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The physics of high altitude lightning /
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GEOPHYSICAL RESEARCH LETTERS, VOL. 22, NO. 2, PAGES 85-88, JANUARY 15, 1995
On the physicsof high altitude lightning
G. M. Milikh and K. Papadopoulos
Departmentsof Physicsand Astronomy,University of Maryland
C. L. Chang
ScienceApplicationInternationalCorporation,McLean, VA
Abstract. Pastandrecentobservations
indicatethepresence distance
L abovea perfectly
con,
ducting
ground
is assumed
of lightningat altitudesin excessof 30 kin. The phenome- (Figure1). ThecurrentdensityJ(7, t) is modeled
as
non is manifestedas a high altitudeopticalflash,correlated
with the presenceof giant thunderstorms
in the atmosphere
below. This letter presentsthe first physical model of the
process. The model is basedon low frequencyRF break-
downof the upperatmosphere,
ignitedby the upwardpropagatingelectromagnetic
pulsesdue to conventionallow altitude lightning. Horizontal intercloudlightningstrokesform
the optimal configuration. Horizontal lightningdischarges
with cloud-to-cloudmomentcharge~ 6,000 -- 8,000 C-kin
accountfor the observedlevel of opticalemissions.
Introduction
•(•, t) - Io(e-•' - e-•')H•[H•- H315(y)5(z
- Zo)i (1)
whereHi&3 • the Heavisideunit functionswith arguments
t, x + L/2, x -- L/2 correspondingly,
5(x) is the delta
function, and i is the unit vector in x direction. Ground
reflectionis treatedas an image chargeand currentinduced
below a perfectly conductingground. Using the charge
density,derivedfrom the chargeconservation
equation,and
the currentdensitygiven by eq. (1) as a source,the electric
field inducedby the lightningdischargeat a point z on the
z-axisis Ey- E•.- 0 and
A number of authors [Boeket al., 1992; Franz et al.,
Ex= E•e-"• {H(r)H(7)F(r/,r•- 1)
1990; Winckleret al., 1993; Vaughamand Vonnegut,1989;
Vaughamet al., 1992;Lyons,1994] reportedthe presenceof
optical flashesat high altitudesabove thunderstormclouds.
SentmanandWescott[1993] describethe phenomenon,
often
called high altitudelightning(HAL), as a luminouscolumn
stretchedbetween30 and 80 kin, with peak luminosityin
(2)
y
the vicinity of 60 kin. HAL correlateswith massivethunF(V,y)-f e"/•l
+ x'dx,7 - •1 + g'- r,
derstorms,
andoccurswith a frequencybetween0.5 and5% Hem
comparedto the total numberof lightningstrokes.It is the • •e fo•ow•g dime•io•ss p•e•m
we• us•
objectiveof this letterto producethe firstquantitativemodel
of the HAL process.
The modelis basedon upperatmospheric
electronenergizationor breakdown
causedby an electromagnetic
pulse
(3)
-H(v)
{ r? (1+•7')
a/' 1+•' + F(r/,•)))
o
fi(Z-Zo),r---,•-
E•=Io
inducedby low altitudelightning. It is on_alogous
to the
thetransient
field,whiletherest
recentlystudiedionospheric
andatmospheric
breakdown
by In eq. (2) F (r/,y) represents
fielddueto thepointcharges.For times
ground
based
RFtransmitters
[Borisov
etal.,1986;
TSang
et is thequasi-static
onlyby
al., 1991;Papadopoulos
et al., 1993]. Emphasis
is placed r < x/1 + •2 thefieldat thepointz is determined
on intercloudlightningstrokessince,as shownbelow,they the transient field.
produceoptimalconfiguration
for upperatmospheric
breakThetotalelectricfieldontheaxesis givenby
clown.
E,o,(Z,Io,r) = Ex(z- Zo,Io,r) - Ex(z+ Zo,-Io, r)
(4)
Spatial Distribution of the Electric
Field and Energy Deposition
The secondterm on the right hand side representsthe contribution of the electricfield due to the groundreflection.
From eqs. (2)-(4) we find that, in practicalunits, the
electricfield inducedby a lightningstrokewith peakcurrent
We study first upper atmospheric
breakdownby an
electromagnetic
pulsedueto a horizontal
lightningstroke.A
simplemodelof lightning,as a line currentconnecting
two
is
discharging
cloudslocatedat altitudeZoand separated
by Io, andpulsedurationtp in the upperatmosphere
Copyright1995 by the AmericanGeophysical
Union.
E(\•
-) -0.2Io(kA)(m.••)
•tot(r/(tp),
•,r)
(5)
Papernumber94GL02733
where•,o, = IE,o,I/E• istheabsolute
valueofthedimen-
0094-8534/95/94GL-02733503.00
85
$6
MI•
ET AL.: ON THE PHYSICS
-L/2
+J
LIG•G
From the dependonceof • on altitude (Figure 4) we
e•t
emissionsin the visible at altitudesz>50 km in the
presenceof ambientelectrons. Their intensitydependson
the densityprofile. When ? exceedsthe ionizationthreshold
strongemissionswill result even for low ambientelectron
density. The ? dependenceis universaland indicatesthat the
L/2
I
OF HIGH ALTITUDE
2
Zo
value of ? maximi?cs at altitudes of about 70 kin.
This is
dueto thefactthatv ,,, e-"/•' whileE ,-, 1/za. Foraltitudes
in excessof 70 kin, where v < g, the quiver energydrops
sharply. When ? > 0.2 eV self absorptionbecomesimportant. In the presentpaperwe concentrateon HAL at lower
altitudes,z _<70 kin, whereself absorptionis negligible.
3
.j
4
Optical Emissionsand Ionization
Figure 1. Schematicof the proposedmodel.
sionlesselectric field. Finally, the horizontaldistributionof
the electric field due to a long pulse, r; < < 1, is given by
The calculationof optical cmi.ssionsfollows the well
established
procedureof ionosphericbreakdown[Borisovet
al., 1986; Tsanget al., 1991; Papadopouloset al., 1993].
First, the local kinetic equationthat describesthe evolution
of the electron distributionf(v,0 in the presenceof an am-
E(x, z)
1
1+
bient electric field and inelastic collisions, is solved and the
(6)
stationarystateis found. This equationis given by
x
O• = 0•v
•ov
• 1 0f(v,t))
•' -•,(f(v))(8)
0f(v,t)
eE
0(•+a
We commentbriefly on the key dependencesof eqs. (2)(5), which give the distributionof the maximumfield value
induce• by low altitude lighming at an altitude z, as a
whereL(f(v)) is anoperator
thatdescribes
theeffects
of the
collisions.
Theformof L(f(v)) foraircanbefound
functionof thepeakcurrentIo, pulsewidthtp=l//•, discharge inelastic
in the above references.For a given altitude and value of
length L, and lighming altitude Zo. The electric field scales
sothatf(v) becomes
_l.jnearly
withIo. Figure2 illustrates
thetemporal
evolution
of E(z), a steadystateis quicklyestablished
E(r) - Ex(r)/E• for valuesof theparameter
r• - 1 and independentof time and is only a functionof z, and E(z).
r• - 0.25. A nonlinear,thresholdlike behaviorof the field Second,the excitation
rateV?xof thej electronic
levelof
on r• is apparent.Of particularimportanceis the nonlinear species by electronimpact,definedby
dependonce
onthepulsewidthtp.In addition,
Figure2 shows
that increasing
the parameter
• - 0.5 L/(z- Zo) leadsto
(9)
js_4•'Ns
f f(vE)vacp•x(v)dv,
/dex
larger transientelectricfields. As a resultstrongerfieldsare
producedby lighmingstrokeswith long path lengths.
•
Figure 3 showsthe temp_oml
evolutionof the total electric field amplitude Etot for two values of
X - (z - Zo)/(z+ Zo).
0.4
The parameterX determines
the delay time between the signals arriving at z from the
dischargecurrentand from its groundreflection.For short
delays(solidcurve)the signaldue to the reflectedfield significantlyreducesthe total field. For longerdelays(broken
curve) the reflectedfield appearsas a secondpeak in Etot,
followingthe decayof the field due to the dischargecurrent.
The criticalparameter[Papadopoulos
et al., 1993] that
controlsenergydepositionand the subsequent
observables
is the value of the generalizedelectron quiver energy •
0.3
0.2
0.1
T
1.5
2
2.5
3
which,for pulseswith durationtp>>l/v, 1•, wherev is
the frequencyof electron-neutralcollisionGurevich[1978]
and g the electroncyclotronfrequency,is given by
1
•- -
e2E 2
(7)
For values of ?.< 0.02 eV most of the energy absorbedby
the electronsexcitesthe low lying vibrationallevels of nitrogen, and emissionsin the visible rangecannotbe excited.
For 0.02 eV < ? < 0.1 eV the electronenergyresultsin excitation of optical emissionsand moleculardissociation.For
Figure2. Thefield• asa function
of r for(a)r•= 1,(b)r•
? > 0.1 eV breakdown is initiated.
= 0.25. From top to bottom• = 0.5, 0.3 and 0.2.
1.5
2
2.5
3
MI•
ET AL.: ON THE PHYSICS OF HIGH ALTITUDE
•t
LIG•G
87
= 10 kin. We consider the optical emissionsdue to three
nitrogen
bands,1stpositive(BaIIg--• AaEu+),2ndpositive
(CalIu• BalIg),and1stnegative
band
(B2Eu
+ --•X2Eg+),
0.25
with lifetimesof 8 ps, 38 ns and 63 ns correspondingly.To
computethe optical emissions,we first calculatethe peak
electric field per unit currentas a functionof altitude. This
0.2
0.15
0.1
"-...
I
electric field is used in the Fokker-Planck code, which solves
-.
quenching
rate,andv{'f theeffectiveionizationratedefined
numericallyeq. (8), to computethe ionizationandexcitation
rates for the three nitrogenbandsof interest. The intensity
of the opticalemissionsas a functionof altitudeis computed
by usingthe ionizationand excitationratesin eq. (10). The
quenchingrates of the excited electroniclevels are taken
from Jones [1974]. The analysiswas performedassuming
a middlelatitudemodelof the neutralatmosphere[Brasseur
and Solomon, 1984] and two profiles of the electrondensity representing"tenuous"and "dense"nighttimeD regions
[Taranenkoet al., 1993]. Figure 5 presentsthe altitudedistribution of the total intensityof the artificial opticalemissions
as a function of the peak electric currentof the intercloud
lighming stroke. It indicatesthat optical emissionswith intensityof tensof kR can be inducedby lightningpulseswith
duration20 ms and peak cur•nt 3040 kA at 60-70 km for
denseionosphere,or at 65-70 km for tenuousionosphere.
These values are consistentwith the observationsby Sentman and Wescott[1993] of 10-50 kR brightnessof upper
atmosphericoptical flashes.The value of the thresholddectric field requiredat 60-70 km can be found from Figure 2
and eq. (5). For the valuesquotedabove it corresponds
to
as the difference between the ionization and attachmentrates,
130-250 V/re.
are solvedunderthe following initial conditions
The mostimportantparameterin the model is the value
of cloud-to-cloud(CC) momentchargeM. To produce10-50
kR brightnessMm6-8x 103 C-kin is required. This should
be comparedto averagevaluesof M~200400 C-kin [Wang,
1963; Hatakeyama, 1958]. It is difficult to determinethe
probabilityof occurrenceof HAL, althoughsomedata are
becoming available [Lyons, 1994]. We can speculatethat
0.05
1.5
2
2.5
3
Figure 3. The total field Etot as a functionof time r for r/
= 1, • = 0.3, and X = 0.6 (solidline), X = 0.4 (brokenline).
whereN, is thedensity
of species, anda•(v) theexcitation cross-section,
is computed.Third, the equationswhich
describe
thedensities
of theexcited
molecules
N• andelectrons N•
ddt
Nj%= vex
J'Ne N
j% Ns
Tjs 17q
js
dNe
ef
(lO)
dt = l/i Ne'
whererj, is thelifetimeof theexcited
molecules,
v•q
• the
Ne(t
- O,z)
- No(z
)
(11)
5(t - o,,,) - o,
whereNo(z) is the densityof the ambientelectrons.
We pr•
nextto calculatethe expectedHAL optical if the value of M scales as a self similar fractal [Bak et
emissionsunder giant thunderstormconditions. In such a al., 1987] M c• with c•=-l, then HAL will occur in a few
casecloudsextend up to 15-20 km [Uman, 1984]. In addi- percentof CC dischargesover giant thunderstorms.There
tion, the HAL seemsto be correlatedwith lightningpulses are, however, a number of measurementsindicating that
with duration,,- 20 ms, called"longevents"[Winckleret al., few hundredsof C are frequentlydischargedin CC events.
1993]. We considerthe particularcaseof horizontalinter- For example,Workmanet al. [1942] notedthat CC charge
cloudlightningat Zo=10kin, with t•=10 and20 ms,andL
transfer reached, on occasion, 100 C. Smith [1957] observed
that in almosthorizontaldischarges
chargebetween130-250
C was transferred,and Takeuti [19651 noted that charge
Figure 5. Altitude depend0nce
of the total intensityof the
opticalemissionsdueto a lightningstrokelocatedat Zo= 10
410
•0
6•0
7•0
8tO
Altitude (km)
kin, pulsewidthtp - 20ms, andpathlengthL = 10 kin,
for differentvaluesof the peak electriccurrent. Numbers
1, 2 and 3 correspondto !o = 20, 30 and 40 kA. Solid and
Figure 4. Altitudedependanteof • for Zo=10kin, L=10 kin,
broken lines show emissions for tenuous and dense electron
9,=8.5x106l/s, !o=30kA, andh,=20ms.
density models.
88
MI•
ET AL.: ON THE PHYSICS OF HIGH ALTITUDE
LIGHTN•G
transferin CC dischargesoften ex•
100 C. Suchvalues Hatakeyama,H., The distributionof the suddenchangeof electric field on the earth'ssurfacedue to lightningdischarge,
in
with Lm 10 km are consistentwith the requirementsof the
model.
L.G. Smith(ed.),RecentAdvancesinAtmospheric
Electricity,
pp. 289-298, PergamonPress,New York, 1958.
The intensityof the optical emissionschangesdrasticallywith smallvariations
(_<10- 15%)in thelocalelec- Iones, A. V., Aurora, D. Reidel PublishingCompany,Boston,
tric field.
This allows us to estima_tethe horizontal extend
1974.
of the optical flashesby using eq. (6). In fact, at z = 60 Lyons, W. A., Characteristics
of luminousstructures
in the
lcmthe electricfield amplitudedrops 10-15% when moving
stratosphere
abovethunderstorms
as imagedby low-light
14-16 km away from thepeakpointlocatedabovethemiddle
video,Geophys.Res. Letts.,21, 875-878, 1994.
point of the lightning,reducingthe opticalemissionsbelow
K., G. Milikh, A. Gurevich,A. Drobot,andR.
the observationalthreshold. This spatialscaleis consistent Papadopoulos,
Shanny,Ionizationrates for atmosphericand Ionospheric
with the observationsof Sentman and Wescott [1993] who
breakdown,J. Geophys. Res., 98(A10), 17,593-17,596,
reportedopticalflasheswith horizontalextentof 10-50 kin.
1993.
We finally note that the flashesoften appearin pairs
of upperat[Franz et al., 1990]. The presenttheorynaturallyaccounts Sentman,D. D., and E. M. Westort,Observations
mospheric
optical
flashes
recorded
from
an
aircraft,
Geophys.
for this relying on ignitionby the refiectexlpulse. A slope
Res. Letts., 20, 2857-2860, 1993.
in the groundsurfaceor deviationof the lightningpath from
the horizontaldirection,leadsto lateraldisplacement
of the Smith, L. G., Intracloudlightning discharges,Quart. J. Roy.
Meterol. Soc., 83, 103-111, 1957.
flash inducedby the groundreflection.
Takeuti,
T.• Studies
in thunderstorm
electricity,
(1)Cloud
discharge, J. Geomagnetism
and Geoelectricity,17, 59-68,
Conclusions
1965.
Taranenko, Y. N., U.S. Inan, and T. F. Bell, The interaction
A modelof HAL basedon electronenergizationinduced
with the lower ionosphereof electromagnetic
pulsesfrom
by horizontallong lightningpulseshasbeenpresented.The
lightning:
excitation
of
optical
emissions,
submitted
to Geomodel naturally accountsfor the observedcorrelationof
phys. Res. Letts.,20(23), 2675-2678, 1993.
HAL with giantthunderstorms,
implyinghorizontalscalesin
A. Drobot,P. Vitello,T. Wallace,
excessof 10 kin, pulsedurationin excessof 20 msandfor the Tsang,K., K. Papadopoulos,
andR. Shanny,RF ionizationof the lowerionosphere,
Radio
appearanceof correlatedpairsof flashes.The intensityof the
$ci., 20(5), 1345, 1991.
observedemissionsrequiresdischargecurrentsof the order
of 30-40 leA, or a cloud-to-cloudmomentchargein excess Uman, M. A., Lightning,Dover, New York, 1984.
of 6-8 x 103C-kin. Detailedanalysis
of thephenomenon
and Vaugham,Jr., andB. Vonnegut,Recentobservations
of lightning
its environmentalimplicationswill be publishedelsewhere.
discharges
from the top of a thundercloud
into the air above,
J. Geophys.Res.,94, 13179-13182,1989.
Acknowledgments,The work hasbeensupported
by the Office
Vaugham,Jr., O. H., R. Blakeslee,W. L. Boeck,B. Vonnegut,
of Naval ResearchgrantN0001490K201.
M. Brook, and J. McKune, Jr., A cloud-to-space
lightning
as recordedby the SpaceShutfiepayloadbay TV cameras,
Mort. Wea. Rev., 120, 1459-1461, 1992.
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View publication stats
G. M. Milikh and K. Papadopoulos,
Universityof Maryland, Departmentsof Physicsand Astronomy,CollegePark,
MD
20742.
C. L. Chang,ScienceApplicationInternationalCorporation,
McLean, VA 22102.
(ReceivedMay 2, 1994;revisedJuly26, 1994;
accepted
September26, 1994.)