FINITE ELEMENT ANALYSIS OF THE MESOSPHERE?S ELECTROMAGNETIC
RESPONSE TO LARGE SCALE LIGHTNING ASSOCIATED WITH
SPRITES AND OTHER TRANSIENT LUMINOUS EVENTS
Except where reference is made to the work of others, the work described in this thesis is
my own or was done in collaboration with my advisory committee. This thesis does not
include proprietary or classified information.
_______________________________
Michael David Allgood
Certificate of Approval:
______________________________ ______________________________
Lloyd S. Riggs Michael E. Baginski, Chair
Professor Associate Professor
Electrical and Computer Engineering Electrical and Computer Engineering
______________________________ ______________________________
Stuart M. Wentworth Joe F. Pittman
Associate Professor Interim Dean
Electrical and Computer Engineering Graduate School
FINITE ELEMENT ANALYSIS OF THE MESOSPHERE?S ELECTROMAGNETIC
RESPONSE TO LARGE SCALE LIGHTNING ASSOCIATED WITH
SPRITES AND OTHER TRANSIENT LUMINOUS EVENTS
Michael David Allgood
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Master of Science
Auburn, Alabama
May 10, 2008
iii
FINITE ELEMENT ANALYSIS OF THE MESOSPHERE?S ELECTROMAGNETIC
RESPONSE TO LARGE SCALE LIGHTNING ASSOCIATED WITH
SPRITES AND OTHER TRANSIENT LUMINOUS EVENTS
Michael David Allgood
Permission is granted to Auburn University to make copies of this thesis at its discretion,
upon request of individuals or institutions and at their expense. The author reserves all
publication rights.
______________________________
Signature of Author
______________________________
Date of Graduation
iv
THESIS ABSTRACT
FINITE ELEMENT ANALYSIS OF THE MESOSPHERE?S ELECTROMAGNETIC
RESPONSE TO LARGE SCALE LIGHTNING ASSOCIATED WITH
SPRITES AND OTHER TRANSIENT LUMINOUS EVENTS
Michael Allgood
Master of Science, May 10, 2008
(B.E.E., Auburn University, 2001)
82 Typed Pages
Directed by Michael E. Baginski
In this research, a numerical investigation of high altitude sprites and other
mesospheric Transient Luminous Events induced by lightning is presented. A Finite
Element Model is created using equations based on a modified form of Maxwell?s
equations and includes the effects of ionization on the upper atmosphere. Results will
first be shown for standard models with constant ambient conductivity which will be
verified based on prior research. A model will then be introduced which includes
ionization effects altering the electron conductivity in a nonlinear manner. These results
will be compared to previously published research.
v
ACKNOWLEDGEMENTS
The author would like to express his appreciation to Dr. Michael Baginski for the
many years of advice and patience shown to me throughout this entire process. He would
also like to thank both Dr. Lloyd Riggs and Dr. Stuart Wentworth for their friendship and
assistance. He would like to acknowledge the Department of Defense for the reception of
a National Defense Science and Engineering Fellowship without which this work never
would have started. His company, Dynetics Inc., provided financial support through the
last few years. The Alabama Supercomputing Authority provided the necessary hardware
and software for completion of this research. He would also like to recognize Dr. V. P.
Pasko for providing data and a figure from his previous work. Finally, special thanks go
to his wife, Sydney, without whose support, patience, and motivation, this research would
have ended many years ago.
vi
Style manual or journal used IEEE Transactions on Microwave Theory and
Techniques.
Computer software used Microsoft Word 2003 and 2007.
vii
TABLE OF CONTENTS
LIST OF IGURES xi
1 INTRODUCTION 1
1.1 Characteristics of Sprites...................................................................................1
1.2 Additional Transient Luminous Events..............................................................2
1.3 Sprite Observations...........................................................................................3
1.3.1 First Recorded Image.............................................................................3
1.3.2 Space Shuttle Images.............................................................................3
1.3.3 Images Recorded from Aircraft..............................................................4
1.3.4 Further GroundBased Recordings.........................................................4
1.4 Overview and Historical Perspective .................................................................5
1.4.1 Measurements in the Middle Atmosphere and Ionosphere......................5
1.4.2 Finite Element Model using Ambient Conductivity Profiles ..................5
1.4.3 NonLinear Conductivity Profile............................................................6
1.4.4 Additional Numerical Methods..............................................................7
1.5 Thesis Outline...................................................................................................7
2 PROBLEM FORMATION 8
2.1 Overview ..........................................................................................................8
2.2 Derivation from Modified Maxwell?s Equations................................................8
2.3 Modeling of Lightning Discharge....................................................................10
2.4 Atmospheric Conductivity Modeling...............................................................11
2.4.1 Standard Models..................................................................................11
2.4.2 General Conductivity Model................................................................13
2.4.2.1 Ambient Values ....................................................................13
2.4.2.2 Electron Mobility ..................................................................15
2.4.2.3 Electron Density....................................................................16
2.4.2.4 Ionization and Attachment Coefficient ..................................17
3 FINITE ELEMENT MODEL 20
3.1 Overview ........................................................................................................20
3.2 Development of Equation................................................................................20
3.3 Geometry of the Region ..................................................................................21
3.3.1 Boundary Conditions...........................................................................22
3.4 Additional FEM Parameters............................................................................22
viii
4 SIMULATION RESULTS 24
4.1 Overview ........................................................................................................24
4.2 Ambient Conductivity Profiles........................................................................24
4.2.1 Total Electric Field Results..................................................................24
4.2.2 Vertical and Horizontal Electric Field Results......................................31
4.3 NonLinear Conductivity ................................................................................38
4.3.1 Total Electric Field Results..................................................................38
4.3.2 Vertical and Horizontal Electric Field Results......................................45
4.3.3 Conductivity and Electron Density Results ..........................................50
5 CONCLUSIONS 63
5.1 Future Work....................................................................................................64
BIBLIOGRAPHY 65
ix
LIST OF FIGURES
2.1 Ambient ion conductivity profiles with respect to altitude.....................................12
2.2 Ambient electron number density profile ..............................................................14
2.3 Number density of air molecules (N) as a function of altitude ..............................14
2.4 Conductivity profile as a function of altitude ........................................................15
2.5 Electron mobility (?
e
) at 50 and 80 km .................................................................16
2.6 Ionization and attachment coefficients for 50 and 80 km.......................................18
2.7 Characteristic air breakdown field with respect to altitude.....................................19
2.8 Ionization and attachment coefficients normalized to E
k
for 70 km........................19
3.1 Geometry of model ...............................................................................................21
4.1 Conductivity profiles for ambient simulations.......................................................26
4.2 Electric field simulations at z = 40 km and ? = 10 km for ambient conductivity
profiles .................................................................................................................26
4.3 Electric field simulations at z = 50 km and ? = 10 km for ambient conductivity
profiles .................................................................................................................27
4.4 Electric field simulations at z = 60 km and ? = 10 km for ambient conductivity
profiles .................................................................................................................27
4.5 Electric field simulations at z = 70 km and ? = 10 km for ambient conductivity
profiles .................................................................................................................28
4.6 Electric field simulations at z = 80 km and ? = 10 km for ambient conductivity
profiles .................................................................................................................28
4.7 Electric field simulations at z = 90 km and ? = 10 km for ambient conductivity
profiles .................................................................................................................29
x
4.8 Electric field simulations using Gish conductivity profile at z = 80 km for
radial distances from 0 to 50 km ...........................................................................29
4.9 Electric field simulations using exponential conductivity profile at z = 80 km
for radial distances from 0 to 50 km......................................................................30
4.10 Electric field simulations using profile 3 at z = 80 km for radial distances from
0 to 50 km.............................................................................................................30
4.11 Vertical electric field simulations at z = 40 km and ? = 10 km for ambient
conductivity profiles .............................................................................................32
4.12 Horizontal electric field simulations at z = 40 km and ? = 10 km for ambient
conductivity profiles .............................................................................................32
4.13 Vertical electric field simulations using Gish conductivity profile at z = 40 km
for radial distances from 0 to 50 km......................................................................33
4.14 Horizontal electric field simulations using Gish conductivity profile at z = 40
km for radial distances from 0 to 50 km ................................................................33
4.15 Vertical electric field simulations using exponential conductivity profile at z =
40 km for radial distances from 0 to 50 km ...........................................................34
4.16 Horizontal electric field simulations using exponential conductivity profile at z
= 40 km for radial distances from 0 to 50 km ........................................................34
4.17 Vertical electric field simulations using Gish conductivity profile at z = 80 km
for radial distances from 0 to 50 km......................................................................35
4.18 Horizontal electric field simulations using Gish conductivity profile at z = 80
km for radial distances from 0 to 50 km ................................................................35
4.19 Vertical electric field simulations using exponential conductivity profile at z =
80 km for radial distances from 0 to 50 km ...........................................................36
4.20 Horizontal electric field simulations using exponential conductivity profile at z
= 80 km for radial distances from 0 to 50 km ........................................................36
4.21 Vertical electric field simulations using profile 3 at z = 80 km for radial
distances from 0 to 50 km .....................................................................................37
4.22 Horizontal electric field simulations using profile 3 at z = 80 km for radial
distances from 0 to 50 km .....................................................................................37
xi
4.23 Electric field simulations using the nonlinear conductivity profile for ? = 0 km
at altitudes of 6090 km ........................................................................................39
4.24 Electric field simulations using the nonlinear conductivity profile for ? = 10
km at altitudes of 6090 km...................................................................................40
4.25 Electric field simulations using the nonlinear conductivity profile for ? = 20
km at altitudes of 6090 km...................................................................................40
4.26 Electric field simulations using the nonlinear conductivity profile for ? = 30
km at altitudes of 6090 km...................................................................................41
4.27 Electric field simulations using the nonlinear conductivity profile for ? = 40
km at altitudes of 6090 km...................................................................................41
4.28 Electric field simulations using the nonlinear conductivity profile for ? = 50
km at altitudes of 6090 km...................................................................................42
4.29 Electric field simulations using the nonlinear conductivity profile for z = 60
km at radial distances of 050 km..........................................................................42
4.30 Electric field simulations using the nonlinear conductivity profile for z = 70
km at radial distances of 050 km..........................................................................43
4.31 Electric field simulations using the nonlinear conductivity profile for z = 80
km at radial distances of 050 km..........................................................................43
4.32 Electric field simulations using the nonlinear conductivity profile for z = 90
km at radial distances of 050 km..........................................................................44
4.33 Electric field simulations at ? = 0 km and z = 80 km for the exponential and the
nonlinear conductivity profiles.............................................................................44
4.34 Electric field simulations at ? = 0 km and z = 6090 km for the exponential and
the nonlinear conductivity profiles at a time of 0.5 ms .........................................45
4.35 Vertical electric field simulations using the nonlinear conductivity profile for
60 km altitude at radial distances of 050 km ........................................................46
4.36 Horizontal electric field simulations using the nonlinear conductivity profile
for 60 km altitude at radial distances of 050 km...................................................46
4.37 Vertical electric field simulations using the nonlinear conductivity profile for
70 km altitude at radial distances of 050 km ........................................................47
xii
4.38 Horizontal electric field simulations using the nonlinear conductivity profile
for 70 km altitude at radial distances of 050 km...................................................47
4.39 Vertical electric field simulations using the nonlinear conductivity profile for
80 km altitude at radial distances of 050 km ........................................................48
4.40 Horizontal electric field simulations using the nonlinear conductivity profile
for 80 km altitude at radial distances of 050 km...................................................48
4.41 Vertical electric field simulations using the nonlinear conductivity profile for
90 km altitude at radial distances of 050 km ........................................................49
4.42 Horizontal electric field simulations using the nonlinear conductivity profile
for 90 km altitude at radial distances of 050 km...................................................49
4.43 Maximum electric field strengths over time for altitudes from 6090 km at
radial distances from 050 km compared to the characteristic air breakdown
field ......................................................................................................................50
4.44 Conductivity profiles using the nonlinear model at 1 ms for radial distances
from 030 km. The ambient profile is shown for reference ....................................52
4.45 Conductivity profiles using the nonlinear model at 100 ms for radial distances
from 030 km. The ambient profile is shown for reference ....................................53
4.46 Electron density profiles using the nonlinear conductivity model at 1 ms for
radial distances from 030 km. The ambient profile is shown for reference ...........53
4.47 Electron density profiles using the nonlinear conductivity model at 100 ms for
radial distances from 030 km. The ambient profile is shown for reference ...........54
4.48 Electron density changes corresponding to three ambient electron density
models for 200 C CG stroke, provided by Dr. V. P. Pasko [37].............................54
4.49 Conductivity profile using the nonlinear model at 60 km altitude for radial
distances from 050 km.........................................................................................55
4.50 Electron mobility using the nonlinear conductivity model at 60 km altitude for
radial distances from 050 km...............................................................................55
4.51 Electron number density using the nonlinear conductivity model at 60 km
altitude for radial distances from 050 km.............................................................56
xiii
4.52 Difference between ionization and attachment coefficients, ?
i
 ?
a
, using the
nonlinear conductivity model at 60 km altitude for radial distances from 050
km ........................................................................................................................56
4.53 Conductivity profile using the nonlinear model at 70 km altitude for radial
distances from 050 km.........................................................................................57
4.54 Electron mobility using the nonlinear conductivity model at 70 km altitude for
radial distances from 050 km...............................................................................57
4.55 Electron number density using the nonlinear conductivity model at 70 km
altitude for radial distances from 050 km.............................................................58
4.56 Difference between ionization and attachment coefficients, ?
i
 ?
a
, using the
nonlinear conductivity model at 70 km altitude for radial distances from 050
km ........................................................................................................................58
4.57 Conductivity profile using the nonlinear model at 80 km altitude for radial
distances from 050 km.........................................................................................59
4.58 Electron mobility using the nonlinear conductivity model at 80 km altitude for
radial distances from 050 km...............................................................................59
4.59 Electron number density using the nonlinear conductivity model at 80 km
altitude for radial distances from 050 km.............................................................60
4.60 Difference between ionization and attachment coefficients, ?
i
 ?
a
, using the
nonlinear conductivity model at 80 km altitude for radial distances from 050
km ........................................................................................................................60
4.61 Conductivity profile using the nonlinear model at 90 km altitude for radial
distances from 050 km.........................................................................................61
4.62 Electron mobility using the nonlinear conductivity model at 90 km altitude for
radial distances from 050 km...............................................................................61
4.63 Electron number density using the nonlinear conductivity model at 90 km
altitude for radial distances from 050 km.............................................................62
4.64 Difference between ionization and attachment coefficients, ?
i
 ?
a
, using the
nonlinear conductivity model at 90 km altitude for radial distances from 050
km ........................................................................................................................62
1
CHAPTER 1
INTRODUCTION
On a night in July of 1989, two frames of video captured flashes of light
discharging between the top of clouds and the ionosphere [1]. Before this night, there
were scattered reports of mysterious lights high above thunderclouds. For years pilots
would also occasionally observe luminous events above storms [1]. Without video
evidence, however, only a handful of researchers gave a second thought to this unusual
occurrence. After that fateful night, more observations were documented leading to a
surge of investigators trying to determine the underlying cause and properties of this
phenomenon, known as a sprite.
1.1 Characteristics of Sprites
A sprite is an electrical phenomenon occurring above the thunderclouds in the
upper atmosphere immediately following an intense lightning discharge. This
phenomenon appears as an optical flash which has been reported at altitudes ranging
from ~ 60 km up to ~ 95 km, in the region known as the mesosphere [26]. Sprites are
usually caused by a positive CloudtoGround (CG) stroke [7, 8] while some sprite
observations have been linked to negative CG strokes also [9]. These CG strokes create
large quasielectrostatic (QE) fields which produce the sprites [10]. Sprites tend to be red
in color near its top due to nitrogen ionization (first positive emission) [7] and blue in
color near its bottom due to nitrogen ionization (first negative emission) [7]. Simulations
2
have indicated that the conditions allowing sprite development occur only at night [11].
The spatial structure of a sprite event can have vertical extents of up to 30 km
with lateral extents of up to 100 km in diameter [1214]. While sprites typically occur in
a large region of the upper atmosphere, it has been shown that it is in a narrow area from
7075 km where they initiate [15, 16]. This has been confirmed by models indicating
large electric fields develop at these altitudes [15]. Within this region, the sprites are
composed of filamentary columns, numbering up to 20, each with a diameter of 0.55 km
[14, 17, 18]. They travel downwards from the upper atmosphere to the thunderclouds at
velocities up to over 10
7
m/s [19]. There is a slight temporal delay after the CG lightning
discharge of up to 200 ms until the sprite becomes visible [14, 20].
1.2 Additional Transient Luminous Events
Sprites are not the only atmospheric phenomenon which has optical emissions in
the upper atmosphere. Another type of Transient Luminous Event (TLE) are known as
elves, normally described as disklike [17, 21]. These appear at altitudes ranging from ~
75105 km with lateral extents from 200660 km [9, 17, 21]. The elves are believed to be
produced by the heating of ambient electrons due to EMP fields [9, 14, 17]. These TLEs
tend to start ~ 100200 ?s after the lightning discharge and only last approximately ~ 1
ms [9, 21].
A third distinct type of TLE are blue jets which are described as ?blue conical
shapes? [13]. As opposed to the sprites, these events propagate upwards from the
thunderclouds at lower velocities [13, 22]. The lateral extents of the blue jets appear to
range from 3540 km [22, 23]. Gigantic jets have also been discovered which reach
altitudes of ~ 90 km with diameters of ~ 40 km [22].
3
The most recent type of TLE discovered are known as sprite halos, originally
mistaken for elves. These appear to be produced by QE fields, not EMP fields which are
related to the creation of elves [14]. The sprite halo occurs before the formation of the
sprites and is seen as a glow at the sprites? vertical extent [14]. Halos can have lateral
extents of less than 100 km in diameter and only have a duration of ~ 1 ms [24, 25].
1.3 Sprite Observations
For over a century, reports have circulated of these lighting discharges originating
from the clouds and traveling upwards [1]. It has only been in the past couple of decades
however that recorded images have appeared. A brief summary of these observations
follows.
1.3.1 First Recorded Image
The first images of an electrical discharge flowing upwards from thunderstorm
cloud tops were recorded in Minnesota using a ?lowlightlevel TV camera? the night of
July 5, 1989 [1]. Over two frames of film, a team observed twin flashes of light initiated
in cloud tops and dissipating in the upper atmosphere [1]. The flashes were calculated to
have a vertical extent of ~ 20 km with a separation of ~ 4 km [1]. The light was also
calculated to be 50 to 100 times as intense as the CG discharges recorded around the
same time [1].
1.3.2 Space Shuttle Images
In an effort to collect data concerning lightning events, cameras mounted on the
space shuttle?s payload bay where used in a project called the Mesoscale Lightning
Experiment (MLE) [2629]. After the first mission, control of the cameras was given to a
ground crew allowing for around the clock operation [26]. During the time period from
4
1989 through 1991, video from this experiment has identified 17 separate occurrences of
TLEs [2729]. Based on the number of CG events recorded over the same period, it is
estimated that 1 out of every 5000 of CG discharges results in a sprite [27, 28]. In
addition to the sprite events, one instance of a blue jet and elve were also recorded [29].
1.3.3 Images Recorded from Aircraft
There have been multiple attempts to capture TLEs using cameras mounted
aboard aircraft flying above thunderstorms. In July 1993, 19 events were captured over
the Midwest United States aboard a NASA owned DC8 [2]. The duration of these events
was estimated to be approximately 16 ms with an occurrence rate of 1 out of every 200
300 CG discharges [2]. Over a two week period during the summer of 1994, observations
of upper atmospheric TLEs were conducted in the Sprites94 aircraft campaign [2].
During this campaign, approximately 500 sprite events and 56 blue jets were recorded
[2]. The sprites appeared both alone and in clusters of two or more, with clusters being
more prevalent [2].
1.3.4 Further GroundBased Recordings
Since the first recorded images of sprites in 1989, thousands of sprites and other
TLEs have been observed during numerous campaigns. In October of 1997, high speed
recordings captured 42 sprite clusters, along with 4 sprite halos which have a temporal
resolution on the order of ~ 1 ms [30]. On August 29, 1998, two sprites were detected in
Mexico resulting from negative CG discharges [31]. A campaign during the early
summer of 2000 called STEPS, centered at Goodland, KS, recorded 1237 TLEs with
approximately 90% being sprites [32].
5
1.4 Overview and Historical Perspective
Along with all the visual evidence of Sprites, there have been numerous attempts
to theoretically predict and describe these events. The earliest paper that suggests the
possibility of sprites occurring was published in 1925 by C. T. R. Wilson [33]. Wilson
mentions what he calls both a ?critical value? and a ?sparkling limit.? When the electric
field at some altitude above the thundercloud exceeds this value, ionization occurs,
resulting in the possibility of an electrical discharge [33]. Wilson assumes ?the critical
field to remain proportional to the pressure? [33] which in turn allows a small electric
field at higher altitudes to result in a discharge.
1.4.1 Measurements in the Middle Atmosphere and Ionosphere
Measurements presented in a 1984 paper by Hale [34] showed some of the
peculiarities of the electric field?s behavior in the middle to upper atmosphere which
could be associated to Sprites. The measurements showed the relaxation time at higher
altitudes to be comparable to that at the source of the lightning perturbation [34]. This
differed from the theories at the time which predicted local relaxation times several
orders of magnitude shorter. Hale also suggests that the energy dissipated in this region is
nearly equivalent to that of the lightning stroke [34].
1.4.2 Finite Element Model using Ambient Conductivity Profiles
Baginski [35] used a finite element model to predict the resulting electric fields in
the upper atmosphere due to charge perturbations associated with lightning. His model
solved the complete set of Maxwell?s Equations. Baginski uses three different
conductivity profiles which include the Gish model, an exponential model, and a model
based on measured data.
6
The results from Baginski?s model confirm Hale?s observations of longer
temporal duration in the middle and upper atmosphere [35]. The electric fields show a
sharp drop after the initial peak, followed by a slow, steady decay in the latetime region.
He concludes that it is this latetime duration which increases the amount of energy
dissipated in the region [35].
1.4.3 NonLinear Conductivity Profile
An iterative approach using a conductivity profile developed by Pasko [36, 37]
was introduced which explained the breakdown of the electrical properties in the upper
atmosphere as proposed by Wilson. Pasko?s model involves using the electron?s number
density and the ionization and attachment coefficients in order to calculate the
conductivity of the region. Equations for these variables where developed based on
measured data and the previous work by Papadopoulos et al [38]. This conductivity
profile provides a more realistic estimate of the electric fields and energy dissipation.
Pasko also describes what he calls the characteristic air breakdown field, the
point at where the electric field will cause the air to breakdown electrically [36, 37, 39,
40]. It is when the electric field exceeds this breakdown threshold ionization occurs,
which according to Wilson will allow an electrical discharge. As altitude increases, this
characteristic breakdown field decreases in magnitude, which confirms Wilson?s theory
of a smaller electric field being necessary at higher elevations [33].
BarringtonLeigh [40] provided a modified set of equations for the ionization and
attachment coefficients. These equations offer slight differences from those given by
Pasko but are less numerically cumbersome and will be used in the work presented in this
7
research. The atmospheric electron number density and ambient conductivity values
provided by Pasko will be used.
1.4.4 Additional Numerical Methods
There have been many other numerical methods developed to simulate the effects
of a lightning perturbation on the upper atmosphere. Taranenko et al. developed a model
based on electron dynamics and Maxwell?s equations to determine the excitation of
optical emissions [41]. Like Pasko?s work, this was based on the heating of electrons due
to changes in the electron density. It showed an increase in ionization at altitudes greater
than 85 km with a decrease below that altitude [11].
A ?particle model? based on a quasielectrostatic design was developed by Tong
et al, indicating the number density of electrons can reach values doubled that of ambient
conditions [6]. Transmissionline models where develop by Dowden et al. which showed
that at 70 km, conductivities of over 30 ?S/m and electron densities of ~ 10
10
e

/m
3
are
needed for sprite development [42, 43].
1.5 Thesis Outline
The remainder of this thesis is outlined as follows. Chapter 2 presents the
formulation of the equations used in this model. This includes those based on a modified
form of Maxwell?s equations, ambient conductivity models, and ionization equations
based on prior research of Pasko [36, 37] and BarringtonLeigh [40]. Chapter 3 presents
specifics of the Finite Element Model and chapter 4 includes the results of the
simulations along with discussions. Chapter 5 is a summary with conclusions and
possible suggestions for future work.
8
CHAPTER 2
PROBLEM FORMATION
2.1 Overview
In this chapter equations are derived based on a modified form of Maxwell?s
equations and atmospheric constitutive parameters which will describe the effects of
ionization on the atmosphere resulting from the reconfiguration of charge due to a
lightning event. Equations describing the ambient ion conductivity are presented
followed by those that will simulate the effect that the electric field has on the electron
component of the conductivity. If significant ionization occurs, this will cause photon
emissions in the optical band [7, 3637]. Therefore, identification of significant ionization
through modeling should identify the presence of a sprite.
2.2 Derivation from Modified Maxwell?s Equations
The effects of ionization are characterized in an equation which is derived from a
modified form of Maxwell?s equations and atmospheric constitutive parameters. The
equations used are as follows in differential form:
SC
J
D
JH +
?
?
+=??
t
(2.1)
ED
0
?=
(2.2)
EJ
C
?=
(2.3)
9
t
f
?
?
?=??
?
S
J (2.4)
?
?
?
?
?
?
?
?
?
?
+??=
t
A
E V (2.5)
For all equations, the SI system of units is used where H is magnetic intensity (A/m), D is
the electric flux density (C/m
2
), E is the electric field (V/m), V is the electric potential
(V), J
S
is the source current density associated with the return stroke current (A/m
2
) [44
46], J
C
is the conduction current density (A/m
2
) [47], ? is the conductivity (mho/m), ?
0
is
the permittivity of free space (F/m), and ?
f
is the source charge density associated with
the return stroke current (C/m
3
). As study focuses on the latetime component of the
sprite, a quasistatic problem is assumed where dA/dt = 0 [45, 46].
The equations required for the simulation are derived from Eqns. (2.1)  (2.5) as
follows: Eqns. (2.2) and (2.3) are inserted into Ampere?s law (2.1) and the divergence is
taken resulting in (2.6). Eqns. (2.4) and (2.5) are then applied resulting in the continuity
equation (2.7).
()
S
J
E
EH ??+
?
?
??+??=????=
t
0
0 ?? (2.6)
0
V
V
0
=
?
?
?
?
?
?
?
?
?
?
?
?
??
+???
t
?
t
??
f
(2.7)
This equation is used in the Finite Element Model (FEM) discussed in Chapter 3. In the
next section, the source charge density which is used to approximate the effects of the
lighting discharge is developed. Afterwards, a number of conductivity profiles will be
introduced, and the effects of each on the electric field signatures examined.
10
2.3 Modeling of Lightning Discharge
The source of a sprite event is an intense, high current lightning discharge causing
charge reconfiguration [44, 49]. The model described here does not take in account the
effects of the propagating fields. As mentioned earlier, this study focuses on the latetime
or quasistatic field behavior [45, 46].
The total charge transferred at time t is the integral of the return stroke current
given by Baginski [35, 44, 50] as:
() ( )
?
?=
t
Rf
itQ
0
??
(2.8)
where ()?
R
i = lightning return stroke current
()tQ
f
= total displace charge due to the return stroke
The temporal structure of this displaced charge is expressed as [35, 44, and 50]:
()
() ( ) ( )()btatIti
t
tQ
R
f
???==
?
?
expexp
0
(2.9)
where a = 1 x 10
4
and b = 5 x 10
5
s
1
.
The charge deposition is expressed using a modified spherical Gaussian profile
provided by Baginski [44, 50]:
()
()
5.1
2
2exp
),(
),(
),,(
??
?
?
R
zrf
zrf
t
Q
t
tzr
ff
?
=
?
?
=
?
?
(2.10)
where
()
22
zzrR ??+= (2.11)
11
and z' is the altitude of charge perturbation (m), and ? is the standard deviation (? = 6000
m
2
). There is a large degree of latitude allowed in choosing the standard deviation as the
total displaced charge is the primary factor in the electric field characteristics at the
altitudes of interest [40, 44, and 50]. In this study, the positive charge center is assumed
located at z' = 10 km [37, 40, 44].
2.4 Atmospheric Conductivity Modeling
The main focus of this research is to investigate the electric field signatures and
ionization levels for different conductivity profiles. The total conductivity used in Eqn.
(2.7) consists of two components, the positive ion and electron conductivities shown by:
ei
??? +=
(2.12)
where ?
i
is the ion conductivity and ?
e
is the electron conductivity. The first profiles
introduced are for standard models [35, 37, 50, 51] based on ambient ion conductivity.
The model is then modified to include the enhanced electron conductivity component.
2.4.1 Standard Models
Two ambient ion conductivities profiles are used as standard baseline models. The
first is the Gish model [35, 50] shown below:
( )
()
()
)/(106
1021.1exp369.0
1075.3exp39.1
4000,0
4000,105.4exp94.2
321
14
4
3
4
2
1
3
1
mSFFF
zF
zF
mzF
mzzF
++?=
?=
?=
>==
60 km. This vertical component of
the conductivity is represented by Eqn. (2.15) using the ambient values for the electron
mobility and the electron density. This profile will be referred to as ?profile 3?. Fig. 2.4
shows the exponential conductivity model and profile 3 for altitudes up to 100 km.
14
40000
45000
50000
55000
60000
65000
70000
75000
80000
85000
90000
1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01 1.E+02 1.E+03
Ne (e

/cm
3
)
A
l
ti
tu
d
e
(
m
)
Figure 2.2: Ambient electron number density profile
0
10000
20000
30000
40000
50000
60000
70000
80000
90000
100000
1.E+19 1.E+20 1.E+21 1.E+22 1.E+23 1.E+24 1.E+25 1.E+26
N (m
3
)
A
ltit
u
d
e
(
m
)
Figure 2.3: Number density of air molecules (N) as a function of altitude.
15
0
20000
40000
60000
80000
100000
1.E14 1.E12 1.E10 1.E08 1.E06 1.E04 1.E02
Conductivity (S/m)
A
ltitu
d
e
(m
)
Exponential
Profile 3
Figure 2.4: Conductivity profile as a function of altitude.
2.4.2.2 Electron Mobility
The electron mobility model is a functional fit based on experimental data and
given by [36, 37]:
()
mVNNENN
mVNNExaN
e
i
i
ie
/1062.1/,36.1
/1062.1/,log
3
00
3
0
2
0
?<=
??=
?
=
?
?
(2.17)
where ()NEx /log= , a
0
= 50.970, a
1
= 3.0260, and a
2
= 8.4733 x 10
2
. Fig. 2.5 shows
the electron mobility for 50 and 80 km altitude as a function of electric field strength. As
can be seen from the figure, for low electric field values the electron mobility tends to the
ambient values.
16
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E04 1.E02 1.E+00 1.E+02 1.E+04
E (V/m)
e
(m
2
/V
/
s
)
50 km
80 km
Figure 2.5: Electron mobility (?
e
) at 50 and 80 km.
2.4.2.3 Electron Density
The electron number density is described according to the following differential
equation [36, 37, and 40]:
()
2
eeai
e
NN
td
dN
??? ??=
(2.18)
where ?
i
is the ionization coefficient (1/s),?
a
is the attachment coefficient (1/s) and ? is
the effective recombination coefficient [36]. The last term of
2
e
N? is used to model the
propagation channels to altitudes <50 km resulting from ionization breakdown [36]. This
term is neglected in this study as the value is not well known for high altitudes and the
solution becomes very numerically intensive.
17
2.4.2.4 Ionization and Attachment Coefficient
The ionization and attachment coefficients are solved using functions provided by
BarringtonLeigh [40]. The ionization coefficient is given by:
?
= ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
=
<=
3
0
0
10
0
1
0
log
otherwise,10
,mV1122000for0
i
i
i
P
i
i
NN
E
ap
N
N
NN
E
?
?
(2.19)
with a
0
= 624.68, a
1
= 239.60, a
2
= 32.878, and a
3
= 1.4546. The attachment coefficient
is given by:
?
= ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
=
<=
4
0
0
10
0
1
0
log
otherwise,10
,mV316200for0
i
i
i
P
a
a
NN
E
bp
N
N
NN
E
?
?
(2.20)
with a
0
= 3567.0, a
1
= 1992.68, a
2
= 416.601, a
3
= 38.7290, and a
4
= 1.35113. These
coefficients are shown as a function of the electric field for altitudes of 50 and 80 km in
Fig. 2.6. It is of note to mention that when compared to the electron mobility, it is not
until significant electric field strengths occur before the coefficient values change the
electron number density, which in turn affects the conductivity.
The electric field intensity at which ?
i
= ?
a
corresponds to E
k
, the characteristic air
breakdown field (V/m) [40]. This breakdown field is described by the equation [36, 37]:
0
6
102.3
N
N
E
k
?= (2.21)
18
The electric field must exceed the electrical breakdown strength of the atmosphere,
shown by Fig. 2.7, for sprites to occur [53]. Fig. 2.8 shows the net ionization (?
i
 ?
a
) as a
function of electric field strength at 70 km.
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
1.E+01 1.E+02 1.E+03 1.E+04
E (V/m)
?
i,
a
(s
1
)
vi  50 km
va  50 km
vi  80 km
va  80 km
Figure 2.6: Ionization and attachment coefficients for 50 and 80 km.
19
0
10000
20000
30000
40000
50000
60000
70000
80000
90000
100000
1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07
E
k
(V/m)
A
l
tit
u
d
e
(
m
)
Figure 2.7: Characteristic air breakdown field with respect to altitude.
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
E/Ek
?
(s
1
)
vi
va
vi  va
Figure 2.8: Ionization and attachment coefficients normalized to E
k
for 70 km
20
CHAPTER 3
FINITE ELEMENT MODEL
3.1 Overview
The equations developed describing the electrical effects on the atmosphere due
to a CG lightning discharge will be solved using a Finite Element Model (FEM). This
will allow all the parameters to be solved simultaneously as opposed to using an iterative
approach as done in previous research. In this chapter, Eqn. (2.7) will be modified to the
form used in the code. The geometry of the region will then be described along with the
corresponding boundary conditions used. The FEM will be solved using cylindrical
coordinates.
3.2 Development of Equation
Equation 2.7 is described in cylindrical coordinates as follows:
0
V
V
0
=
?
?
?
?
?
?
?
?
?
?
?
?
??
+???
t
?
t
??
f
(3.1)
Using the following vector identities
()
z
z
A
A
A
?
?
+
?
?
+
?
?
=??
??
?
??
?
?
11
A
(3.2)
() www ??+??=?? AAA
(3.3)
21
(3.1) is the differential form of the equation describing the electrical behavior of the
atmosphere when subject to charge perturbations. The region for the FEM is assumed to
be azimuthally symmetric about the zaxis, resulting in all derivatives with respect to ?
going to zero.
()
tzz
t
tz
f
??
?
+
?
?
=
??
?
+
?
?
=
=
?
?
+
?
?
+
?
?
VV
B
VV
A
0
BA1
2
0
2
0
??
?
?
?
?
?
?
?
?
(3.4)
3.3 Geometry of the Region
The FEM will be solved in a region depicted by a cylinder with a radius of 80 km
and a height of 95 km (Fig. 3.1) consistent with earlier models [35, 44, and 50].
Figure 3.1: Geometry of model. Filled square denote points at which measurements
are recorded.
+
10 km
Source of Charge
40 km
80 km
z
95 km
50 km
50 km
?
22
3.3.1 Boundary Conditions
Upper and Lower Boundaries ? The lower boundary of the earth?s surface is
modeled as a perfect electrical conductor. With the earth?s surface having conductivity
values of 10
3
to 10
2
S/m and the atmosphere having values of 10
14
to 10
13
S/m [44, 50].
This is a difference of at least 10 orders of magnitude makes the earth?s surface appear as
a perfect electrical conductor.
The upper boundary is assumed to be a perfect electrical conductor. The
approximation of 95 km is appropriate since the simulations of interest are at lower
altitudes and the conductivity is increasing at approximately an exponential rate. Tests
have shown that increasing this boundary does not result in significant differences in the
electric field in the area of interest. For both the upper and lower boundary, the horizontal
electrical fields are set equal to zero and the electron number density is set to the ambient
values.
Outer Boundary ? The outer radial boundary extends out to 80 km. While this
distance is not limited by any physical constraint, the accuracy of the FEM?s solution will
be greater if the discretized volume is kept to a minimum. Simulations have shown that
increasing this boundary past 80 km does not result in differences in the area of interest.
3.4 Additional FEM Parameters
Two additional parameters defined in the FEM include the density of the
triangulation (mesh) and the time step. The triangulation defines the step sizes and the
limits of the model.
The triangulation density controls both the number and relative density of the
mesh. The area where the mesh density is the greatest is in the proximity of the charge
23
perturbation. As the distance increases from the charge center, the triangle density
steadily decreases. The mesh density is also increased in the vicinity of the sprite event
(6090 km.)
To determine the acceptable number of triangles to be used in the FEM
simulation, multiple runs were conducted and the results compared. The simulations
included runs with the number of triangles ranging from 1000 to 4000 triangles in 1000
triangle increments. While there were differences in the results from 1000 to 3000
triangles, it was determined that between 3000 and 4000 triangles, the differences were
negligible and therefore 3000 triangles was chosen.
The time step used in the simulation is determined by both the duration of the
simulation and the required number of time steps to accurately simulate the event. The
minimum time step used was 10
8
seconds with a duration of 100 seconds. The
maximum time used is 100 seconds as it was determined that this allowed the equations
to run its course in the model. The simulations were divided into multiple runs, each
consisting of 100 steps per decade (i.e., 10
2
10
1
has 100 time steps). This allows for very
accurate short time analysis without prohibitively long simulations.
24
CHAPTER 4
SIMULATION RESULTS
4.1 Overview
The FEM results are presented in two sections with figures and comments given
for both. The first section contains results those for the cases where the ambient
conductivity profiles were used. Simulations of the total electric field in addition to the
horizontal and vertical components are shown in Figs. 4.24.22.
The second section contains the results for the case of where the ionization is
included in conductivity calculations. The simulated electric fields, conductivity profiles,
electron number densities, electron mobilities, and the ionization and attachment
coefficients are shown in Figs. 4.234.64.
4.2 Ambient Conductivity Profiles
The three ambient conductivity profiles used in the research are shown in Fig. 4.1.
They include the Gish profile, the exponential profile, and ?profile 3? which includes
high altitude electron ionization.
4.2.1 Total Electric Field Results
Figs. 4.24.7 present the simulated electric fields at a radial distance of 10 km and
altitudes of 4090 km. Each of these plots contrasts the effects of the various
conductivities. The results for an altitude of 70 km and radial distances of 050 km are
25
shown in Figs. 4.84.10. Characteristics of the simulations are given below with
discussed explanations.
1. The Gish model has a significantly lower conductivity than the other models [35,
50] and, as expected, the associated electric fields have a longer temporal
duration.
2. As the altitude increases, the peak magnitude of the electric fields occurs at earlier
times. At a given altitude, the peak value of the electric field occurs at
approximately the same time [Figs. 4.84.10].
3. The peak magnitude and duration of the electric field decreases as the altitude
increases for all profiles.
4. For altitudes of 40 and 50 km, the electric fields are virtually identical for the
exponential conductivity model and ?profile 3?. This is due to the electron
ionization only modifying the conductivity at altitudes >~ 60 km.
5. At the altitudes of 80 and 90 km, the peak value of the electric field for the
simulation using the Gish conductivity exceeds the other.
26
40000
50000
60000
70000
80000
90000
1.E11 1.E10 1.E09 1.E08 1.E07 1.E06 1.E05 1.E04
Conductivity (S/m)
Al
titud
e (
m
)
Gish
Exponential
Profile 3
Figure 4.1: Conductivity profiles for ambient simulations
0
100
200
300
400
500
600
700
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01 1.E+02
Time (s)
E (
V
/
m
)
Gi sh
Exponential
Pr of i l e 3
Figure 4.2: Electric field simulations at z = 40 km and ? = 10 km for ambient
conductivity profiles.
27
0
100
200
300
400
500
600
700
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01 1.E+02
Time (s)
E (
V
/
m
)
Gi sh
Exponential
Pr of i l e 3
Figure 4.3: Electric field simulations at z = 50 km and ? = 10 km for ambient
conductivity profiles.
0
100
200
300
400
500
600
700
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01 1.E+02
Time (s)
E (
V
/
m
)
Gi s h
Exponential
Pr of i l e 3
Figure 4.4: Electric field simulations at z = 60 km and ? = 10 km for ambient
conductivity profiles.
28
0
100
200
300
400
500
600
700
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01 1.E+02
Time (s)
E (
V
/
m
)
Gi sh
Exponential
Pr of i l e 3
Figure 4.5: Electric field simulations at z = 70 km and ? = 10 km for ambient
conductivity profiles.
0
100
200
300
400
500
600
700
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01 1.E+02
Time (s)
E (
V
/
m
)
Gi sh
Exponential
Pr of i l e 3
Figure 4.6: Electric field simulations at z = 80 km and ? = 10 km for ambient
conductivity profiles.
29
0
100
200
300
400
500
600
700
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01 1.E+02
Time (s)
E (
V
/
m
)
Gi sh
Exponential
Pr of i l e 3
Figure 4.7: Electric field simulations at z = 90 km and ? = 10 km for ambient
conductivity profiles.
0
10
20
30
40
50
60
70
80
90
100
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01 1.E+02
Time (s)
E (
V
/
m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.8: Electric field simulation using Gish conductivity profile at z = 80 km for
radial distances from 0 to 50 km.
30
0
10
20
30
40
50
60
70
80
90
100
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01 1.E+02
Time (s)
E (
V
/
m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.9: Electric field simulation using exponential conductivity profile at z = 80 km
for radial distances from 0 to 50 km.
0
10
20
30
40
50
60
70
80
90
100
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01 1.E+02
Time (s)
E (
V
/
m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.10: Electric field simulation using profile 3 at z = 80 km for radial distances
from 0 to 50 km.
31
4.2.2 Vertical and Horizontal Electric Field Results
The vertical and horizontal electric fields at a radial distance of 10 km and an
altitude of 40 km are shown in Figs. 4.11 and 4.12 respectively. Figs. 4.134.21 show the
horizontal and vertical electric fields at altitudes of 40 and 80 km and radial distances
from 050 km for the various conductivity models. Some of the noteworthy
characteristics of these simulations are described below.
1. For all altitudes and radii of interest, the peak magnitudes of the vertical electric
fields are greater than those of the horizontal electric fields.
2. Unlike the vertical electric fields, the peak magnitudes of the horizontal electric
fields do not occur on the radial axis. At 40 km altitude, the peak magnitude of the
electric field occurs at a radial distance of ~ 20 km. For 80 km altitude, the peak
value of the electric field occurs between 30 and 40 km.
3. At an altitude of 80 km and radial distance of 50 km, the horizontal electric fields
show a temporary field reversal for ?profile 3?.
4. The horizontal electric field magnitudes are not strongly affected by the
conductivity profiles while the temporal effects are similar to that of the vertical
electric fields.
32
700
600
500
400
300
200
100
0
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01 1.E+02
Time (s)
E
z
(V
/m
)
Gi sh
Exponential
Pr of i l e 3
Figure 4.11: Vertical electric field simulations at z = 40 km and ? = 10 km for ambient
conductivity profiles.
250
200
150
100
50
0
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01 1.E+02
Time (s)
E
(V
/m
)
Gi sh
Exponential
Pr of i l e 3
Figure 4.12: Horizontal electric field simulations at z = 40 km and ? = 10 km for ambient
conductivity profiles.
33
800
700
600
500
400
300
200
100
0
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01 1.E+02
Time (s)
E
z
(V
/m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.13: Vertical electric field simulation using Gish conductivity profile at z = 40
km for radial distances from 0 to 50 km.
300
250
200
150
100
50
0
50
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01 1.E+02
Time (s)
E
(V/
m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.14: Horizontal electric field simulation using Gish conductivity profile
at z = 40 km for radial distances from 0 to 50 km.
34
800
700
600
500
400
300
200
100
0
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01 1.E+02
Time (s)
E
z
(V
/
m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.15: Vertical electric field simulation using exponential conductivity profile
at z = 40 km for radial distances from 0 to 50 km.
300
250
200
150
100
50
0
50
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01 1.E+02
Time (s)
E
(V
/m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.16: Horizontal electric field simulation using exponential conductivity profile at
z = 40 km for radial distances from 0 to 50 km.
35
100
90
80
70
60
50
40
30
20
10
0
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01 1.E+02
Time (s)
E
z
(V
/m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.17: Vertical electric field simulation using Gish conductivity profile at z = 80
km for radial distances from 0 to 50 km.
25
20
15
10
5
0
5
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01 1.E+02
Time (s)
E
(V
/m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.18: Horizontal electric field simulation using Gish conductivity profile
at z = 80 km for radial distances from 0 to 50 km.
36
100
90
80
70
60
50
40
30
20
10
0
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01 1.E+02
Time (s)
E
z
(V
/m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.19: Vertical electric field simulation using exponential conductivity profile
at z = 80 km for radial distances from 0 to 50 km.
25
20
15
10
5
0
5
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01 1.E+02
Time (s)
E
(V
/
m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.20: Horizontal electric field simulation using exponential conductivity profile
at z = 80 km for radial distances from 0 to 50 km.
37
100
80
60
40
20
0
20
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01 1.E+02
Time (s)
E (
V
/
m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.21: Vertical electric field simulations using profile 3 at z = 80 km for radial
distances from 0 to 50 km.
25
20
15
10
5
0
5
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01 1.E+02
Time (s)
E (
V
/
m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.22: Horizontal electric field simulations profile 3 at z = 80 km for radial
distances from 0 to 50 km.
38
4.3 NonLinear Conductivity
In this section, electric fields, conductivities, electron mobilities, electron number
densities, and ?
i
and ?
a
resulting from CG lightning are presented for the nonlinear
conductivity discussed in Section 2.4.2. All simulations presented in this section will be
with respect to the nonlinear conductivity and limited As stated earlier, the level of
electron ionization does not modify the conductivity at altitudes below 60 km; therefore
all simulations presented in this section will be limited to altitudes from 6090 km and
will be with respect to the nonlinear conductivity profile.
4.3.1 Total Electric Field Results
Figs. 4.234.28 show the electric field simulations for a constant radial distance at
altitudes varying from 6090 km. Figs. 4.294.32 show the electric field simulations at a
constant altitude for radial distances of 050 km. Figs. 4.334.34 show comparisons of
these simulations to the electric field signatures found using the exponential conductivity
model. Following are some characteristics of the electric fields:
1. Fig. 4.31 shows the rate of decay of the electric field for an altitude of 80 km is
more rapid closer to the radial axis. For the other altitudes this trend is not
observed.
2. Fig. 4.33 shows that the value of the peak electric field is larger for the nonlinear
conductivity simulations than the ambient exponential conductivity simulations at
80 km altitude. The rate of decay of the electric field following the onset of the
peak value is greater for the nonlinear case.
3. Fig. 4.34 compares the electric fields for the nonlinear conductivity profile and
the exponential conductivity model at 0.5 ms at a radial distance of 0 km. The
39
nonlinear conductivity?s peak electric field is larger at altitudes less than ~ 75
suggesting the high altitude ionization effectively ?shorts? the electric field.
0
50
100
150
200
250
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01
Time (s)
E (
V
/
m
)
60 km
70 km
80 km
90 km
Figure 4.23: Electric field simulations using the nonlinear conductivity profile for
? = 0 km at altitudes of 6090 km.
40
0
50
100
150
200
250
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01
Time (s)
E (
V
/
m
)
60 km
70 km
80 km
90 km
Figure 4.24: Electric field simulations using the nonlinear conductivity profile for
? = 10 km at altitudes of 6090 km.
0
50
100
150
200
250
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01
Time (s)
E (
V
/
m
)
60 km
70 km
80 km
90 km
Figure 4.25: Electric field simulations using the nonlinear conductivity profile for
? = 20 km at altitudes of 6090 km.
41
0
50
100
150
200
250
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01
Time (s)
E (
V
/
m
)
60 km
70 km
80 km
90 km
Figure 4.26: Electric field simulations using the nonlinear conductivity profile for
? = 30 km at altitudes of 6090 km.
0
50
100
150
200
250
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01
Time (s)
E (
V
/
m
)
60 km
70 km
80 km
90 km
Figure 4.27: Electric field simulations using the nonlinear conductivity profile for
? = 40 km at altitudes of 6090 km.
42
0
50
100
150
200
250
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01
Time (s)
E (
V
/
m
)
60 km
70 km
80 km
90 km
Figure 4.28: Electric field simulations using the nonlinear conductivity profile for
? = 50 km at altitudes of 6090 km.
0
50
100
150
200
250
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00
Time (s)
E (
V
/
m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.29: Electric field simulations using the nonlinear conductivity profile for
z = 60 km at radial distances of 050 km.
43
0
20
40
60
80
100
120
140
160
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00
Time (s)
E (
V
/
m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.30: Electric field simulations using the nonlinear conductivity profile for
z = 70 km at radial distances of 050 km.
0
10
20
30
40
50
60
70
80
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00
Time (s)
E (
V
/
m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.31: Electric field simulations using the nonlinear conductivity profile for
z = 80 km at radial distances of 050 km.
44
0
1
2
3
4
5
6
7
8
9
10
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00
Time (s)
E (
V
/
m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.32: Electric field simulations using the nonlinear conductivity profile for
z = 90 km at radial distances of 050 km.
0
10
20
30
40
50
60
70
80
1.E06 1.E05 1.E04 1.E03 1.E02 1.E01
Time (s)
E (
V
/
m
)
80 km  NonLinear
80 km  Exponential
Figure 4.33: Electric field simulations at ? = 0 km and z = 80 km for the exponential and
the nonlinear conductivity profiles.
45
60000
65000
70000
75000
80000
85000
90000
0 50 100 150 200 250
E (V/m)
A
l
ti
tu
d
e
(m
)
NonLinear 0.5 ms
Exponential 0.5 ms
Figure 4.34: Electric field simulations at ? = 0 km and z = 6090 km for the exponential
and the nonlinear conductivity profiles at a time of 0.5 ms.
4.3.2 Vertical and Horizontal Electric Field Results
Figs. 4.354.42 show the vertical and horizontal electric fields for the nonlinear
conductivity simulation. Each of these components is shown at elevations varying from
6090 km with radial distances ranging from 050 km. Observations from these results
follow.
1. The electric field is primarily vertically oriented. This trait is seen for the ambient
conductivity simulations also.
2. For the vertical electric fields, the maximum peak value occurs on the zaxis
(vertical) for all altitudes with the magnitude decreasing as the radial distance
increases. For the horizontal electric fields, the maximum peak value occurs at a
radial distance of 3040 km for all cases of interest.
46
250
200
150
100
50
0
1.00E06 1.00E05 1.00E04 1.00E03 1.00E02 1.00E01 1.00E+00
Time (s)
E
z
(V
/m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.35 Vertical electric field simulations using the nonlinear conductivity profile
for 60 km altitude at radial distances of 050 km.
60
50
40
30
20
10
0
10
1.00E06 1.00E05 1.00E04 1.00E03 1.00E02 1.00E01 1.00E+00
Time (s)
E
(V
/m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.36 Horizontal electric field simulations using the nonlinear conductivity profile
for 60 km altitude at radial distances of 050 km.
47
160
140
120
100
80
60
40
20
0
1.00E06 1.00E05 1.00E04 1.00E03 1.00E02 1.00E01
Time (s)
E
z
(V/
m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.37: Vertical electric field simulations using the nonlinear conductivity profile
for 70 km altitude at radial distances of 050 km.
30
25
20
15
10
5
0
5
10
1.00E06 1.00E05 1.00E04 1.00E03 1.00E02 1.00E01
Time (s)
E
(V
/m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.38: Horizontal electric field simulations using the nonlinear conductivity profile
for 70 km altitude at radial distances of 050 km.
48
80
70
60
50
40
30
20
10
0
10
1.00E06 1.00E05 1.00E04 1.00E03 1.00E02
Time (s)
E
z
(V
/m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.39: Vertical electric field simulations using the nonlinear conductivity profile
for 80 km altitude at radial distances of 050 km.
10
8
6
4
2
0
2
4
6
8
10
1.00E06 1.00E05 1.00E04 1.00E03 1.00E02
Time (s)
E
(V
/m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.40: Horizontal electric field simulations using the nonlinear conductivity profile
for 80 km altitude at radial distances of 050 km.
49
10
9
8
7
6
5
4
3
2
1
0
1.00E06 1.00E05 1.00E04 1.00E03 1.00E02
Time (s)
E
z
(V
/m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.41: Vertical electric field simulations using the nonlinear conductivity profile
for 90 km altitude at radial distances of 050 km.
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
1.00E06 1.00E05 1.00E04 1.00E03 1.00E02
Time (s)
E
(V
/
m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.42: Horizontal electric field simulations using the nonlinear conductivity profile
for 90 km altitude at radial distances of 050 km.
50
4.3.3 Conductivity and Electron Density Results
This section discusses the behavior of the nonlinear conductivity used in the
simulation. According to Eqns. (2.12) and (2.15), the total conductivity is given by
eeei
Nq ??? += where ?
i
is the ambient ion exponential model, ?
e
, the electron
mobility, and N
e
, the electron number density.
Fig. 4.43 shows the simulated electric fields for altitudes from 6090 km at radial
distances varying from 050 km compared to the characteristic breakdown field. The
electric field values shown are calculated to be the maximum values at each location over
the entire time span of the simulation. The region in altitude and radial distance
corresponding to the electric field values greater than the breakdown field is where
ionization occurs. This results in photon emissions in the optical band, identifying the
presence of a sprite [7, 36, 37].
60000
65000
70000
75000
80000
85000
90000
0 50 100 150 200 250
E (V/m)
A
l
ti
tu
d
e
(
m
)
Ek
Max E  0 km
Max E  10 km
Max E  20 km
Max E  30 km
Max E  40 km
Max E  50 km
Figure 4.43: Maximum electric field strengths over time for altitudes from 6090 km at
radial distances from 050 km compared to the characteristic air breakdown field.
51
Figs. 4.44 and 4.45 show the conductivity profiles for altitudes from 6095 km at
radial distances varying from 030 km for 1 and 100 ms times. Figs. 4.464.47 show the
corresponding electron number densities. The ambient profiles have been added to the
four plots for comparison. The electron densities computed in this simulation are
confirmed by the results of earlier work shown by Fig. 4.48 [37]. These results show the
electron density changes for three ambient electron density models [37]. While greater
change is shown in the earlier work, a much larger total charge was also used.
The remaining figures are broken up into groups of four at different altitudes.
Figs. 4.4952 are at 60 km, Figs. 4.5356 at 70 km, Figs. 4.5760 at 80 km, and Figs.
4.6164 at 90 km. In each group, the first plot is the conductivity, second is the electron
mobility, third is the number density of electrons, and last is the difference between the
ionization and attachment coefficients. Each plot presents results for the simulations at
radial distances from 050 km. The following are some observations and explanations of
these results.
1. The change in conductivity due to the electric field increased with altitude. This is
a direct result of the level of ionization increasing as altitude is increased. At 60
km, the conductivity changes by less than one percent while at 90 km, it changes
by over an order of magnitude.
2. At a given altitude, the temporal behavior of the conductivity tends to track the
behavior of the electron mobility.
3. The value for the electron number density after the transient slightly differs from
the ambient values for the cases shown. If the electric fields do not reach the
52
characteristic breakdown field, ?
i
 ?
a
stays negative and therefore the final value
of the electron number density will be slightly less than the ambient value.
Likewise, when the electric field exceeds the breakdown field, the number density
will increase for that period of time. This is a second order effect that is due to the
omission of the
2
e
N? term, discussed in Section 2.4.2.3, from the electron
ionization equation. However, the inclusion of this term makes the solution very
numerically intensive. Additionally, the value is not well known for high
altitudes.
60000
65000
70000
75000
80000
85000
90000
95000
1.E09 1.E08 1.E07 1.E06 1.E05 1.E04 1.E03
Conductivity (S/m)
A
l
tit
u
d
e (m
)
ambient
0 km
10 km
20 km
30 km
Figure 4.44: Conductivity profiles using the nonlinear model at 1 ms for radial distances
from 030 km. The ambient profile is shown for reference.
53
60000
65000
70000
75000
80000
85000
90000
95000
1.E09 1.E08 1.E07 1.E06 1.E05 1.E04 1.E03
Conductivity (S/m)
A
l
tit
u
d
e
(m
)
ambient
0 km
10 km
20 km
30 km
Figure 4.45: Conductivity profiles using the nonlinear model at 100 ms for radial
distances from 030 km. The ambient profile is shown for reference.
50000
60000
70000
80000
90000
1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 1.E+10
N
e
(m
3
)
A
l
t
i
t
ude
(
m
)
ambient
0 km
10 km
20 km
30 km
Figure 4.46: Electron density profiles using the nonlinear conductivity model at 1 ms for
radial distances from 030 km. The ambient profile is shown for reference.
54
50000
60000
70000
80000
90000
1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 1.E+10
N
e
(e

/m
3
)
A
l
tit
u
d
e (m
)
ambient
0 km
10 km
20 km
30 km
Figure 4.47: Electron density profiles using the nonlinear conductivity model at 100 ms
for radial distances from 030 km. The ambient profile is shown for reference.
Figure 4.48: Electron density changes corresponding to three ambient electron density
models for 200 C CG stroke, provided by Dr. V. P. Pasko [37].
55
1.095E09
1.097E09
1.099E09
1.101E09
1.103E09
1.105E09
1.107E09
1.109E09
1.111E09
1.113E09
1.115E09
1.E08 1.E07 1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01
Time (s)
C
o
n
d
u
c
tiv
ity
(S
/m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.49: Conductivity profile using the nonlinear model at 60 km altitude for radial
distances from 050 km.
1.E+02
1.E+03
1.E+04
1.E+05
1.E08 1.E07 1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01
Time (s)
?
e
(m
2
/V
/s
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.50: Electron mobility using the nonlinear conductivity model at 60 km altitude
for radial distances from 050 km.
56
5.20E+03
5.25E+03
5.30E+03
5.35E+03
5.40E+03
5.45E+03
5.50E+03
5.55E+03
5.60E+03
1.E05 1.E04 1.E03 1.E02
Time (s)
N
e
(e

/m
3
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.51: Electron number density using the nonlinear conductivity model at 60 km
altitude for radial distances from 050 km.
6.0E+01
5.0E+01
4.0E+01
3.0E+01
2.0E+01
1.0E+01
0.0E+00
1.0E+01
1.E05 1.E04 1.E03 1.E02 1.E01
Time (s)
?
i
?
?
a
(
s
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.52: Difference between ionization and attachment coefficients, ?
i
 ?
a
, using the
nonlinear conductivity model at 60 km altitude for radial distances from 050 km.
57
5.4E09
5.6E09
5.8E09
6.0E09
6.2E09
6.4E09
6.6E09
6.8E09
1.E08 1.E07 1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01
Time (s)
Cond
uc
t
i
vi
t
y
(
S
/
m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.53: Conductivity profile using the nonlinear model at 70 km altitude for radial
distances from 050 km.
1.E+02
1.E+03
1.E+04
1.E+05
1.E08 1.E07 1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01
Time (s)
?
e
(m
2
/V
/s
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.54: Electron mobility using the nonlinear conductivity model at 70 km altitude
for radial distances from 050 km.
58
0.0E+00
5.0E+04
1.0E+05
1.5E+05
2.0E+05
2.5E+05
1.E05 1.E04 1.E03 1.E02
Time (s)
N
e
(e

/m
3
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.55: Electron number density using the nonlinear conductivity model at 70 km
altitude for radial distances from 050 km.
1.4E+03
1.2E+03
1.0E+03
8.0E+02
6.0E+02
4.0E+02
2.0E+02
0.0E+00
2.0E+02
1.E05 1.E04 1.E03 1.E02
Time (s)
?
i
?
?
a
(
s
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.56: Difference between ionization and attachment coefficients, ?
i
 ?
a
, using the
nonlinear conductivity model at 70 km altitude for radial distances from 050 km.
59
1.E08
1.E07
1.E06
1.E05
1.E08 1.E07 1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00 1.E+01
Time (s)
Co
nduc
t
i
vi
t
y
(
S
/
m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.57: Conductivity profile using the nonlinear model at 80 km altitude for radial
distances from 050 km.
1.E+03
1.E+04
1.E+05
1.E+06
1.E08 1.E07 1.E06 1.E05 1.E04 1.E03 1.E02 1.E01 1.E+00
Time (s)
?
e
(m
2
/V
/s
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.58: Electron mobility using the nonlinear conductivity model at 80 km altitude
for radial distances from 050 km.
60
0.0E+00
2.0E+07
4.0E+07
6.0E+07
8.0E+07
1.0E+08
1.2E+08
1.4E+08
1.6E+08
1.8E+08
2.0E+08
1.E05 1.E04 1.E03
Time (s)
N
e
(e

/m
3
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.59: Electron number density using the nonlinear conductivity model at 80 km
altitude for radial distances from 050 km.
5.0E+03
0.0E+00
5.0E+03
1.0E+04
1.5E+04
2.0E+04
2.5E+04
3.0E+04
1.E05 1.E04 1.E03
Time (s)
?
i
?
?
a
(
s
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.60: Difference between ionization and attachment coefficients, ?
i
 ?
a
, using the
nonlinear conductivity model at 80 km altitude for radial distances from 050 km.
61
1.0E07
1.0E06
1.0E05
1.0E04
1.E08 1.E07 1.E06 1.E05 1.E04 1.E03 1.E02
Time (s)
Cond
uc
t
i
vi
t
y
(
S
/
m
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.61: Conductivity profile using the nonlinear model at 90 km altitude for radial
distances from 050 km.
1.E+03
1.E+04
1.E+05
1.E+06
1.E08 1.E07 1.E06 1.E05 1.E04 1.E03 1.E02 1.E01
Time (s)
?
e
(m
2
/V
/s
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.62: Electron mobility using the nonlinear conductivity model at 90 km altitude
for radial distances from 050 km.
62
2.380E+08
2.385E+08
2.390E+08
2.395E+08
2.400E+08
2.405E+08
2.410E+08
2.415E+08
2.420E+08
1.E06 1.E05 1.E04 1.E03
Time (s)
N
e
(e

/m
3
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.63: Electron number density using the nonlinear conductivity model at 90 km
altitude for radial distances from 050 km.
1.0E+02
5.0E+01
0.0E+00
5.0E+01
1.0E+02
1.5E+02
2.0E+02
2.5E+02
3.0E+02
1.E06 1.E05 1.E04 1.E03
Time (s)
?
i
? ?
a
(
s
)
0 km
10 km
20 km
30 km
40 km
50 km
Figure 4.64: Difference between ionization and attachment coefficients, ?
i
 ?
a
, using the
nonlinear conductivity model at 90 km altitude for radial distances from 050 km.
63
CHAPTER 5
CONCLUSIONS
This thesis presented simulations using a Finite Element Model with multiple
conductivity profiles to add to this body of research to confirm the possibility of a high
level of ionization and therefore the presence of a sprite.
In Chapter 3, a Finite Element Model was described that effectively models
transient electric field behavior in the atmosphere. These equations are simultaneously
solved as opposed to an earlier derivative approach [36, 37, 40].
In Chapter 4, simulations of the transient electric field are presented for a variety
of conductivities. The research showed that the upper atmosphere is strongly affected by
the quasistatic effects of a positive CG stroke. For the ambient profiles, the electric field
reaches its peak value in a matter of milliseconds, and then slowly decays over time. The
results agree well with simulations done by Baginski et al. [35, 45, 46, 50]. It was shown
at high altitudes the maximum electric field strength decreased in magnitude and
occurred at shorter times. When nonlinear effects were included in the conductivity, an
increase of up to two orders of magnitude in the overall conductivity was observed. At 90
km altitude, this effectively ?shorted out? the electric field.
Results for the nonlinear conductivity profile were then presented. The electron
component of the conductivity does not modify the results at altitudes below 60 km [37,
52]. At higher altitudes, the electric field results had differences compared to the ambient
64
conductivity models. A sharp increased rate of decay in electric field strength at 80 km
was observed. Fig. 4.43 show the region of the simulation where ionization occurred,
identifying the presence of a sprite.
Figs. 4.46 and 4.47 show the electron density profiles with respect to the ambient
values at two time periods. These figures confirmed the results of the simulation when
compared to previous work shown by Fig. 4.48 [37]. While the earlier work showed a
greater divergence from the ambient values, [37] used a much larger total. At 80 km
altitude, the electron density reached values over an order of magnitude greater than the
ambient. This led to an increase in the conductivity which directly resulted in the sharp
decrease of the electric field strength.
5.1 Future Work
The research presented here suggests future research should focus on possible
inclusion of the ?N
e
2
term to the model. Ideally all the transient conductivities should
return to the ambient values. A denser mesh can be added to the region where the sprite is
expected to occur. Visual evidence shows, sprites are similar to individual columns with
diameters on the order of just a few kilometers [14, 17, 18] suggesting a more discretized
model be used in the vicinity of the sprite.
65
BIBLIOGRAPHY
[1] R. C. Franz, R. J. Nemzek, and J. R. Winckler, ?Television image of a large upward
electrical discharge above a thunderstorm system,? Science, vol. 249, no. 4964, pp.
4851, July 1990.
[2] D. D. Sentman and E. M. Wescott, ?Red sprites and blue jets: Thunderstorm
excited optical emissions in the stratosphere, mesosphere, and ionosphere,? Phys.
Plasmas, vol. 2, no. 6, pp 25142522, June 1995.
[3] V. P. Pasko, U. S. Inan, and T. F. Bell, ?Sprites as evidence of vertical gravity wave
structures above mesoscale thunderstorms,? Geophysical Research Letters, vol. 24,
no. 14, pp. 17351738, July 1997.
[4] S. B. Mende and D. D. Sentman, ?Lightning between earth and space,? Scientific
American, vol. 277, iss. 2, pp. 5659, Aug. 1997.
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