JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. D9, 10.1029/2000JD000244, 2002
Simulated three-dimensional branched lightning
in a numerical thunderstorm model
Edward R. Mansell1
Department of Physics and Astronomy and Cooperative Institute for Mesoscale Meteorological Studies (CIMMS)
University of Oklahoma, Norman, Oklahoma, USA
Donald R. MacGorman and Conrad L. Ziegler
National Severe Storms Laboratory, Norman, Oklahoma, USA
Jerry M. Straka
School of Meteorology, University of Oklahoma, Norman, Oklahoma, USA
Received 8 December 2000; revised 27 August 2001; accepted 7 September 2001; published 2 May 2002
[1] Lightning discharges are simulated by using a stochastic dielectric breakdown model within a
numerical thunderstorm model with extensive parameterizations of electrification mechanisms. The
lightning model simulates the macroscopic bidirectional extension of discharges as a step-by-step
stochastic process. Discharge channels are propagated on a uniform grid, and the direction of
propagation (including diagonals) for a particular step is chosen randomly, with the probability for
choosing a particular direction depending on the net electric field. After each propagation step the
electric fields are recomputed via Poisson’s equation to account for the effect of the conducting
channel. The lightning parameterization produces realistic looking, three-dimensional, branched
lightning discharges. A variety of lightning types have been produced, including intracloud
discharges, negative cloud-to-ground (CG) lightning, and positive CG lightning. The model
simulations support the hypothesis that negative CG flashes occur only when a region of positive
charge exists below the main negative charge region. Similarly, simulated positive CG flashes were
found to occur only in regions of storms where the two significant charge layers closest to ground
had roughly a ‘‘normal dipole’’ structure (i.e., positive charge above negative).
INDEX TERMS:
3300 Meteorology and Atmospheric Dynamics; 3304 Meteorology and Atmospheric Dynamics:
Atmospheric electricity; 3324 Meteorology and Atmospheric Dynamics: Lightning; 3337
Meteorology and Atmospheric Dynamics: Numerical modeling and data assimilation; KEYWORDS:
lightning, thunderstorm electrification, numerical thunderstorm model
1. Introduction
[2] Lightning parameterizations are required for simulations of
the electrical evolution of strongly electrified storms. Without
lightning flashes the electric field can quickly reach magnitudes
far greater than any that have been measured in actual storms. A
number of lightning parameterizations have been presented in the
literature, and each has some advantages as well as shortcomings.
This paper presents a lightning parameterization derived from the
dielectric breakdown model that was developed by Niemeyer et al.
[1984] and Wiesmann and Zeller [1986] to simulate electric
discharges through gases. Previous studies [e.g., Petrov and
Petrova, 1993, 1999; Pasko et al., 2000] have also used the
Wiesmann-Zeller breakdown model to study electrical discharges
in the atmosphere, but they have been limited to two-dimensional
slab-symmetric or axisymmetric domains and did not include cloud
models. The present work extends the use of the Wiesmann-Zeller
breakdown model in two ways. First, the model has been extended
to a three-dimensional domain, which is required for the most
realistic representation of electric field and thunderstorm dynamics.
Second, this new lightning parameterization has been incorporated
1
Also at National Severe Storms Laboratory, Norman, Oklahoma, USA.
Copyright 2002 by the American Geophysical Union.
0148-0227/02/2000JD000244$09.00
ACL
within a full simulation numerical cloud model, in which it has
been able to produce a variety of lightning types and structures.
[3] Section 2 contains an overview of lightning parameterizations in the literature. Section 3 describes the Wiesmann-Zeller
dielectric breakdown model and our adaptations to make it a
lightning parameterization. In section 4, examples of simulated
unipolar discharges are presented and qualitatively compared with
laboratory discharges. Some examples of parameterized lightning
flashes are also presented. We conclude with a description of some
lightning behaviors observed in modeled storms.
2. Background
[4] Lightning parameterizations have fallen roughly into two
categories: those creating ‘‘bulk’’ flashes and those with explicit
lightning channels. An example of a bulk scheme is the parameterization of Rawlins [1982], which simply reduced the charge
densities everywhere in the cloud model domain. Takahashi
[1987] simulated lightning flashes by removing charge from the
regions of highest charge density. Ziegler and MacGorman [1994]
distributed charge to regions where the net charge density exceeded
a threshold value. Such bulk schemes are relatively simple to
implement but lack realism. Most parameterizations with explicit
channels treat only a one-dimensional or unbranched channel. The
branching of lightning channels would be extremely difficult to
treat analytically because of the lack of symmetry and the incomplete understanding of the processes involved.
2-1
ACL
2-2
MANSELL ET AL.: SIMULATED 3-D BRANCHED LIGHTNING
[5] The lightning parameterization of Helsdon et al. [1992] was
the first to apply the concept of the bidirectional leader [Kasemir,
1960]. The parameterization could produce a single, unbranched
channel that traced the electric field line from an initiation point.
The channel was extended in both directions along the field line
until the ambient field magnitude fell below a certain threshold at
the locations of the channel endpoints. An idealized analytical
model was used to calculate the positive and negative linear charge
densities induced on the channel by the ambient electric field. The
charge distribution at each end of the channel was adjusted to
maintain net charge neutrality of the flash, consistent with the
theory of Kasemir [1960]. The charge carried by the channel was
deposited in the small ion category of the storm model, which
explicitly treated ion capture by hydrometeors [Helsdon and
Farley, 1987]. Flashes parameterized by Helsdon et al. [1992]
were capable of reversing the sign of the net charge density at
affected points. Two limitations of the Helsdon et al. parameterization were that cloud-to-ground (CG) flashes were not treated and
development of the channel structure ignored the contribution to
the electric field from the charge on the channel itself.
[6] The independent parameterizations of Solomon and Baker
[1996] and Mazur and Ruhnke [1998] produced only a single,
vertical channel, but did treat the contribution to the electric field
by the charge induced on the channel itself. Thus they used the net
electric field rather than the ambient field to control propagation.
(The net electric field is the sum of the ambient field plus the
‘‘local’’ field contribution from the conducting channel.) This is an
important feature for a lightning parameterization, because the
enhancement of the electric field at channel tips is what is thought
to allow lightning propagation into regions of weak electric field.
[7] Solomon and Baker [1996] used analytical equations to
calculate the contribution to the electric field from the charge
induced on a one-dimensional channel. Their parameterization
produced both intracloud (IC) and negative cloud-to-ground
(CG) flashes when used with simple charge models [Solomon
and Baker, 1996] or within a 1.5-dimensional cloud model [Solomon and Baker, 1998]. Mazur and Ruhnke [1998] assumed an
axisymmetric ‘‘tripole’’ charge arrangement for the storm (a main
negative region with a main positive region above and a smaller
positive region below), but their model could easily be adapted to
an axisymmetric cloud model such as the one used by Solomon and
Baker [1998]. Mazur and Ruhnke simulated IC and CG flashes
and estimated the electric current through the channel as the flash
developed.
[8] MacGorman et al. [2001] presented a lightning parameterization that may be considered to be an extension of the Helsdon
et al. [1992] parameterization in conjunction with some methods of
the bulk lightning parameterization employed by Ziegler and
MacGorman [1994]. As in the model of Helsdon et al., it created
an initial lightning channel by tracing the electric field line passing
through the chosen initiation point. At the endpoints, however,
branching volumes were determined from the ambient charge
densities and electric potentials.
[9] Petrov and Petrova [1993, 1999] used the dielectric breakdown model of Wiesmann and Zeller [1986] to simulate branched
unipolar lightning flashes moving away from one or two charged
conductors on a two-dimensional grid. (The Wiesmann-Zeller
model is also used in the present study and is described more
fully in section 3.) An advantage of the Wiesmann-Zeller model is
that it determines the contribution to the electric field from the
lightning channels as the flash develops. The effects of the charge
induced by the ambient field on the conducting channels are to
enhance the electric field around channel extremities and to reduce
the electric field around interior branches (i.e., the ‘‘screening’’
effect). A second important feature of the Wiesmann-Zeller model
is that it produces branched channel structures.
[10] The studies by Petrov and Petrova [1993, 1999] used
simple two-dimensional arrangements of charged conductors to
produce the ambient electric field. Flashes began at the surface of a
conductor, and channels propagated through the surrounding
neutral regions. Petrov and Petrova [1999] also examined the
effect of using a pressure-dependent threshold, instead of a constant threshold, for conventional breakdown. When the electric
field threshold for propagation decreased with height, channels
extended to higher altitudes than they did when the threshold was
constant. However, the simple arrangement of charged volumes
limited the applicability of the results to real thunderstorms.
[11] Pasko et al. [2000] applied the Wiesmann-Zeller dielectric
breakdown model to simulate the bidirectional development of
sprite discharges high in the stratosphere. This model used an
axisymmetric domain and produced realistic looking sprite structures. Discharges were initiated via a simple approximation of the
effect of a CG flash on the electrostatic field at high altitudes.
[12] The lightning model of Hager et al. [1989, 1998] also
produced bidirectional, axisymmetric IC and CG flashes. Hager’s
model was completely deterministic, and channels were extended
from all points where the electric field exceeded the breakdown
threshold in any given step. The only substantive difference
between Hager’s model and the breakdown model of Niemeyer
et al. [1984] is the method used to solve Poisson’s equation and the
process of choosing where to extend the channels. However, the
deterministic growth process could be unrealistic: Garik et al.
[1987] showed that for simulations of diffusion-limited aggregation (a similar process) a deterministic approach failed to simulate
the proper geometry, which could be achieved only by having an
element of random selection.
3. Electrical Discharge Models
[13] The lightning parameterization presented here is derived
from the dielectric breakdown model developed by Niemeyer et al.
[1984] (hereinafter called NPW) and Wiesmann and Zeller [1986]
(hereinafter called WZ). Dielectric breakdown models simulate
only the macroscopic properties of discharge channels and do not
deal with the microscopic processes of breakdown. Although the
formulation is rather simple, the models produce complex structures that resemble actual electrical discharges. Our first step,
described in section 3.1, was to use the unidirectional breakdown
model to simulate the discharge experiments of Williams et al.
[1985]. We then extended the model to parameterize bidirectional
lightning in thunderstorms, as described in section 3.2.
3.1. Dielectric Breakdown Model
[14] The NPW model was developed to examine the branched
structures of dielectric breakdown. WZ made the NPW model
more physical by adding a critical electric field threshold (Ecrit) for
propagation and allowing for an internal electric field (Eint) in the
discharge channel. The internal field simply specifies the voltage
drop along the channel segment and accounts for the resistance of
the channel. The most common application of the WZ model has
been to simulate breakdown through insulating material between
two conductors (e.g., flat plates or concentric spheres) to which a
voltage difference is applied [e.g., Fowler et al., 1998]. Dissado
and Sweeney [1993] formulated a different model specifically for
solid dielectrics and achieved results similar to those of the WZ
model. They simulated two-dimensional electrical channels that
originated bidirectionally from a defect within the dielectric and
were driven by an oscillating electric field.
[15] The WZ model simulates the step-by-step propagation of a
unipolar discharge from an initiation point by successively creating
new breakdown extensions, or bonds, from the discharge structure.
Figure 1 shows a sample two-dimensional grid with part of a
discharge. Each new extension is chosen randomly from all
possible new extensions. An extension from a grid point on the
channel to an adjacent grid point is possible (i.e., has a nonzero
ACL
MANSELL ET AL.: SIMULATED 3-D BRANCHED LIGHTNING
Figure 1. Sample grid showing a portion of a discharge in two
dimensions. Open circles represent grid points not connected to the
discharge. Solid circles and solid lines indicate the discharge path.
Dashed lines indicate possible new bonds. In three dimensions,
connections are allowed along all of the unit cube directions.
probability) if the electric field vector favors propagation toward
the adjacent grid point and the electric field magnitude between the
two points exceeds a critical value, Ecrit. Once all possible
extensions have been identified, the probability p of choosing a
particular extension is given by
1
pi ð E Þ ¼
F
ðEi Ecrit Þh
0
for Ei > Ecrit
for Ei Ecrit ;
ð1Þ
where h is a weighting exponent and Ei is the magnitude of the
electric field component between the ith pair of channel and
adjacent nonchannel points. ThePdivisor F is the sum
P of the
unnormalized probabilities, F ¼ k ðEk Ecrit Þh , so
pi ¼ 1.
Each probability is transformed into a range of width pi within the
range (0, 1):
pi !
X
k¼1;i1
pk ;
X
!
pk ;
2-3
[18] Wiesmann [1988] found that introducing Ecrit > 0 (with h =
1) has a similar effect as setting h > 1, since both tend to limit side
branching. Pietronero and Wiesmann [1988] argued that h can be
derived from the relationship between the electron avalanche
velocity and the local electric field in gas discharges. However,
Dissado and Sweeney [1993] pointed out the difficulty in physically justifying any exponent other than h = 1. We have therefore
assumed a linear relationship between pi and Ei (i.e., h = 1) as well
as assuming Ecrit > 0.
[19] In the WZ model, electrical resistance of the channels is
represented by an internal electric field Eint. A perfect conductor
would have Eint = 0, and its surface would have a constant
potential, while a conductor with resistance has Eint > 0, and the
potential varies along its length. A nonzero internal field acts to
reduce the electric field between the channel and nonchannel
points and thus reduces the overall extent of the discharge structure
(all other values being held equal). When a new grid point is added
to the discharge, the electric potential fnew at the new channel point
is calculated from the potential fb of the point from which the
channel segment extends:
fnew ¼ fb sEint d;
ð2Þ
where d is the length of the new segment and s is the sign of charge
carried by the channel. Thus the potential at a point that is n
segments away from the initiation point can be written in terms of
the reference potential fref of the starting point, so that (2) becomes
fðnÞ ¼ fref s
n
X
Eint ð jÞdj ;
ð3Þ
j¼1
where the sum is along the path from the initiation point to
segment n. Each channel point has a unique path from the initiation
point, because no branch is allowed to rejoin to another branch.
Unipolar discharges are generally initiated from a ground plane, so
that fref = 0. For bidirectional discharges (section 3.2) the reference
potential is taken as the average of the ambient potentials of the
two initial grid points.
[20] The electric potential f at grid points not on the channel
must be updated after each extension of the conducting channel.
The potential is found by solving Poisson’s equation,
r
r2 f ¼ ;
e
ð4Þ
k¼1;i
and a random number selects one new extension.
[ 16 ] Most implementations of the WZ discharge model
(including this one) add only one new extension at a time and
then update the potential of the surrounding grid for the effect of
extending the conducting channel. In reality, growth may occur
simultaneously on different branches. For example, Femia et al.
[1993] and Fowler et al. [1998] have attempted to deal with
simultaneous growth.
[17] NPW had a critical field of zero (Ecrit = 0) in their model
and found that the density of branching decreases with increasing
values of h. The outermost channel points (i.e., exposed channel
tips) tend to have the largest electric field magnitudes (i.e., largest
probabilities), while inner branches tend to have lower probabilities
because the outer branches form a partial Faraday cage that reduces
the electric field inside. Choosing h > 1 tends to increase the
probabilities of possible extensions from the outermost points
relative to those from interior points, so the interior becomes
shielded more quickly. However, when h < 1 the probabilities
for exterior and interior points are more nearly equal. Thus, with
Ecrit = 0, NPW found that ‘‘bush’’ type (densely packed) discharges
result for h 1, but ‘‘branched’’ structures result for h 3.
where r is the net charge density at points not on the channel and e
is the electric permittivity. The electric potential is already defined
by (3) along the discharge structure, which is treated as a boundary
with Dirichlet conditions when solving (4). Because of the complex geometry of the discharge boundary, Poisson’s equation is
most easily solved with an iterative technique such as successive
overrelaxation (SOR), as described in Appendix A. The charge
density induced on the channel at a grid point can be found simply
from Poisson’s equation (r = er2f). The charge on the channel is
not needed for solving (4), because the effect of this charge on f is
included by satisfying the boundary condition imposed by the
channel potential (equation (3)). For unipolar discharges the
channel charge is not calculated until the discharge extinguishes
(i.e., all growth probabilities become zero). We do calculate the
channel charge periodically during development of bidirectional
discharges, however, to check for overall flash neutrality
(described in section 3.2.3).
[21] No timescale is assumed for the discharge development.
Therefore currents are not calculated. The sequence in which
channel segments are added can be considered a kind of time
ordering, but it is somewhat problematic to assign an actual
timescale to the simulated discharge. Part of the difficulty is that
ACL
2-4
MANSELL ET AL.: SIMULATED 3-D BRANCHED LIGHTNING
real discharges would be expected to have simultaneous extensions
from different branches, but the present model does not accommodate simultaneous growth. For lightning simulations needing a
timescale, one could estimate the elapsed time by using average
measurements of positive and negative leader velocities [Mazur
and Ruhnke, 1998], but this has not been attempted here.
3.2. Stochastic Lightning Model
[22] The stochastic lightning model (SLM) is an application of
the WZ model to simulate bidirectional discharges in regions of
varying net charge density (e.g., in an electrified storm). The
bidirectional model of lightning propagation is adopted here
because of support from both theory and observation [e.g., Kasemir, 1960; Mazur and Ruhnke, 1993, 1998]. The main difference
from the unidirectional WZ model is that the discharge channel
may be extended from both ends (positive and negative) of an
initial channel. This section describes the details of the SLM and
the procedure for simulating lightning flashes in the context of a
thunderstorm model.
[23] When used within the electrified numerical thunderstorm
model (section 4.2), the SLM propagates discharges in a domain
whose overall size is the same as that of the thunderstorm model
but whose grid spacing is the same in all three dimensions. The
grid spacing for the SLM is kept less than or equal to the smallest
spacing in the thunderstorm domain. The net charge density is
interpolated linearly in three dimensions from the thunderstorm
domain to the SLM grid. The equal spacing not only simplifies the
numerics but also reduces the bias in the electric field calculations
in different directions. For wide domains (80 80 km or larger)
we find that 500-m spacing is the highest practical resolution while
maintaining computational efficiency. Smaller domains (smaller
storms) would allow better resolution of the lightning, but our
focus has been on larger storms. Note that the stepped leaders
photographed beneath clouds typically advance in segments on the
order of 50 m, and tortuosity in lightning channels is also observed
on even smaller scales. Although 500-m resolution obviously
cannot simulate the branching and tortuosity observed on smaller
scales, the larger-scale tortuosity and structure is simulated reasonably well, because of the fractal nature of lightning channels.
Also, one of the main functions of the lightning parameterization is
the redistribution of charge, and this is adequately performed.
[24] Computationally, the channel diameter is the same as the
grid spacing, an issue that we have not seen addressed in the
literature. A channel diameter of 500 m is certainly not realistic,
but it appears that the channel can be thought of as having a
smaller diameter (and correspondingly higher charge density)
within the larger grid. This idea was partially tested in the
following manner. A flash was first propagated at 500-m resolution. The channel and the net ambient charge density were then
interpolated onto a grid with doubled resolution (250-m spacing).
(Each channel segment thus became two half-length segments.)
When the Poisson equation was again solved around the channel, it
turned out that the charge carried by the channels changed by no
more than 4 – 10%. Tests at even higher resolution are planned but
have not been carried out. (A grid with adaptive resolution would
help but has not been attempted.)
3.2.1. Lightning initiation. [25] The decision process for
initiating a lightning flash follows the procedure of MacGorman
et al. [2001]. A flash occurs when the electric field magnitude
exceeds the initiation threshold Einit anywhere in the model
domain. Our default choice for the initiation threshold is the
height-varying ‘‘breakeven’’ or ‘‘runaway electron’’ electric field
given by Marshall et al. [1995] [see also Gurevich et al., 1992],
Ebe ð zÞ ¼ 167ra ð zÞ
ð5Þ
ra ð zÞ ¼ 1:208 expðz=8:4Þ;
ð6Þ
where Ebe has units of kilovolts per meter, and ra is the air density
(in kilograms per cubic meter) as a function of altitude, z (in
kilometers). In the present study, Ebe was modified to have a lower
bound of 30 kv m1 and an upper limit of 125 kV m1. Marshall
et al. [1995] found that the runaway electron threshold was a better
fit than the conventional breakdown threshold to the maximum
electric field magnitudes seen in many thunderstorm soundings.
The actual processes of natural lightning initiation are not
understood well. Whether or not the runaway electron hypothesis
correctly describes lightning initiation, it at least provides a good
match to the maximum electric field magnitudes observed as a
function of height. We would expect similar height distributions of
lightning initiation and structure if we used a pressure-dependent
threshold for conventional breakdown, though the magnitudes of
charge density and electric field in the storm would be affected.
Recent tests using a constant value of Einit throughout the domain
have also produced reasonable results.
[26] Following MacGorman et al. [2001], the lightning initiation
point is chosen randomly from among all the points where the
electric field magnitude is greater than a lower threshold 0.9Einit.
The lower threshold and randomized choice are an attempt to allow
lightning to occur over a larger range of locations, to account for
some of the natural variability of lightning initiation due to unresolved subgrid-scale fluctuations in a cloud model. The discharge is
started as a conducting leader channel between two grid points. The
first point is chosen as described above, and the second is the
adjacent grid point that is closest to the direction of the electric field
vector. Positive and negative leaders are propagated from opposite
ends of the initial channel. Positive leaders carry net positive charge
and tend to propagate toward and through regions of lower electric
potential and negative charge. Negative leaders do the opposite,
carrying net negative charge and extending preferentially into and
through regions of net positive charge and higher electric potential.
3.2.2. Channel characteristics. [ 27 ] Although the
breakdown physics are different for negative and positive leader
development, both types of leaders are treated the same way in the
model. The positive and negative parts of the flash are propagated
independently so that up to two new channel segments (one
positive and one negative) may be added per step. Both ends
have default initial propagation thresholds of Ecrit = 0.75 Einit(z).
The factor of 0.75 was chosen based on qualitative comparisons
between flashes propagated on test grids of 250- and 500-m grid
spacing. On the 250-m grid a flash was made using Ecrit = Einit(z)
and used as a control run. On the 500-m grid, Ecrit was reduced
until the flash attained nearly the same overall spatial extent as on
the 250-m grid. This procedure is reasonable, because the coarser
resolution does not represent gradients in the potential as well as
the finer resolution, so the calculated electric field magnitudes tend
to be smaller on the coarser grid.
[28] Some recent observations indicate that lightning channels
have at least a small resistance [Rakov et al., 1998]. Therefore the
simulated lightning channels are assumed to have a nonzero
internal electric field (Eint). The value of Eint may be set to a
constant value or to a fraction of the initiation threshold. The
example flashes shown in section 3.2.3 all used constant values of
Eint = 500 V m1.
3.2.3. Flash neutrality. [ 29 ] Following the theory of
Kasemir [1960] and Mazur and Ruhnke [1998], we assume that
the channel structure should maintain overall charge neutrality as
long as neither end reaches ground. With their very simple storm
charge distribution, Mazur and Ruhnke [1998] maintained charge
neutrality by adjusting the electric potential of their vertical onedimensional channel. For IC flashes the change in potential
typically was small, but for CG flashes it was much larger. The
amount of change in channel potential depends on the distribution
of ambient potential, which of course depends on the storm charge
distribution, so the charge distribution affects the characteristics of
their simulated lightning.
MANSELL ET AL.: SIMULATED 3-D BRANCHED LIGHTNING
[30] For computational simplicity our parameterization maintains near-neutrality (within 5%) by a technique other than adjusting the reference potential of the channel (although that method is
also being tested). Instead, we parameterize the effect that adjusting
the reference potential would have on the growth of the lightning
structure. In our channel propagation scheme we assume that the
main effect of changing the potential would be to alter the electric
field surrounding the channel, thereby altering the probabilities
used to extend the flash. To mimic this effect, therefore, we adjust
the electric field threshold for propagation at each end of the flash
and also allow the end with less charge to add extra points, if
possible. Of course, if a flash becomes a CG, then neutrality is no
longer assumed.
[31] In our scheme for enforcing rough neutrality, the charge
carried by positive and negative channel segments is computed
periodically (e.g., after every fifth propagation step) by using
Poisson’s equation (equation (4)). The magnitude of charge on
positive segments is then compared with that on negative segments. If the magnitude of one polarity is more than 5% larger than
that of the other, the electric field threshold for propagation is
increased for that polarity, thereby retarding subsequent growth of
the corresponding end of the flash. Likewise, growth is promoted
for channels of the polarity having less charge by reducing the
propagation threshold for that polarity. Additionally, growth is
temporarily stopped for the channels with greater charge so that the
opposite end can ‘‘catch up.’’ If the difference in charge subsequently falls below 5%, then the propagation thresholds are
changed in increments back toward the default values. Adjustments
are made in increments of 0.05Einit(z), and the minimum allowed
threshold is 0.3 Einit(z). (These values were chosen arbitrarily and
were assumed to be reasonable given the lack of knowledge about
the propagation of actual lightning channels.) No maximum limiting value is specified. If growth cannot be forced, even at the
minimum threshold, the flash is simply halted. Otherwise, the flash
develops until all probabilities for growth fall to zero.
3.2.4. Treatment of CG flashes. [32] Until one of the
leaders reaches ground, every flash is treated as a cloud flash in
all respects. Once a leader reaches ground, however, propagation is
stopped for all the branches of the same polarity as the one that
reached ground. A negative leader reaching ground results in a
CG flash, which effectively lowers negative charge to ground
from the storm. Similarly, a positive leader connecting to ground
makes a positive cloud-to-ground (+CG) flash.
[33] Our present treatment of CG flashes makes several simplifications. Chief among these is that it does not change the
channel potential when contact to ground is made. This is admittedly an oversimplified treatment, because the potential of the
channel connection should be brought to ground potential, and this
should become the reference potential. A treatment of the change
in potential of CG flashes is an obvious target for future improvements to the parameterization. A second simplification is that the
charge on branches of the polarity that connects to ground (i.e., the
downward propagating branches) is assumed to become zero, to
mimic the lowering of charge to the ground and the loss of charge
from the model domain. The upward growing (i.e., opposite
polarity) leaders, however, are allowed to continue propagating
until they stop naturally (or reach a domain boundary). This
continued growth of the upward growing portion of a CG flash
is not intended to mimic individual strokes or interstroke processes
but to simulate the combined upward structure of all processes in a
CG flash.
[34] The classification of a flash as a CG has some complications. In the initial tests of the lightning parameterization within the
thunderstorm model, it was noticed that a branch often would
extend downward below 3 km altitude without continuing all the
way to ground. A number of shortcomings may cause this apparent
failure to make a CG flash. For example, the coarse resolution of
the channel grid results in significant underprediction of the
ACL
2-5
Figure 2. Discharges through plastic with uniform charge
densities Williams et al. [1985, Figure 1]. Slabs were 35.0 cm long.
electric field at channel tips. This underprediction has greater
consequences at lower altitudes, where Ebe (and therefore Ecrit) is
larger. Also, the charge balancing procedure may underestimate the
effect of the changing channel potential for developing CG flashes.
Thus, for simplicity, a flash is presently classified as a CG when a
leader branch reaches down to 1.75 km altitude (AGL). We will
continue to examine this issue, especially in light of observations
indicating that IC flashes can occur at relatively low altitude
[Proctor, 1991; Koshak and Krider, 1989; Murphy et al., 1996].
Although IC:CG ratios are obviously affected by this uncertainty in
identifying CG flashes, the total lightning flash rate and dominant
CG polarity are not.
3.2.5. Charge redistribution. [35] The redistribution of
lightning charge begins with the transfer of the charge from the
fine grid to the coarser grid. Then the lightning charge field is
smoothed to avoid sharp gradients in the final charge distributions.
Since the thunderstorm model does not explicitly treat ion
attachment processes, the ions released by lightning are assumed
to be instantaneously distributed directly to the hydrometeor
categories. Each hydrometeor category receives charge in
proportion to its total surface area, regardless of preexisting
charge, as given by Ziegler and MacGorman [1994]:
si
dri ¼ P
dr;
k sk
ð7Þ
where dr is the charge density at a grid point on the lightning
channel and dri is the charge density deposited on the ith
hydrometeor category having a total surface area si. If branches
extend outside the cloud to regions with no hydrometeors, the
charge on branches outside the cloud is considered lost from the
storm and is not tracked by the model.
4. Simulated Discharges
4.1. Unipolar Discharges
[36] An initial test of the WZ model was to simulate unipolar
discharges produced in the laboratory by Williams et al. [1985].
Williams et al. exposed plastic slabs to an electron beam, thereby
creating two-dimensional horizontal charge distributions in the
plastic. They found that the discharges became more extensive as
the charge density in the plastic increased (Figure 2).
[37] Figures 3 and 4 show simulations of discharges into slabs
with horizontally uniform charge. The only difference between the
two is that in Figure 3 the channels have zero internal electric field
(i.e., the channels are treated as perfect conductors) and in Figure 4
ACL
2-6
MANSELL ET AL.: SIMULATED 3-D BRANCHED LIGHTNING
(a)
(b)
(c)
(d)
Figure 3. Numerical simulations of unidirectional discharges
into a rectangular region of charge. Charges densities are (a) 0.5,
(b) 0.7, (c) 0.9, and (d) 1.1 nC m3. Simulated slabs are 30 grid
points wide and 240 grid points long. Ecrit = 100.0 kV m1 and
Eint = 0.0 kV m1 (no internal field).
they have a small internal field (i.e., the channels have nonzero
resistance). Figure 4 compares well qualitatively with the laboratory discharges (Figure 2). As the charge density was increased, the
maximum propagation distance and degree of branching increased
in both the experiment and the simulation. Both also exhibit a
narrowing structure with increasing distance from the initiation
point.
[38] Williams et al. [1985] also presented discharges in slabs
that had small regions of one charge density embedded in a
region of different charge density (Figure 6 of Williams et al.
[1985], reproduced here in Figure 5). One experiment had
pockets of higher charge density surrounded by a region of lower
charge density, and a second trial had the same pattern but with
switched charge densities (i.e., lower density charge pockets
surrounded by a region of higher density charge). They observed
preferential propagation and denser branching of the discharge in
the regions of larger charge density. The corresponding simulations with enhanced or depleted regions showed very similar
behavior (Figure 6).
[39] The simulations in Figures 3, 4, and 6 were scaled to
atmospheric dimensions. The grid spacing was 125 m in all three
dimensions. The charge densities were within the range of values
determined from electric field and particle charge measurements in
thunderstorms. The charge regions were confined to a layer that
was 7 grid points in thickness. The central layer had the given
charge density and was 3 grid points in thickness. The charge
density decreased linearly above and below the central layer to zero
at the top and bottom of the layer. This distribution approximates
the assumed gaussian distribution of charge in the laboratory
experiments [Williams et al., 1985]. The simulated discharges were
initiated at a ‘‘needle’’ conductor extended from the ground plane.
The propagation threshold was set at 100 kV m1.
(a)
(b)
(c)
(d)
Figure 4.
As in Figure 3, but with Eint = 1.0 kV m1.
Figure 5. Discharges through plastic with two charge densities
Williams et al. [1985, Figure 6]. (a) Isolated regions (innermost
contours) of higher charge density (1.2 C m3) are surrounded by
lower charge density (0.6 C m3). (b) Isolated regions of lower
charge density (0.6 C m3) are surrounded by higher charge
density (1.2 C m3). Outermost region has zero net charge.
[40] One might ask the question, how comparable are discharges in plastic slabs to lightning discharges? There is no doubt
that they are different. It probably is reasonable to expect that some
general behaviors are similar, such as faster and more branched
propagation in regions of higher charge density [Williams et al.,
1985]. It appears less likely to us that the internal field for a small
laboratory discharge would be comparable to that of a lightning
leader. Certainly there is no one value of internal field valid for the
complete development of any particular discharge, and the choice
for our parameterization is necessarily a very rough estimate.
4.2. Example Simulated Lightning Flashes
[41] The lightning model has been integrated into a full
cloud simulation model, which was used to produce examples
of simulated lightning for this paper. The numerical cloud
model [Straka and Anderson, 1993] is three-dimensional and
includes detailed bulk microphysics, with separate categories for
cloud water, rain, cloud ice (columns, plates, and rimed), snow
aggregates, frozen drops, three graupel densities, and two size
ranges of hail. Details of the model dynamics are given by
Carpenter et al. [1998]. Electrification parameterizations include
noninductive (independent of the ambient electric field) and
inductive (field-dependent) charge separation. The examples
shown here used a noninductive scheme based on the laboratory
work of Saunders and Peck [1998] for charge separation by
rebounding collisions between ice crystals and actively riming
(a)
2-7
ACL
MANSELL ET AL.: SIMULATED 3-D BRANCHED LIGHTNING
cloud boundaries are poorly resolved, with an assumed thickness of 500 m.
[42] The storm simulations that produced the results presented
here used atmospheric soundings that produced multicell or supercell storms. The multicell storm simulation used the analytical
thermodynamic and shear profiles of Weisman and Klemp [1982,
1984], and a half-circle hodograph with arc length (Us) of 10 m
s1. The supercell storm used a composite of two atmospheric
soundings from the Texas panhandle on 2 June 1995 [Gilmore,
2000]. Further details of the cloud model and environments are
given by Mansell [2000].
[43] Two simulated intracloud flashes are shown in Figures 7
and 8. The intracloud flash in Figure 7 is from the multicell storm
40 min after the first lightning flash. The second IC flash
(Figure 8) is from the supercell storm (30 min after it produced
↑
N
45
(b)
40
Top View
40
45
50
45
40
14
12
Figure 6. Simulations of unidirectional discharges with regions
of enhanced or reduced charge density in which (a) small square
regions have r = 1.2 nC m3 surrounded by r = 0.6 nC m3
and (b) charge densities are exchanged. Discharges were
propagated in a three-dimensional domain and had an internal
electric field of 1 kV m(1). Initiation point, at the top edge of the
lower left small square in each plot, is from a needle protrusion
from the simulated ground plane. Area shown is 100 by 100 grid
points (12.5 12.5 km).
precipitation ice particles. Three other laboratory-based noninductive collisional charging parameterizations are also optionally available, based on Takahashi [1978], Ziegler et al. [1991],
or Brooks et al. [1997]. Inductive charging is allowed in
rebounding collisions of graupel or hail with cloud droplets as
shown by Ziegler et al. [1991]. The model also includes a
parameterization for electrical screening layer development [Ziegler et al., 1991]. It should be noted that the screening layers at
10
8
↑
N
40
6
Side View
45
50
4
Figure 7. An intracloud (IC) flash from the multicell storm
simulation at 3795 s, viewed from the side and from above.
Initiation point at 8.25 km altitude is indicated by the diamond
between the positive (shaded) and negative (black) branches.
Magnitude of the dipole moment of the flash was 173 C km, and
the flash deposited 36.3 C of charge at each end. Axes show
distance in kilometers.
2-8
ACL
MANSELL ET AL.: SIMULATED 3-D BRANCHED LIGHTNING
↑
N
65
65
60
60
55
55
Top View
15
35
55 60
40
45
50
↑
N
50
50
Top View
12.5
45
10
30
7.5
15
5
↑
N
50
Side View
35
40
45
its first lightning flash). Both IC flashes exhibit two-layer
(‘‘bilevel’’) horizontal structure associated with two regions of
charge in the storm: an upper positive charge region and a lower
negative region (the ‘‘main positive’’ and ‘‘main negative’’ charge
regions). Similar lightning structure has been observed by several
investigators, including MacGorman et al. [1981] and Shao and
Krehbiel [1996].
[44] Both positive and negative CG flashes have been produced
by the storm simulations. An example of a CG flash from the
multicell storm simulation is shown in Figure 9. The flash began at
50
10
36
38
40
6
4
Top View
34
8
46
↑
N
42
44
50
48
46
44
↑
N
34
36
Side View
40
38
40
45
50
10
Figure 8. An IC flash at 3000 s from the simulated supercell
storm. This flash had a dipole moment magnitude of 241 C km and
deposited 52.4 C of each charge polarity. Flash initiated at an
altitude of 7.25 km (diamond). Other details are as in Figure 7.
48
45
35
65
50 55 60
2
42
Figure 9. A negative cloud-to-ground (CG) flash at 3810 s in the
multicell storm that lowered 30.1 C to ground. Flash began at an
altitude of 6.75 km (diamond). Negative leader (black) went
through a lower positive charge region and then to ground. Positive
leaders (shaded) branched into the main negative charge region.
Axis labels show distance in kilometers.
5
↑
N
Side View
30
35
40
45
50
Figure 10. A positive CG flash at 2795 s in the supercell
simulation. Flash initiated at an altitude of 7.25 km (diamond). The
positive leader (shaded) propagated through the main negative
charge region to ground. The negative leaders (black) branched
extensively into the main positive charge region. Flash lowered
82.3 C to ground. Axis labels show distance in kilometers.
an altitude of 6.75 km, between the main negative region and a
lower positive charge region near the base of the updraft. The CG
occurred at approximately the same time as the IC flash in Figure 7.
The downward propagating negative leader has noticeable horizontal extent in this example, but the amount of horizontal
branching varies from flash to flash. The upward positive leaders
have branching that is nearly exclusively horizontal, as in the
simulated IC flashes.
[45] Figure 10 shows an example of a positive ground flash
(+CG) from the supercell storm simulation. The downward positive leader shows some extensive horizontal branching before
connecting to ground. The upward negative leaders also branched
quite extensively, up to 20 km away from the initiation point. This
+CG began at an altitude of 7.25 km, between the main positive
and negative charge regions. In many respects it is similar to the IC
flash shown in Figure 8, which occurred 205 s later.
5. Discussion
[46] Lightning flashes in the model appear always to originate
between regions of opposite charge, as expected, where the electric
MANSELL ET AL.: SIMULATED 3-D BRANCHED LIGHTNING
ACL
2-9
Figure 11. Lightning composites (left) and contour slices (right) for (a, b) simulated multicell storm with CG flash
(c, d) and supercell storm with +CG flash. Composites (Figures 11a and 11c) are surfaces over the regions of positive
(blue) and negative (red) lightning leaders during a 2.5-min period. Green surfaces indicate initiation point locations.
Slices (Figures 11b and 11d) again show lightning activity regions within 3 km of the charge contour plane (red for
negative leaders, blue for positive leaders, and green for initiation locations) overlaid with charge density contours
(solid for positive and dashed for negative). Contour values start at ±0.1 nC m3 with intervals of 0.3. Cloud
boundaries (droplets and ice crystals) are indicated by gray surfaces (left) or thick gray line (right). See color version
of this figure at back of this issue.
field reaches the largest magnitudes (Figure 11). This is true not
only for IC and CG flashes (Figures 11a and 11b), consistent
with observations, but also for +CG flashes (Figures 11c and 11d).
The initiation locations are always found at or near the boundary
between positive and negative charge. Some initiation points
appear farther away from the charge boundaries in Figure 11,
because the slices show lightning activity in a slab that is 7 km
(7 grid points) wide, and the charge contours are taken at the center
of the slab.
[47] Another notable effect of the simulated lightning is the
occasional polarity reversal produced by lightning in the storm’s
net charge density. This effect is evident in Figure 11b where a
pocket of negative charge appears within the main positive
region (e.g.. at z = 11 km and x = 43 km) and positive charge
pockets appear in the negative region (e.g., at z = 7 km and
x = 39 km). The net charge polarity reversals can be caused by
single flashes or by the cumulative effect of repeated flashes.
Similar reversals were produced by the lightning parameter-
izations of Helsdon et al. [1992] and Solomon and Baker
[1998].
[48] An interesting model result is that IC lightning flashes
often are initiated in the same region as CG flashes within a few
minutes of each other or less (e.g., Figures 8 and 10). One must
ask whether this is natural behavior or an artifact of our model.
Observations indicate that IC and CG lightning can initiate from
similar altitudes. Proctor [1991] reported that the height distribution of lightning initiations for South African storms was
bimodal. The upper group of initiations was composed almost
exclusively of IC flashes, but the lower group had both CG and
IC flashes. Koshak and Krider [1989] and Murphy et al. [1996]
found that for Florida thunderstorms, IC flashes occasionally
occurred low in the storms, and they hypothesized that these
IC flashes were discharges between a lower positive charge
region and the main negative charge region tapped by CG
flashes. What remains to be tested is whether our parameterization produces the correct proportion of CG flashes and low IC
ACL
2 - 10
MANSELL ET AL.: SIMULATED 3-D BRANCHED LIGHTNING
flashes, though relatively few observational studies are available
for this comparison.
[49] In our model, CG flashes always began between the
main negative charge and lower positive charge (Figure 11b), as
inferred by Clarence and Malan [1957]. The modeling study of
Solomon and Baker [1998] also found that CG flashes
occurred only after the development of a sufficiently strong
lower positive charge region. Similarly, +CG flashes also always
began between opposite charges in our simulations and occurred
only where the lowest significant charge region near the initiation point was negative. This result for +CG flashes is consistent
with the finding by Takeuti et al. [1978] and Brook et al. [1982]
that winter thunderstorms in Japan that produced +CG flashes
had a normal dipole structure (positive charge above negative).
More recently, Carey and Rutledge [1998] inferred from corona
point sensor measurements that +CG flashes emanated from
regions of a storm where the lowest significant charge region
was negative.
[50] The model results concerning +CG flashes offer a possible
clarification of the conclusions of Takeuti et al. [1978] and Brook
et al. [1982]. Takeuti et al. inferred from surface electric field
measurements that the gross charge structure of +CG-producing
thunderstorms had a positive charge region above and displaced
horizontally from a lower negative charge region (the ‘‘tilted
dipole’’). A common interpretation of this charge configuration is
that the key to +CG flashes is the ‘‘unshielding’’ of positive charge
(i.e., the removal of the negative charge from beneath the positive
charge) so that the positive charge would become ‘‘exposed to
ground.’’ The simulation results strongly suggest that even if this
unshielded charge arrangement were to occur, a negative charge
region would still be needed to initiate +CG lightning. Another
important factor may be the altitude of the positive charge region
involved in +CG flashes. Brook et al. [1982] found that the source
charge heights for +CG flashes tended to be only slightly greater
than heights for CG flashes. The present thunderstorm model
offers the opportunity to further investigate the circumstances of
+CG flashes.
6. Conclusions
[51] The lightning parameterization presented here, an extension of the Wiesmann-Zeller model of dielectric breakdown, has
been found to be a useful tool in the study of lightning
characteristics in numerical thunderstorm models. The successful
simulation of laboratory discharges and the realistic structure and
location of simulated lightning flashes lend credence to the
model and to comparisons with lightning behavior in observed
storms.
[52] The stochastic lightning model can produce a wide variety
of lightning structures, including both CG flashes and bilevel IC
flashes often observed by lightning mapping systems [e.g., Shao
and Krehbiel, 1996; MacGorman et al., 1981; Rison et al., 1999].
Simulated flashes also reproduced observed features such as cloudto-air discharges in which part of the flash extends beyond the
cloud boundary. Brook et al. [1985] reported on photographs
showing lightning channels extending above thunderstorms, particularly in the vicinity of overshooting tops. In our simulations
both positive and negative channels may extend above the cloud
near the overshooting top.
[53] The model has also produced all major types of lightning,
including IC, CG, and +CG flashes. The type of flash and the
polarity of CG lightning depend on the charge distribution, which
in turn depends on storm type, history, and other factors. The CG
flashes produced by the model are consistent with the hypothesis
that a lower positive charge region is critical for their development
[e.g., Clarence and Malan, 1957; Williams, 1989]. This hypothesis
has been in the literature for many years and is widely accepted.
One of the results of our storm simulations is that a similar
hypothesis should apply to positive CG flashes: a negative charge
region is needed below a positive charge region to promote
positive CG flashes.
[54] The results reported here support the idea of using lightning mapping data to infer some of the gross features of a
thunderstorm’s charge distribution. For example, simulated negative leaders tend to propagate through positive net charge, and
positive leaders tend to propagate through negative net charge, as
hypothesized by MacGorman et al. [1981], Williams et al.
[1985], and Shao and Krehbiel [1996]. Although the region
inhabited by a given polarity of leaders differs significantly from
the region inhabited by the opposite polarity of thunderstorm
charge (especially in the vicinity of recent lightning), the two
correspond well enough in our simulations that regions filled with
leaders are expected to give a rough idea of the location of net
charge. This relationship and the tendency for lightning to begin
on or near the boundary between oppositely charged regions can
be used to provide considerable information about thunderstorm
charge.
[55] Having three-dimensional lightning mapping data with
other storm data from a wide variety of storm types also is essential
for our ongoing modeling research. Present lightning mapping
technologies [e.g., Krehbiel et al., 2000] offer ample opportunities
for our present comparisons with model results. Such data enable
us both to test various aspects of our cloud model and lightning
parameterization and to investigate the storm processes responsible
for observed lightning characteristics. Though interpreting storm
model results requires an accounting for the limitations of even the
most sophisticated models, simulations of observed storms are
required to examine, in a self-consistent way, how the various
individual electrical behaviors predicted by theory or observed in
the laboratory affect the electrical properties of the very complex
system that is a thunderstorm.
Appendix A:
SLM Procedure
[56] Before a discharge begins the electric potential is calculated from Poisson’s equation equation (4) for the model
domain using the FISHPACK package (developed by J. Adams,
P. Swarztrauber, and R. Sweet), which applies fast Fourier
transforms in the horizontal directions and solves the resulting
tridiagonal system of equations in the vertical direction. The
bottom of the domain (ground) is set at zero potential (Dirichlet
boundary condition), and the top and sides have the Neumann
condition @f/@n = 0. The potential is solved using extended
lateral boundaries to increase the accuracy of the solution by
reducing the influence of the mirror charges that necessarily
result from the boundary condition. A future improvement will
be to extend the top boundary as well.
[57] Iterative methods are commonly used to solve the
Poisson equation (equation (4)). In our lightning parameterization the standard technique of successive overrelaxation
(SOR) is used to update the solution in a subdomain around
the developing lightning flash. The unidirectional version of our
parameterization used the same SOR methods, but the solution
was updated throughout the entire domain. After each channel
extension, one or more SOR iterations are required to recompute
the electric potential, using the solution from the previous
extension as a first guess. At the beginning of each relaxation
sweep through the grid, the SOR routine calculates the residual
r (the estimated error in the solution) at each grid point by
using
r ¼ ðsÞ2 r2 fn þ r=e ;
ðA1Þ
where s is the uniform grid spacing, fn is the solution to the
potential from the most recent nth iteration, and r and e are the
MANSELL ET AL.: SIMULATED 3-D BRANCHED LIGHTNING
net charge density and electric permittivity, respectively. The
Laplacian (r2f) is calculated by a standard seven-point finite
difference formulation (second-order accuracy), which is the
sum of the second derivatives in each direction:
r2 f 3
X
@2f
;
@x2i
i¼1
ðA2Þ
where
@ 2 f f½xi xi 2f½xi þ f½xi xi ffi
:
@x2i
ðxi Þ2
ðA3Þ
Each SOR sweep updates the solution to the potential from the
residual by
1
nþ1
¼ fnijk þ wrijk ;
fijk
6
ðA4Þ
where w is the overrelaxation parameter and has an experimentally determined value in the range 1 to 2. After updating, r is
calculated at each grid point, and the maximum fractional
residual e = Max[r]/Max[f] is determined. Relaxation iterations
continue until e 5 103.
[58] An increase in computational efficiency was achieved by
adjusting the domain in which the Poisson equation is updated
during the development of a lightning flash. When a new
channel segment is added to the discharge structure, the electric
potential will be significantly affected at the nearby points
(assuming a 1/r dependence). It was found to be sufficient to
recalculate the potential by the SOR method in a relatively small
subdomain around the new channel point. After every ten
propagation steps, relaxation is also performed in a subdomain
that encompasses the entire lightning flash. This ‘‘local/global’’
method is effective in solving for the important short wavelengths (i.e., the high wave number Fourier components of the
solution to the potential). The method is similar to, though it
was developed independently of, the one employed by Sañudo
et al. [1995]. The long wavelengths are not as relevant to the
process since the solution at points far from the flash is not
important to the propagation of the discharge. The local solution
is performed in a subdomain that is 21 grid points (10 km) wide
in each direction and is centered on the new point. The global
solution is calculated in a subdomain that contains the entire
flash and is increased in size by two grid points on each side
after each SOR sweep. The subdomain boundaries usually use
Dirichlet conditions. However, if the boundary of a subdomain
is within 10 grid points of a model domain boundary, then the
smaller subdomain is expanded all the way to that boundary and
uses the corresponding condition for the model domain (Neumann condition for top and sides and Dirichlet condition for
bottom). This strategy of using a limited subdomain does not
appear to introduce any obvious instabilities to the solution.
Only rarely has a parameterized flash shown behavior that seems
pathological (e.g., horizontal extension well beyond the cloud
boundary).
[59] Acknowledgments. Support for this research was provided
under National Science Foundation grants ATM 9617318, ATM 0003869,
and ATM 9807179. This work was also supported in part by a grant from
the National Oceanic and Atmospheric Administration for high-performance computing resources at the Arctic Region Supercomputing Center. The
National Severe Storms Laboratory (Norman, Oklahoma, USA) provided
most of the computational resources used for this work. Most of the
computational work was carried out on the multiprocessor SGI Origin
2000 (Vortex) at NSSL. We also thank V. Mazur for his comments, which
ACL
2 - 11
improved this manuscript, and M. Uman and an anonymous reviewer for
additional beneficial comments.
References
Brook, M., M. Nakano, P. Krehbiel, and T. Takeuti, The electrical structure
of the Hokuriku winter thunderstorms, J. Geophys. Res., 87, 1207 – 1215,
1982.
Brook, M., C. Rhodes, O. H. Vaughan Jr., R. E. Orville, and B. Vonnegut,
Nighttime observations of thunderstorm electrical activity from a high
altitude airplane, J. Geophys. Res., 90, 6111 – 6120, 1985.
Brooks, I. M., C. P. R. Saunders, R. P. Mitzeva, and S. L. Peck, The effect
on thunderstorm charging of the rate of rime accretion by graupel, Atmos.
Res., 43, 277 – 295, 1997.
Carey, L. D., and S. A. Rutledge, Electrical and multiparameter radar observations of a severe hailstorm, J. Geophys. Res., 103, 13,979 – 14,000,
1998.
Carpenter, R. L., K. K. Droegemeier, and A. M. Blyth, Entrainment and
detrainment in numerically simulated cumulus congestus clouds, I, General results, J. Atmos. Sci., 55, 3417 – 3432, 1998.
Clarence, N. D., and D. J. Malan, Preliminary discharge processes in
lightning flashes to ground, Q. J. R. Meteorol. Soc., 83, 161 – 172,
1957.
Dissado, L. A., and P. J. J. Sweeney, Physical model for breakdown structures in solid dielectrics, Phys. Rev. B., 48, 16,261 – 16,268, 1993.
Femia, N., L. Niemeyer, and V. Tucci, Fractal characteristics of electrical
discharges: Experiments and simulation, J. Phys. D Appl. Phys., 26,
619 – 627, 1993.
Fowler, H. A., J. E. Devaney, J. G. Hagedorn, and F. E. Sullivan, Dielectric
breakdown in a simplified parallel model, Comput. Phys., 12(5), 478 –
487, 1998.
Garik, P., K. Mullen, and R. Richter, Models of controlled aggregation,
Phys. Rev. A, 35, 3046 – 3055, 1987.
Gilmore, M. S., Observed and simulated 2 – 3 June 1995 west Texas panhandle supercells, Ph.D. dissertation, 250 pp., Texas A&M Univ., College
Station, 2000.
Gurevich, A. V., G. M. Milikh, and R. Roussel-Dupre, Runaway electron
mechanism of air breakdown and preconditioning during a thunderstorm,
Phys. Lett. A, 165, 463 – 468, 1992.
Hager, W. W., A discrete model for the lightning discharge, J. Comput.
Phys., 144, 137 – 150, 1998.
Hager, W. W., J. S. Nisbet, J. R. Kasha, and W.-C. Shann, Simulations of
electric fields within a thunderstorm, J. Atmos. Sci., 46, 3542 – 3558,
1989.
Helsdon, J. H., Jr., and R. D. Farley, A numerical modeling study of a
Montana thunderstorm, 2, Model results versus observations involving
electrical aspects, J. Geophys. Res., 92, 5661 – 5675, 1987.
Helsdon, J. H., Jr., G. Wu, and R. D. Farley, An intracloud lightning parameterization scheme for a storm electrification model, J. Geophys. Res.,
97, 5865 – 5884, 1992.
Kasemir, H. W., A contribution to the electrostatic theory of a lightning
discharge, J. Geophys. Res., 65, 1873 – 1878, 1960.
Koshak, W. J., and E. P. Krider, Analysis of lightning field changes during
active Florida thunderstorms, J. Geophys. Res., 94, 1165 – 1186, 1989.
Krehbiel, P. R., R. J. Thomas, W. Rison, T. Hamlin, J. Harlin, and M. Davis,
GPS-based mapping system reveals lightning inside storms, Eos Trans.
AGU, 81(3), 21, 2000.
MacGorman, D. R., A. A. Few, and T. L. Teer, Layered lightning activity,
J. Geophys. Res., 86, 9900 – 9910, 1981.
MacGorman, D. R., J. M. Straka, and C. L. Ziegler, A lightning parameterization for numerical cloud models, J. Appl. Meteorol., 40, 459 – 478,
2001.
Mansell, E. R., Electrification and lightning in simulated supercell and nonsupercell thunderstorms, Ph.D. dissertation, 211 pp., Univ. of Okla., Norman, 2000.
Marshall, T. C., M. P. McCarthy, and W. D. Rust, Electric field magnitudes
and lightning initiation in thunderstorms, J. Geophys. Res., 100, 7097 –
7103, 1995.
Mazur, V., and L. H. Ruhnke, Common physical processes in natural and
artificially triggered lightning, J. Geophys. Res., 98, 12,913 – 12,930,
1993.
Mazur, V., and L. H. Ruhnke, Model of electric charges in thunderstorms
and associated lightning, J. Geophys. Res., 103, 23,299 – 23,308, 1998.
Murphy, M. J., E. P. Krider, and M. W. Maier, Lightning charge analysis in
small Convection and Precipitation Electrification (CaPE) experiment
storms, J. Geophys. Res., 101, 29,615 – 29,626, 1996.
Niemeyer, L., L. Pietronero, and H. J. Wiesmann, Fractal dimension of
dielectric breakdown, Phys. Rev. Lett., 52(12), 1033 – 1036, 1984.
Pasko, V., U. S. Inan, and T. F. Bell, Fractal structure of sprites, Geophys.
Res. Lett., 27, 497 – 500, 2000.
ACL
2 - 12
MANSELL ET AL.: SIMULATED 3-D BRANCHED LIGHTNING
Petrov, N. I., and G. N. Petrova, Physical mechanisms for intracloud lightning discharges, Tech. Phys., 38, 287 – 290, 1993.
Petrov, N. I., and G. N. Petrova, Physical mechanisms for the development
of lightning discharges between a thundercloud and the ionosphere, Tech.
Phys., 44, 472 – 475, 1999.
Pietronero, L., and H. J. Wiesmann, From physical dielectric breakdown to
the stochastic fractal model, Z. Phys. B, 70, 87 – 93, 1988.
Proctor, D. E., Regions where lightning flashes began, J. Geophys. Res., 96,
5099 – 5112, 1991.
Rakov, V. A., et al., New insights into lightning processes gained from
triggered lightning experiments in Florida and Alabama, J. Geophys.
Res., 103, 14,117 – 14,130, 1998.
Rawlins, F., A numerical study of thunderstorm electrification using a
three-dimensional model incorporating the ice phase, Q. J. R. Meteorol.
Soc., 108, 779 – 800, 1982.
Rison, W., R. J. Thomas, P. R. Krehbiel, T. Hamlin, and J. Harlin, A GPSbased three-dimensional lightning mapping system: Initial observations
in central New Mexico, Geophys. Res. Lett., 26, 3573 – 3576, 1999.
Sañudo, J., J. B. Gómez, F. Castaño, and A. F. Pacheco, Fractal dimension
of lightning discharge, Nonlinear Processes Geophys., 2, 101 – 106,
1995.
Saunders, C. P. R., and S. L. Peck, Laboratory studies of the influence of
the rime accretion rate on charge transfer during crystal/graupel collisions, J. Geophys. Res., 103, 13,949 – 13,956, 1998.
Shao, X. M., and P. R. Krehbiel, The spatial and temporal development of
intracloud lightning, J. Geophys. Res., 101, 26,641 – 26,668, 1996.
Solomon, R., and M. B. Baker, A one-dimensional lightning parameterization, J. Geophys. Res., 101, 14,983 – 14,990, 1996.
Solomon, R., and M. B. Baker, Lightning flash rate and type in convective
storms, J. Geophys. Res., 103, 14,041 – 14,057, 1998.
Straka, J. M., and J. R. Anderson, Numerical simulations of microburstproducing storms: Some results from storms observed during COHMEX,
J. Atmos. Sci., 50, 1329 – 1348, 1993.
Takahashi, T., Riming electrification as a charge generation mechanism in
thunderstorms, J. Atmos. Sci., 35, 1536 – 1548, 1978.
Takahashi, T., Determination of lightning origins in a thunderstorm model,
J. Meteorol. Soc. Jpn., 65, 777 – 794, 1987.
Takeuti, T., M. Nakano, M. Brook, D. J. Raymond, and P. Krehbiel, The
anomalous winter thunderstorms of the Hokuriku coast, J. Geophys. Res.,
83, 2385 – 2394, 1978.
Weisman, M. L., and J. B. Klemp, The dependence of numerically simulated convective storms on vertical wind shear and buoyancy, Mon.
Weather Rev., 110, 504 – 520, 1982.
Weisman, M. L., and J. B. Klemp, The structure and classification of
numerically simulated convective storms in directionally varying wind
shears, Mon. Weather Rev., 112, 2479 – 2498, 1984.
Wiesmann, H. J., Realistic models of dielectric breakdown, in Fractals’
Physical Origin and Properties, edited by L. Pietronero, pp. 243 – 257,
Plenum, New York, 1988.
Wiesmann, H. J., and H. R. Zeller, A fractal model of dielectric breakdown
and prebreakdown in solid dielectrics, J. Appl. Phys., 60, 1770 – 1773,
1986.
Williams, E. R., The tripole structure of thunderstorms, J. Geophys. Res.,
94, 13,151 – 13,167, 1989.
Williams, E. R., C. M. Cooke, and K. A. Wright, Electrical discharge
propagation in and around space charge clouds, J. Geophys. Res., 90,
6054 – 6070, 1985.
Ziegler, C. L., and D. R. MacGorman, Observed lightning morphology
relative to modeled space charge and electric field distributions in a
tornadic storm, J. Atmos. Sci., 51, 833 – 851, 1994.
Ziegler, C. L., D. R. MacGorman, J. E. Dye, and P. S. Ray, A model
evaluation of noninductive graupel-ice charging in the early electrification of a mountain thunderstorm, J. Geophys. Res., 96, 12,833 – 12,855,
1991.
D. R. MacGorman, Cooperative Institute for Mesoscale Meteorological
Studies, University of Oklahoma, 100 East Boyd, Room 1110, Norman, OK
73019, USA. (don.macgorman@nssl.noaa.gov)
E. R. Mansell and C. L. Ziegler, National Severe Storms Laboratory,
1313 Halley Circle, Norman, OK 73069, USA. (mansell@nssl.noaa.gov;
ziegler@nssl.noaa.gov)
J. M. Straka, School of Meteorology, University of Oklahoma, 100 East
Boyd, Room 1310, Norman, OK 73019, USA. (jstraka@ou.edu)
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. D9, 10.1029/2000JD000244, 2002
Figure 11. Lightning composites (left) and contour slices (right) for (a, b) simulated multicell storm with CG flash
(c, d) and supercell storm with +CG flash. Composites (Figures 11a and 11c) are surfaces over the regions of positive
(blue) and negative (red) lightning leaders during a 2.5-min period. Green surfaces indicate initiation point locations.
Slices (Figures 11b and 11d) again show lightning activity regions within 3 km of the charge contour plane (red for
negative leaders, blue for positive leaders, and green for initiation locations) overlaid with charge density contours
(solid for positive and dashed for negative). Contour values start at ±0.1 nC m3 with intervals of 0.3. Cloud
boundaries (droplets and ice crystals) are indicated by gray surfaces (left) or thick gray line (right).
ACL
2-9