 is the dielectric constant andJT(t ) is a function of the instant cloud-to-ground current (seethe Methods section). Here we assume that the conductivity isdominated by its electronic component, σ = eneµ(z,E/N ), where eis the elementary charge and µ(z,E/N ) is the mobility of electronsat altitude z and reduced field E/N . We use realistic exponentialprofiles for the air density6 and the initial electron density6,7.

Although this model entails important simplifications, we havechecked that it agrees with the more complex mesospheric modelby Luque and Ebert6 (see Supplementary Information). We remarkthat the main uncertainty in the model comes from the initialdensity of free electrons, but this uncertainty is hardly avoidablegiven the natural variability of this background density.

Figure 2 shows the results of a simulationwith a currentmomentobtained by Gamerota et al.10. The resulting reduced electric fieldis at all times and everywhere below the classical breakdown fieldEk/N ≈ 120 Td, but there is a large multiplication of electronsvisible from about t ≈ 50ms at about 75 km altitude. The reason isthe increase in the density of O− ions due to electron dissociativeattachment (2); once this density is high enough, electrons arereleased through associative detachment and breakdown occurs.This process takes some tens of milliseconds.

In the observations by Gamerota et al.10, a sprite is first seen atthe time and altitude marked by a red spot in Fig. 2b and c and by avertical line in Fig. 2a, d, and e. Themodel indicates that the delayedsprite inception may be related to a high concentration of O− andCB+AD breakdown. Fig. 2d indeed shows that at the time of spriteinception (∼55ms) the electron density grows for any value of thereduced electric field. That panel also shows that breakdown ismorecomplex than assumed by conventional breakdownmodels.

After the dielectric breakdown, the atmospheric inhomo-geneities locally enhance the electric field and initiate streamerdischarges. The ignition of sprite streamers is a stochastic processandmay take place some time after the breakdown. As the processesleading to streamer inception cannot be incorporated in the presentzero-dimensional model, the simulation does not provide theprecise altitude and time of inception. However, as described in theMethods section and in Supplementary Fig. S5, the time derivativeof the electron density has a sharply localized maximum at eachaltitude that can be used as a criterion to predict inception altitudeand time. Note also that our calculations are not reliable whenstreamers dominate the dynamics of the discharge.

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LETTERS NATURE GEOSCIENCE DOI: 10.1038/NGEO1314

Electron density (cm

¬3)

hl (

kA k

m)

Alti

tude

(km

)A

ltitu

de (

km) 85

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d)

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d)

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3 ),η

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100806040200

a d

e

b

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75 km

0 10 20 30 40 50 60 70 80

102

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10¬1

0 10 20 30 40 50 60 70 80

Figure 2 |Associative detachment from O− ions explains the delayed inception of a sprite. Current moment (hI; see Methods) waveform (a) of athunderstroke on 3 July 2008, 08:55:11 UT obtained by Gamerota et al.10 and its simulated E/N (b) and electron density (c). The red spot and the straightlines mark the observed inception time and altitude. At the inception altitude (75 km) we show E/N (d) and the densities of electrons and O− with theirratio η (e). In d, the parametric areas of increase/decrease of the electron density, depending on η, are shown with the colours used in Fig. 1.

Ince

ptio

n al

titud

e (k

m)

Inception time (ms)

A

B

80

78

76

74

72

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68

66

640 20 40 60 80 100 120 140

1

1

3

4

2

2

4

3

Figure 3 | Inception times and altitudes of delayed sprites. The bluesquares are the model predictions from the current moment waveforms inref. 10 corresponding to the observations plotted by red circles. The errorbars represent the differences between model and observations. Thetriangles represent other observations in ref. 10. The solid line is anestimation of inception altitudes and times determined by varying thecontinuing current in a parametrized discharge model, with Acorresponding to a continuing current moment hIc=40 kA km and B tohIc= 7 kA km (see Methods); the hashed region marks a plausible range ofvariation of this curve for different initial electron densities.

In Fig. 3 we compare results of our model with the other threeevents for which Gamerota et al.10 provide the current momentwaveform. The differences between the model and the observationsare about 2 km in the inception altitude and about 10ms in theinception time, except in one event where the altitude predictedby the model is about 8 km higher than observed. This is probablydue to the simplification of assuming complete homogeneity in thehorizontal plane: local inhomogeneities as well as the convexity ofa more realistic downward propagating front would enhance theelectric field (see Supplementary Information). Nevertheless, themodel is close enough to the observations to indicate that delayedspritesmay be amanifestation of CB+ADbreakdown.

We also show in Fig. 3 a parametric study where we usedschematic, parametrized current moment waveforms with atriangularly shaped (that is, piecewise linear) return stroke and aconstant continuing current (see Methods section). We found that

the inception altitude and time of delayed sprites depends veryweakly on the return stroke for physically meaningful values of itsintensity and duration. Therefore, the one-parameter dependenceon the continuing current gives a curve in Fig. 3 that approximatelyrelates inception times and altitudes. As the initial electrondensity is usually very uncertain, we plotted also the range ofvariation when the baseline initial electron density6,7 is multipliedor divided by two. If one compares the full set of observationsreported by Gamerota et al.10 with the solid line in Fig. 3 onesees that the observed inception times plotted in Fig. 3 are inmost cases somewhat shorter than the times predicted by themodel. Again, this is probably because, by assuming completehorizontal homogeneity, we underestimate the electric fields andhence overestimate inception times.

The fair weather equilibrium density of O− in the upper atmo-sphere is negligible; it takes some tens ofmilliseconds of a significantelectric field to build up a density of O− that affects dielectric break-down. This explains why sprite inception models based on conven-tional breakdown (refs 6,7,13) have succeeded inmodelling promptsprites. In particular,Hu et al.7 showed that electric fields are close tothe conventional breakdown field in prompt, bright sprites. A shorttimescale is also the likely explanation why estimations of electricfields based on light emission from sprite streamers28 yielded valuesclose to Ek; this was explainedwith conventional breakdownmodelswithout associative detachment (ref. 29).

To conclude, we note that further investigation is needed toclarify the role of associative detachment in other atmosphericelectrical phenomena. Two examples are blue jets and giant bluejets4 and the long-standing problem of the initiation of lightninginside thunderclouds, where measured electric fields are lower thanthe conventional breakdown threshold30.

MethodsReaction rates. In all our calculations the reaction rates of electron impactionization and electron dissociative attachment are obtained by solving theBoltzmann equation with the BOLSIG+ solver developed by Hagelaar andPitchford22, with cross-sections from Phelps and Pitchford23. The reaction rate ofassociative detachment also depends on the reduced electric field and was measuredby Rayment andMoruzzi9. We fitted the experimental data with a rational functionk(E/N )= a(E/N )2/(b2+ (E/N )2), with a= 1.16×10−12 cm3 s−1 and b= 43.5 Td.In all simulations we used dry air with aN2: O2 composition of 80:20.

Zero-dimensionalmodel. Equation (6) can be easily derived from the conservationof charge q, ∂t q=−∇ ·σE, and the Poisson equation q= ε0∇ ·E. If we assumeplanar symmetry this reduces to ∂z (ε0∂tEz +σEz )= ∂z JT = 0, where JT is called

24 NATURE GEOSCIENCE | VOL 5 | JANUARY 2012 | www.nature.com/naturegeoscience

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NATURE GEOSCIENCE DOI: 10.1038/NGEO1314 LETTERSthe total current. This derivation is found in many textbooks and introduces theMaxwell relaxation time τM = ε0/σ . The source of the total current is the timevariation of the cloud charge Q located at an altitude h and its image created bythe conducting surface of the Earth. To estimate JT we assume that the field Ep

created by the electric dipole p= 2hQ is approximately homogeneous in a layer farabove the ground, from z0 = 65 km to z1 = 80 km. We also take the conductivityin the lower boundary of that layer to be low enough that the conduction currentis negligible and JT = ε0∂tEp. The cloud electric field is approximated by averagingthe lower and upper dipolar fields,

Ep=hQπε0

(1z30+

1z31

)(7)

where an extra factor 2 is included that results from the complete screening of thefield at∼80 km.We obtain the total current from (7), noting that dQ/dt = I , whereI is the cloud-to-ground electric current. The model depends on these somewhatarbitrary parameters, which, when changed within a physically meaningful range,produce variations in themodel inception time of the order of 10ms.

In the model, equations (5) and (6) are coupled with the correspondingevolution equations for the densities of positive ions generated by electronimpact ionization and O− ions generated by dissociative attachment. Finally,an assumption in the zero-dimensional model is quasineutrality: in (5) we haveneglected the transport of electrons. This amounts to assuming that everywherethe net charge density (measured as the density of elementary charges) is muchsmaller than the number densities of electrons and positive ions. This assumptionis accurate enough in the pre-streamer stage of sprite inception, but breaks downwhen streamers emerge.

Themodel is cross-checked with a more accurate 2Dmodel that includes fewerof these simplifications (see Supplementary Information).

Criterion for sprite inception. At present, we cannot model the full emergence ofdelayed sprites but we speculate that streamer inception has a higher probabilitywhere there is a fast multiplication of electrons. Supplementary Fig. S5 shows thetime derivative of the electron density at each altitude for the simulation in Fig. 2.At each altitude, dne/dt peaks sharply at a different time. We expect that thesepeaks translate into a high probability of streamer inception and therefore wepredict the likeliest inception at the earliest of these peaks, marked with a red dot inSupplementary Fig. S5. In the Supplemetary Information, we show the inception ofa streamer starting from a local inhomogeneity.

Parametrization of the cloud-to-ground discharge. To calculate the solid linein Fig. 3 we run the zero-dimensional model with a set of discharges with acontinuously varying continuing current intensity. The cloud-to-ground current ismodelled as a piecewise linear (that is, triangular) symmetrical peak of durationτ and height Imax followed by a constant current Ic. We consider that the chargemoment change due to the return stroke is h1Q= hImaxτ/2. For Fig. 3 weuse h1Q= 100C km and τ = 4ms; the curve is not very sensitive to theseparameters. The continuing current moment is varied between hIc = 7 kA kmand hIc = 40 kA km.

Received 9 May 2011; accepted 13 October 2011; published online13 November 2011; corrected online 29 November 2011

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AcknowledgementsThis work was supported by the Spanish Ministry of Science and Innovation, MICINNunder project AYA2009-14027-C05-02 and by the Junta de Andalucía, Proyecto deExcelencia FQM-5965.

Author contributionsF.J.G-V. first included the associative detachment reaction in his kinetic model (ref. 20)and pointed to its relevance in preliminary kinetic simulations in the context ofthis work. A.L. developed the self-consistent model detailed here and proposed theconnection with delayed sprite inception. Evaluation, interpretation and literaturestudies were carried out together.

Additional informationThe authors declare no competing financial interests. Supplementary informationaccompanies this paper on www.nature.com/naturegeoscience. Reprints and permissionsinformation is available online at http://www.nature.com/reprints. Correspondence andrequests for materials should be addressed to A.L.

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This manuscript was originally incorrectly published as an Article. It has now been changed to the correct category of a Letter in all versions.

Mesospheric electric breakdown and delayed sprite ignition caused by electron detachment A. Luque and F. J. Gordillo-Vazquez

Nature Geoscience http://dx.doi.10.1038/ngeo1314 (2011); published online 13 November 2011; corrected online 29 November 2011

ERRATUM

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