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.
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VALDIVIA
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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.)