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MARCH 2001
MACGORMAN ET AL.
459
A Lightning Parameterization for Numerical Cloud Models
DONALD R. MACGORMAN
NOAA/National Severe Storms Laboratory, Norman, Oklahoma
JERRY M. STRAKA
School of Meteorology and Center for the Analysis and Prediction of Storms,
University of Oklahoma, Norman, Oklahoma
CONRAD L. ZIEGLER
NOAA/National Severe Storms Laboratory, Norman, Oklahoma
(Manuscript received 10 August 1998, in final form 31 May 2000)
ABSTRACT
A new lightning parameterization has been developed to enable cloud models to simulate the location and
structure of individual lightning flashes more realistically. To do this, three aspects of previous parameterizations
have been modified: 1) To account for subgrid-scale variations, the initiation point is chosen randomly from
among grid points at which the electric field magnitude is above a threshold value, instead of being assigned
always to the grid point having the maximum electric field magnitude. 2) The threshold value for initiation can
either be constant, as in previous parameterizations, or can vary with height to allow different flash initiation
hypotheses to be tested. 3) Instead of stopping at larger ambient electric field magnitudes, extensive flash
development can continue in regions having a weak ambient electric field but a substantial charge density. This
behavior is based on lightning observations and conceptual models of lightning physics. However, like previous
parameterizations for cloud models, the new parameterization attempts to mimic only the gross structure of
flashes, not the detailed development of lightning channels, the physics of which is only poorly understood.
Though the choice of parameter values affects the dimensions of a flash, the qualitative features of simulated
flash structure are similar to those of observed lightning as long as the parameter values are consistent with the
larger electric field magnitudes measured in storms and with simulated charge densities produced over reasonably
large regions. Initial simulations show that, by permitting development of flashes in regions of substantial charge
density and weak ambient electric field, the new parameterization produces flash structure much like that of
observed flashes, as would be expected from the inferred correlation between observed horizontal lightning
structure and thunderstorm charge.
1. Introduction
A new parameterization of lightning has been developed to improve the ability of numerical cloud models
to simulate more realistically the location and structure
of individual lightning flashes and thereby to improve
estimates of the effect of flashes on the charge distribution and electric field of simulated storms. Lightning
has at least two major effects on storms that make it
essential to include a lightning parameterization in any
numerical cloud model used to simulate electrical properties of storms after the first lightning flash.
First, the overall effect of lightning is to limit the
Corresponding author address: Donald R. MacGorman, Cooperative Institute for Mesoscale Meteorological Studies, University of
Oklahoma, 100 E. Boyd, Room 1110, Norman, OK 73019-0628.
E-mail: [email protected]
q 2001 American Meteorological Society
magnitude of the electric field that can be produced by
a thunderstorm. When the electric field anywhere in the
storm becomes too large, it causes a lightning flash.
Each flash reduces the magnitude of the electric field
in the region of flash initiation and, therefore, tends to
reduce the larger electric field magnitudes of the storm.
Without lightning, the electric field produced by a simulated thunderstorm often builds to unrealistically large
magnitudes (larger than is observed in storms and even
larger than is needed to cause dielectric breakdown of
clear air). Thus, one criterion for judging the success
of a lightning parameterization is that the magnitude of
the electric field in a simulated storm having frequent
lightning remains bounded by magnitudes normally observed in thunderstorms.
Second, lightning flashes redistribute charge in thunderstorms. At a minimum, the result of this redistribution is to limit the magnitudes and relative locations
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of charges to configurations that produce electric field
magnitudes comparable to the threshold for flash initiation. However, investigators also have suggested that
lightning may produce new charged regions, either immediately, at the location of lightning channels (Marshall and Winn 1982; Helsdon et al. 1992), or later, after
opposite polarities of charge are unmasked as snow and
cloud ice in regions permeated by lightning fall at different terminal speeds (Ziegler and MacGorman 1994).
If a lightning parameterization can adequately simulate a storm’s production of lightning, this capability
can be used to study several aspects of storms besides
their electric fields and charge. An obvious application
is to study characteristics of storms and storm environments that affect the production and evolution of lightning, including the type, charge redistribution, and
storm-relative location of lightning. Such studies are
necessary to develop a good understanding of observed
relationships between lightning and storm properties
that hold promise for nowcasting and storm warnings
(e.g., Goodman et al. 1988; Williams et al. 1989;
MacGorman and Burgess 1994). Numerical cloud models may also be used to study effects of lightning on
other storm and environmental processes, such as effects
on the coalescence, levitation, or alignment of hydrometeors caused by lightning-induced changes in a thunderstorm’s electric field (e.g., Moore et al. 1962; Latham
1969; Metcalf 1993; Krehbiel et al. 1996) and the global
impact of NO x produced by lightning (e.g., Ridley et
al. 1996; Stith et al. 1999). Furthermore, because the
electrification mechanisms thought to be important for
electrifying storms are sensitive to several aspects of
microphysics, simulated lightning production could
eventually aid in evaluating these aspects of microphysical parameterizations.
Obviously, one should be as cautious when interpreting results from lightning parameterizations as when
interpreting results concerning many of the other phenomena parameterized for cloud models. Parameterized
lightning differs from real lightning in several respects.
Much of the physics of lightning initiation and propagation is poorly understood, and some aspects of what
is understood about lightning are simplified by parameterizations, as will be discussed later. Still, when used
with caution, model simulations of storm electrification
and lightning can be useful for testing the consequences
of hypothesized electrical processes and for extracting
information complementary to the observations obtainable with present technologies.
Though several lightning parameterizations already
exist, as discussed later, a new parameterization has been
developed to study relationships involving characteristics of lightning not reproduced well by existing parameterizations. To understand this parameterization,
most of our intended audience will need additional background material. Therefore, before describing the new
lightning parameterization, this paper will briefly outline
some basic concepts of lightning physics incorporated
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in existing parameterizations, review previous parameterizations to place the new one in perspective, and
summarize observations of the three-dimensional structure of lightning that are relevant to developing the new
parameterization.
a. Some basic lightning concepts
Several concepts based on theory and laboratory experiments have formed the basis of recent lightning parameterizations for numerical cloud models. Though
there is disagreement about the molecular and microphysical processes responsible for initiating a lightning
flash, there is broad agreement that a flash is initiated
when the ambient electric forces on atmospheric ions
and electrons in some region become strong enough to
force air to create a conducting channel through which
large electric currents flow (e.g., Uman 1969, chapter
7; MacGorman and Rust 1998, chapter 5). Kasemir
(1960, 1984) suggested that, once a flash is initiated,
the spark propagates bidirectionally (initially parallel
and antiparallel to the electric field) from the point of
initiation. Bidirectional propagation appears to be supported by a study of sparks from ungrounded objects
(Kasemir 1984) and by studies of lightning strikes to
instrumented aircraft (e.g., Mazur 1989a,b). Data from
three-dimensional lightning mapping systems indicate
that a lightning flash typically develops both upward
and downward from its initiation point (e.g., Shao and
Krehbiel 1996). However, knowledge of many details
of this development, including the average velocity in
each direction, remains uncertain (e.g., Rakov et al.
1998; Cooray 1998), and these details may affect the
structure and charge distribution of flashes.
Observations have established that a lightning channel consists of hot, ionized air that behaves in many
ways like a good electrical conductor (again see Uman
1969, chapter 7; MacGorman and Rust 1998, chapter
5). Because the surface of a conductor has a constant
electric potential, a long conducting channel distorts any
ambient electric field in which it is embedded in such
a way that it increases the component of the field along
the channel at each end. From spark experiments, several investigators (e.g., Griffiths and Phelps 1976; Williams et al. 1985) have suggested that a lightning channel will continue to lengthen as long as the electric field
at the tip of one or both ends exceeds a threshold for
continued propagation. The value of this threshold is
uncertain, but it is expected to decrease with height.
Griffiths and Phelps (1976) estimated that positive
streamers continue propagating in electric fields greater
than roughly 100 kV m21 at 9 km above mean sea level
(MSL) and 300 kV m21 at 2.5 km MSL.
A primary electrical effect of lightning is to neutralize
some of the thunderstorm charge that produces the electric field causing the flash. Lightning often has been
treated as if a flash physically transports charge from
its initiation point to its termination point. However,
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Kasemir (1960, 1984) suggested that the charge is not
physically transported in this way, but is induced on the
channel by the thunderstorm’s electric field. He modeled
the developing lightning channel as an electrically neutral conductor (i.e., the negative charge induced on one
end is balanced by an equal positive charge induced on
the opposite end). Note that whether charge is transported from some region or is induced, current must
flow to cause any redistribution of charge on a conductor. Induction of charge has been explicitly incorporated into the lightning parameterizations of Helsdon
et al. (1992) and Solomon and Baker (1996).
Charge neutralization by lightning sometimes has
been treated simply by subtracting the lightning charge
from the existing thunderstorm charge in regions affected by the simulated lightning (e.g., Rawlins 1982;
Takahashi 1987). However, on a microphysical scale,
the process probably is more complicated. Moore et al.
(1962) suggested that the charge on a lightning channel
is quickly converted to atmospheric ions as the channel
cools and that these atmospheric ions then are captured
by hydrometeors, which may or may not have been
previously charged. This capture of ions from lightning
was treated explicitly by Helsdon et al. (1992). Since
channels propagating through negative charge tend to
be positively charged and channels propagating through
positive charge tend to be negatively charged, the immediate effect from either a simple or a microphysical
treatment of neutralization is at least to reduce the net
charge of the thunderstorm in the vicinity of lightning.
Helsdon et al. (1992) showed one parameterized flash
that actually reversed the polarity of net charge at some
points along the channel.
The microphysics of neutralization may also affect
the thunderstorm charge distribution in more subtle
ways. For example, Helsdon et al. (1992) and Ziegler
and MacGorman (1994) noted that the total surface area
of small cloud ice particles is much larger than that of
the much sparser and larger snow particles that often
carry much of the positive charge in the upper part of
storms (at temperatures &2208C). Thus, most of the
negative charge left by lightning propagating through
an upper positively charged region is captured by cloud
particles. Ziegler and MacGorman (1994) pointed out
that the immediate effect is to reduce the net charge in
the region containing lightning, but that subsequent sedimentation of snow can leave a region of net negative
charge (dominated by the charge from lightning on
cloud ice particles) above a region of net positive charge
(dominated by the charge on snow).
b. A brief history of lightning parameterizations
Several parameterizations of lightning have been developed for numerical cloud models, but all have involved compromises that fail to reproduce some aspects
of observed lightning flashes. Such compromises are
inevitable in parameterizations and may not interfere
461
with specific applications. Rawlins (1982) was the first
to add a simple lightning parameterization to a numerical cloud model. When the magnitude of the electric
field attained the discharge threshold of 500 kV m21
anywhere within the model domain, his parameterization simply reduced the net charge density and the
charge densities of individual water substance categories
arbitrarily by 70% at each grid point. Note that, because
many regions of a storm contain a mixture of positively
and negatively charged hydrometeors, reducing the
charge density of all water substance categories involves
considerably more charge than reducing simply the net
charge.
Takahashi (1987) developed a somewhat more complex parameterization of a lightning flash for his axisymmetric, two-dimensional model. Whenever the maximum magnitude of the electric field became at least
340 kV m21 (the maximum field observed by Gunn
1948), his parameterization neutralized 20 C of charge
(the mean charge neutralized by lightning flashes studied by Workman et al. 1942) by destroying that amount
in each of two oppositely charged regions. To neutralize
the charge, he found the maximum positive charge density and the maximum negative charge density, determined the smallest region about each of the two maxima
that contained 20 C, and subtracted positive charge
equally from all charged particles in the positive region
and negative charge equally from all charged particles
in the negative region. Focusing the effect of lightning
on regions having substantial charge density was motivated by MacGorman et al.’s (1981) suggestion, based
on observed lightning structure inside clouds, that lightning channels often permeate regions where charge
would be expected inside storms, with relatively few
channels in regions of little charge.
Ziegler and MacGorman (1994) also used a simple
parameterization, but used a three-dimensional cloud
model and treated the bulk effect of lightning (sometimes the effect of multiple flashes) on storm charge in
a time step, instead of treating an individual, localized
flash. Like all other parameterizations, their parameterization initiated lightning when the maximum electric
field magnitude reached a threshold value. Ziegler and
MacGorman then neutralized a fraction of the net charge
at all grid points where the magnitude of the net charge
density exceeded 0.5 nC m23 . The motivation for focusing the effect of lightning on regions of substantial
charge density was the same as that for Takahashi’s
(1987) parameterization; both were based on observations of lightning by MacGorman et al. (1981). To mimic the effect of lightning releasing the charge from its
channels, charge was neutralized at each grid point involved in the lightning event by adding the opposite
polarity of charge to the preexisting charge. The charge
was distributed to each hydrometeor category in proportion to that category’s fraction of the total surface
area of hydrometeors at that grid point, because a hydrometeor’s cross section for ion capture tends to in-
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crease in proportion to its surface area. Though the process of ion capture was not treated in detail and was
assumed to occur within a time step of the model (typically 5 s), the resulting errors are expected to be smaller
than those caused by uncertainties in our knowledge of
how charge from lightning is distributed and in our
knowledge of the number density and size distribution
of hydrometeor categories, which affect time-dependent
capture.
Three parameterizations have computed individual
lightning channels. The first, developed by Helsdon and
Farley (1987) and Helsdon et al. (1992), estimated both
the geometry and charge distribution of an intracloud
lightning flash for a two-dimensional numerical cloud
model. This parameterization has since been extended
to a three-dimensional cloud model. The basic theory
for their parameterization was taken from Kasemir
(1960, 1984). Accordingly, the parameterized lightning
propagated bidirectionally from the point of initial
breakdown.
To simplify the parameterization, Helsdon and Farley
(1987) and Helsdon et al. (1992) used the ambient electric field from thunderstorm charge to control channel
propagation and termination, instead of using the total
electric field, which also includes the contribution from
the developing channel. [Helsdon et al. (1992) acknowledged that omitting the contribution from the channel
was an oversimplification that could affect results.] The
electric field threshold for lightning initiation was chosen to be 400 kV m21 . Once a flash was initiated at the
grid point having the maximum electric field magnitude,
bidirectional propagation was simulated by having one
end of the lightning propagate approximately parallel
to the ambient electric field line passing through that
point and by having the other end propagate approximately antiparallel. Lightning propagation at each end
was terminated at the point where the ambient electric
field magnitude fell below 150 kV m21 .
The direction of lightning development probably was
affected relatively little by following the ambient electric field instead of the total electric field. (The effect
on the choice of the point where propagation stopped,
however, may have been significant.) At each end of an
initially linear channel developing bidirectionally, the
contribution to the electric field from the channel will
be greatest at the ends, along the channel axis. If the
ambient electric field line curves, then the electric field
due to the channel will tend to cause a developing lightning channel to curve less than the ambient field line.
(This discussion ignores the random-walk element of
channel tortuosity.) However, in simulated thunderstorm
charge distributions, the ambient electric field line passing through the location of the maximum field magnitude of $400 kV m21 typically has little curvature in
adjacent regions in which the magnitude of the electric
field is $150 kV m21 , so the forward bias caused by
the channel has little effect on the direction of channel
development in these regions.
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Once the lightning geometry was determined, Helsdon et al. (1992) computed the distribution of linear
charge density induced by the ambient electric field on
the conducting lightning channel, assumed to be electrically neutral overall, as hypothesized by Kasemir
(1960, 1984). To ensure that the channel was electrically
neutral, Helsdon et al. extended the channel four grid
points (800 m in the published simulation) at each end
and adjusted the distribution of linear charge density on
these extensions to force the integral of lightning charge
density for the whole channel to be zero. The linear
charge densities were converted to number densities of
singly charged ions along the channel and at nearby grid
points by assuming that the charge densities decreased
exponentially with distance from the channel. The ion
density produced by lightning at each grid point was
then added to the pre-existing ion density. In subsequent
time steps, the ions released by lightning were captured
by hydrometeors, a process treated in detail by their
model.
Solomon and Baker (1996) developed a second parameterization that determines an individual lightning
channel. Though Baker et al. (1995) had previously added a simple parameterization of lightning to their 1.5dimensional, kinematic cloud model, it did not explicitly
treat channels. Solomon and Baker (1996) developed
an analytical treatment of a vertical, one-dimensional
lightning channel restricted to the axis of the cylindrical
storms produced by their cloud model. A flash was initiated at a grid point on the axis when the electric field
at that grid point exceeded the threshold for initiation.
To determine its vertical extent, the flash was modeled
as a narrow ellipsoidal conductor that grew longer bidirectionally in steps along the vertical axis of the storm.
After each step, the parameterization calculated the electric field at each end of the flash, including the enhancement caused by the conducting lightning channel.
At the end or ends where the electric field still exceeded
the threshold for continued propagation, the channel
then was extended one step further, and the process was
repeated until the electric field at both ends was below
the threshold.
A third parameterization that treats lightning channels
(Mazur and Ruhnke 1998) has not been used in numerical cloud models, but has been used with an idealized charge distribution, to examine the interrelationships of the storm’s charge distribution and electric potential with the developing lightning channel’s extent,
electric potential, charge distribution, and current. Mazur and Ruhnke’s parameterization initiated a flash at a
relative maxima in the electric field. Like Solomon and
Baker’s (1996) parameterization, their parameterization
confined the flash to the vertical axis of a cylindrical
cloud and computed the total electric field at the tip of
the flash when deciding whether to extend the flash
farther. It differed from that of Solomon and Baker
(1996) by looking at more properties of the flash and
by using a numerical approximation instead of an an-
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463
FIG. 2. A vertical cross section of lightning structure reconstructed
from thunder for a flash produced by a Colorado thunderstorm
(MacGorman 1978). The lightning is superimposed on contours of
radar reflectivity (dBZ ) from a vertical cross section roughly through
the middle of the horizontal structure of the flash.
FIG. 1. A vertical cross section of lightning structure reconstructed
from thunder (a) for a flash that was characteristic of an early period
in an Arizona storm (Teer 1973) and (b) for a flash that was characteristic of a later period in the same Arizona storm (MacGorman
et al. 1981). Each dot indicates the reconstructed location of a lightning channel segment that produced a thunder impulse.
alytic solution to obtain the lightning charge and total
electric field. The parameterization could be adapted to
study lightning relationships in the more realistic storm
charge distributions produced by cloud models.
tent of individual flashes tended to grow as the horizontal extent of a storm grew.
MacGorman et al. (1981) hypothesized that each of
the two layers of lightning channels corresponds to a
major region of thunderstorm charge, the two layers
being a reflection of the vertical dipole that was widely
believed to describe the gross charge distribution of
many thunderstorms. As evidence to support their hypothesis, they noted that the heights of the two layers
of channels were similar to the heights of charge inferred
from in situ measurements of the electric field in the
one case where a comparison could be made and in all
cases were similar to the heights of charge neutralized
by lightning in other storms analyzed by Workman et
al. (1942), Reynolds and Neill (1955), Uman et al.
(1978), and Krehbiel et al. (1979). They also suggested
that single layers of lightning structure occur when the
c. Observed lightning structure
The primary goal of the new parameterization is to
improve simulations of the gross features of observed
lightning structure. To explain what the new parameterization is designed to achieve, it will be helpful to
review observations of lightning structure.
Systems that map lightning channel segments have
shown that the channel segments often tend to occur in
one or two layers 2–3 km thick (Lhermitte and Krehbiel
1979; Krehbiel 1981; MacGorman et al. 1981, 1983;
Proctor 1983; Taylor et al. 1984; Mazur et al. 1984,
1986; Maier et al. 1995). Individual lightning flashes
often have extensive horizontal channel structure in either (Fig. 1a) or both (Figs. 1b–3) of the layers (e.g.,
MacGorman et al. 1981; Shao and Krehbiel 1996). A
flash having extensive horizontal structure in each of
two layers typically has a single vertical channel connecting the two layers. In the few storms analyzed for
a substantial portion of their lifetime, the horizontal ex-
FIG. 3. A vertical cross section of lightning structure mapped by
a VHF interferometer for a flash produced by a New Mexico storm
(Shao and Krehbiel 1996). Contours indicate radar reflectivity (dBZ ).
Shading denotes the lightning flash. The indicated temperature levels
are typical values, not measured in the storm environment.
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FIG. 4. Channel segments of one cloud flash (indicated by dots)
mapped by a VHF interferometer superimposed on lightning charge
centers (circles) and dipoles (arrows) for successive processes of a
cloud flash having similar structure but from another storm. Alphabetical labels indicate the sequence of progression of the lightning.
A shaded circle depicts negative charge; an open circle depicts positive charge. The diameter indicates the relative magnitude of charge
neutralized by each process. The length of arrows is proportional to
the corresponding dipole moment, and the direction indicates the
direction of the point dipole in this plane. The indicated temperature
levels are from typical environmental soundings (Shao and Krehbiel
1996).
vertical separation between positive and negative charge
in a thunderstorm is smaller. This might occur, for example, if the charge regions systematically descended
enough with distance downshear from the updraft that
the heights spanned by the two charge regions overlapped.
Additional support for this interpretation was provided by other studies showing that many mapped channel
segments of individual flashes tended to be located in
the vicinity of the charge centers neutralized by the flash
(Krehbiel 1981; Lhermitte and Williams 1985; Nisbet
et al. 1990), although the version of the Lightning Detection and Ranging VHF mapping system used in these
studies sampled too slowly to provide much detail in
the flash structure. Subsequent studies using VHF mapping systems that sampled faster did not have data on
lightning charge centers. However, Shao and Krehbiel
(1996) noted that the heights of the upper positive and
lower negative charge centers of cloud flashes in previous studies corresponded well with the heights of the
two layers of channels mapped in similar storms by
newer VHF systems. A superposition of mapped channel structure and lightning charge centers for two different flashes from similar storms is shown in Fig. 4 to
illustrate the layering of both and the apparently close
relationship between charge centers and channel structure.
To further support their hypothesis, MacGorman et
al. (1981) offered a conceptual model that is illustrated
VOLUME 40
FIG. 5. A conceptual model of the relationship between lightning
structure and the electric field, electric potential (contour lines), and
charge of a thunderstorm (MacGorman and Rust 1998). Arrows indicate electric field vectors in the plane of the figure, and an arrow’s
length indicates the electric field magnitude approximately at its midpoint. After being initiated at the point labeled I, lightning propagates
bidirectionally, roughly along an electric field line, into the positively
charged region and the negatively charged region on either side of
the initiation point. The flash puts positive charge into the negatively
charged region and negative charge into the positively charged region.
in Fig. 5. They noted that a lightning flash is initiated
at a location where the ambient electric field magnitude
is very large. If initiation occurs between two oppositely
charged regions, each having enough charge to reduce
the electric field to 0 kV m21 somewhere inside the
region, the direction of the total electric force tends to
drive each of the two ends of the developing lightning
flash from the initiation point to a region of weak ambient electric field inside one of the charged regions
(thereby roughly maximizing the ambient potential difference between the oppositely charged ends of the
lightning).
One might expect propagation to stop in a region of
weak ambient electric field, but observed lightning typically has had channels in regions below or inside clouds
where the magnitude of the ambient electric field was
inferred to be much less than 100 kV m21 . MacGorman
et al. (1981) suggested that much of the horizontal development of lightning channels inside clouds occurs in
regions of weak ambient electric field within charged
regions, as shown in Fig. 5, because the direction of the
stronger ambient electric field in surrounding regions
tends to constrain subsequent propagation to the region
of weak ambient field. (Equivalently, the direction of
the electric field opposes propagation that would substantially decrease the difference in the ambient electric
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potential between the oppositely charged ends of the
flash.) Thus, if the ambient potential difference between
the ends of the flash is large enough that the field from
the conducting channel can maintain propagation in a
region of weak ambient electric field within a charged
region, most further propagation would remain near or
inside the charged region, in locations where the ambient electric field tends to be relatively small and the
ambient potential difference with an oppositely charged
region tends to be largest.
This conceptual model was developed further by Williams et al. (1985), who studied spark propagation
through plastic blocks, each of which was doped inside
with a layer of negative charge that varied horizontally
in one of several patterns. The doping process also created a layer of positive charge on the surface of the
plastic. If the electric field between these two layers was
large enough, a spark was initiated and propagated into
the two layers of charge. (Initiation typically was aided
by striking the plastic at a point to create a strong local
stress.) The discharge left a visible trace of its path
inside the plastic.
When isolated spots of large charge density were embedded in a background region of small charge density,
Williams et al. (1985) showed that the spark permeated
spots of large charge density with extensive branching,
but propagated through the regions of small charge density between the spots as relatively isolated channels
having much less branching. Conversely, when charge
density was small in isolated spots and large in the
surrounding region, the spark permeated the surrounding region with heavy branching and tended to avoid
the isolated spots of small charge density. Such behavior
might be expected from superposition. Electrostatic
forces are the same if the two densities of a single polarity are replaced by having the larger density in the
whole region and superposing compensating opposite
charge in regions where the smaller density actually
exists. The polarity of charge on the spark also is this
opposite polarity, so the electrostatic influence of the
superposed spots is to inhibit propagation in their vicinity. In discussing the relevance of these results to
thunderstorms, Williams et al. invoked scaling of parameters between plastic and the atmosphere that had
been estimated from previous experiments, to argue that
the spark’s behavior implies that lightning also will
propagate preferentially through regions of larger
charge density.
Williams et al. (1985) also showed that sparks propagated farther and faster as the charge density through
which they propagated was increased. Propagation
ceased when the magnitude of the electric field at the
tip of the spark became too small. The relevant electric
field was the total electric field, which sums the ambient
electric field and the contribution from the channel itself.
As the space charge density increased, the difference in
potential between the positive charge and negative
charge increased, so the length of spark and amount of
465
branching for which a large enough electric field could
be maintained at the propagating tip also increased.
2. A new lightning parameterization
One goal in developing our new parameterization was
to improve simulation of the observed structure of lightning flashes by using the conceptual model of MacGorman et al. (1981) and Williams et al. (1985). From
initiation at a grid point having a large electric field
magnitude, which typically occurs between two oppositely charged regions, the new parameterization propagates a simulated flash both parallel and antiparallel to
the ambient electric field line until the ambient field
becomes weak, as done by Helsdon et al. (1992). However, if the charge density is large enough at either end
of the flash, the new parameterization extends propagation beyond that of Helsdon et al. by allowing the
flash to propagate extensively through a charged region
at that end. This extension mimics the tendency of observed lightning, thereby allowing the parameterization
to produce extensive horizontal lightning structure in
layers.
Like the parameterization of Helsdon et al. (1992),
the new parameterization does not compute the electric
field at the extremities of the flash to determine how far
the flash expands. (It was impractical for our numerical
cloud model to compute the lightning’s contribution to
the field for each new channel increment of a flash having complex, extensive structure.) Instead, the parameterization roughly maximizes the potential difference
between the positive and negative ends of a flash propagating along an electric field line and assumes, as observed for sparks by Williams et al. (1985) and expected
from theory, that in regions of weak ambient electric
field (hence, relatively uniform ambient electric potential), further propagation tends to move through regions
having larger charge densities. Thus, combined criteria
for the ambient electric potential and the charge density
are used as a proxy for a criterion on the instantaneous
electric field at the extremities of the flash, to govern
the spatial extent of simulated lightning.
The variables that control our lightning parameterization are shown in Table 1. Changing the values for
these variables can affect the structure and charge transfer of simulated lightning flashes, as will be discussed.
A flow chart for the parameterization is given in Fig.
6, which describes how the parameterization handles
initiation and initial development of the flash, and in
Fig. 7, which describes how the parameterization determines lightning propagation in regions of weak electric field and how it determines and neutralizes the
charge involved in each flash. Each process in the flow
chart is described in detail in the following subsections.
a. Lightning initiation
As done by all previous parameterizations, lightning
is initiated when the magnitude of the electric field at
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TABLE 1. Variables of the lightning parameterization affecting lightning structure or charge and values used for the variables to produce
the simulated flashes shown in Figs. 10 and 11.
Variable
Einit
dEinit
Value
150 kV m21
10 kV m21
Estop
rchan
15 kV m21
0.5 nC m23
rneut
0.5 nC m23
fr
0.3
zcg
0.5 km
Description
Threshold value of |E| for determining when a flash will occur
Offset from Einit to determine grid points available for lightning initiation
Minimum |E| above which initial channel propagation is allowed
Minimum |r(i, j, k)| for grid points to be involved in lightning beyond
initial propagation
Magnitude of |r(i, j, k)| not available to a flash (threshold for neutralizing charge)
Fraction of the available charge density neutralized at a grid point by a
flash
Height below which a flash is categorized as a cloud-to-ground flash
any grid point within the model domain reaches a threshold value (Einit ). However, once this occurs, our parameterization differs from previous parameterizations by
initiating a lightning flash at a grid point chosen randomly from the set of grid points at which the electric
field magnitude exceeds a slightly smaller threshold
(Einit 2 dEinit ). The main reason for this random choice
is the subgrid-scale variability of the electric fields observed in storms. Observed gradients are large enough
FIG. 6. Flow chart of the initial propagation of a flash in the new
parameterization.
that, if one were trying to locate the maximum electric
field in an actual storm sampled at the resolution of a
model grid, one could not determine with confidence
which of several grid points having comparably large
electric field magnitudes was nearest the maximum. Furthermore, the problem of choosing the point nearest the
maximum in a numerical cloud model is exacerbated,
because models tend to smooth isolated strong gradients. Thus, our parameterization chooses the initiation
point randomly from among qualifying grid points because there is no firm basis for differentiating among
grid points having an electric field near the threshold
for lightning initiation.
The fact that the actual electric field in the vicinity
of a grid point may well have a larger magnitude than
that produced by the model at the grid point also justifies
choosing a smaller initiation threshold than many have
suggested is necessary to produce lightning. Winn and
Byerley (1975) noted that few, if any, reliable measurements of electric field magnitudes larger than 200
kV m21 have been made in thunderstorms. They suggested that, if larger values are required to initiate lightning, these values exist only over distance scales so
small that they rarely are sampled by in situ sensors.
Such distances typically would be subgrid scale in a
cloud model, particularly since models use differences
in the electric potential of adjoining grid points to calculate the electric field at a grid point. Thus, using an
initiation threshold comparable to the larger electric
field magnitudes typically observed in storms (roughly
100–200 kV m21 ) but smaller than many think necessary takes into account that conditions for producing
lightning may exist only on small distance scales that
cannot be reproduced directly by the model.
Note that it is by no means certain that larger electric
field magnitudes are required. For example, several investigators (e.g., McCarthy and Parks 1992; Gurevich
et al. 1992, 1994; Roussel-Dupre et al. 1992) have suggested that, in the presence of relativistic electrons produced by the energetic cosmic rays that are ubiquitous
in the atmosphere, lightning can be initiated at electric
field magnitudes much smaller than needed for conventional dielectric breakdown of clear air. The threshold
MARCH 2001
MACGORMAN ET AL.
FIG. 7. Flow chart of the later stages of the new flash parameterization, including flash extension into regions with weak electric fields
and substantial charge and the subsequent neutralization of charge
within the lightning structure.
field magnitude for initiation in the presence of relativistic electrons is called the breakeven electric field. Both
breakdown thresholds decrease with height. Marshall et
al. (1995) compared the magnitude of the electric field
measured in vertical soundings of many storms with the
breakeven electric field as a function of height. They
found that, in all storms examined, relative maxima in
467
the measured electric field tended to approach and be
bounded by the breakeven electric field at all altitudes.
The random choice of initiation point has consequences for the resulting lightning. After several flashes
have been produced in a simulated storm, the electric
field magnitude is typically near the model’s lightning
initiation threshold at several grid points, some separated a considerable distance from the others. In this
parameterization, the choice of initiation point controls
the type of lightning produced and the region affected
in the storm. Choosing a point with an electric field
magnitude somewhat smaller than the maximum simulated value has often been important in allowing anvil
lightning, positive or negative cloud-to-ground lightning, or lightning involving the screening layer charge
at cloud boundaries.
When the model has used a uniform initiation threshold throughout the storm, preliminary sensitivity tests
have shown that increasing Einit within the range 100
kV m21 , Einit , 200 kV m21 (the only range examined)
typically has little impact on the ensemble of grid points
eligible for initiation of flashes in a simulation. The main
effects usually are to increase the magnitude of the maximum electric field and the maximum charge density of
the storm and to increase the time required to produce
the first flash. All else being equal, this also tends to
increase the average charge neutralized by a flash and,
therefore, tends to decrease flash rates. However, if inductive charging is important for lightning production
in a simulation, the effect of the increase in inductive
charging rates due to larger electric fields may overwhelm the effect of the greater charge neutralized by
lightning, so the net change may be to increase flash
rates. Furthermore, in simulations of some strong
storms, the maximum flash rate becomes as large as
allowed (typically 5–10 flashes per 5-s time step) for
all values of Einit that we have tried.
Though the uniform lightning initiation threshold
used by all previous cloud models remains an option in
our new parameterization, most theory and laboratory
results suggest that conditions affecting lightning initiation change with altitude. For example, the breakeven
electric field and the threshold field for conventional
dielectric breakdown both decrease with height. Thus,
the parameterization includes an option to specify Einit
as a function that varies with altitude. The breakeven
electric field is the only such option now included, but
others are possible. When Einit is specified as a function
of height, dEinit is assumed to be a fixed percentage of
Einit (typical choices have been #10%).
It would be feasible, though such an option has not
yet been implemented, to superimpose a water substance
field as a mask on E(i, j, k) to limit initiation to regions
having particular microphysical attributes. For example,
to achieve conventional dielectric breakdown in cloudy
air, several investigators have suggested that the ambient
electric field must be enhanced by hydrometeors, some
suggesting the extruded filaments between colliding
468
JOURNAL OF APPLIED METEOROLOGY
raindrops are responsible (e.g., Chauzy and Kably 1989)
and others suggesting the sharper appendages of wet ice
(e.g., Griffiths and Latham 1974). To examine these
hypotheses, one could limit lightning initiation to grid
points at which both |E(i, j, k)| $ Einit 2 dEinit and the
mixing ratio of rain or of wet graupel is in a particular
range or is exceeded.
b. Lightning structure
Our parameterization of lightning flash structure proceeds in two stages. The initial development is similar
to that used by Helsdon et al. (1992). Beginning at the
grid point chosen for initiation, a flash traces the electric
field line outward in both directions (parallel and antiparallel) until the magnitude of the ambient electric
field at each end falls below some threshold value (Estop ).
If one end of the channel reaches ground, the parameterization terminates that end, but allows the other end
to continue developing. The electric field line traced by
the flash is interpolated from the field at grid points; the
tracing is not constrained to follow adjacent grid points.
The step size for tracing the electric field is no larger
than 10% of the spacing between grid points to provide
tracing accuracy commensurate with grid resolution.
Some preliminary experiments have been tried to
roughly adjust the direction of channel development to
correct for the difference in direction between the ambient and total electric field. This could be done by
enhancing the component of the electric field that is
locally parallel to the end of the channel, the enhancement added being either fixed or proportional to the
ambient potential difference between the tip of the channel and the initiation point. Though more work is needed
to develop this option, usually the resulting effect on
channel direction is small over much of the length of
the channel during the initial stage of channel development.
The quantity Estop can be set either to a fixed value
or a fixed percentage of Einit when Einit varies with height.
When fixed, it has typically been set to 10–15 kV m21 ,
so that initial flash development tends to be terminated
in clear air, at the ground, or in regions of charge. This
is a much smaller value than the 150 kV m21 used by
Helsdon et al. (1992). Though there is uncertainty about
the threshold for continued propagation and the threshold probably varies with height, the value used by Helsdon et al. is comparable to the threshold normally
thought to be necessary for spark propagation in air at
roughly 6 km MSL (e.g., Griffiths and Phelps 1976).
Our parameterization uses a smaller value, because
the electric field governing cessation of lightning propagation is not the ambient electric field of the thunderstorm, which varies little during a lightning flash, but
the instantaneous, local electric field at the tip of the
channel, which includes the contribution from the developing channel itself (e.g., Williams et al. 1985). As
mentioned previously, many lightning channels have
VOLUME 40
been observed in regions having ambient electric field
magnitudes much less than 150 kV m21 below and inside
thunderstorms. Such observations make sense because
the inferred difference in the ambient electric potential
across the length of mapped ground flashes and cloud
flashes is large, so the contribution of the lightning channel to the electric field at the end can also be large.
Helsdon et al. (1992) noted that omitting the electric
field contributed by lightning was a shortcoming of their
parameterization.
Proceeding further with this approach, the new parameterization continues lightning propagation farther
beyond Estop than was done by Helsdon et al.’s (1992)
parameterization, which adds only a short linear extrapolation of the channel once the end reaches a grid
point with ambient |E(i, j, k)| , Estop . Instead, our parameterization has a second stage, in which it can extend
flash structure considerably beyond this point through
regions of weak ambient electric field that contain larger
net charge densities. Because it would require considerable computer resources for a numerical cloud model
to compute the channel’s contribution to the electric field
for flashes with complex structure, the parameterization
simply assumes that lightning can propagate in regions
with weak ambient electric field magnitudes and significant charge density, as has been inferred from observed lightning and is expected from theory. Flash
propagation is stopped only when it reaches small
charge densities or surrounding regions in which the
larger magnitudes of the ambient electric field oppose
further propagation. This extension is consistent with
the cloud flash structure observed by MacGorman et al.
(1981), Shao and Krehbiel (1996), and others and with
the conceptual model of MacGorman et al. (1981) and
Williams et al. (1985).
It is difficult to trace the ambient electric field accurately through regions where it is small. In any case,
the electric field from the lightning channel dominates
the ambient field under these conditions. Therefore, it
does not make sense to keep tracing the ambient electric
field, when the ambient field is weak. Furthermore, simply tracing the electric field would not simulate
branched channel structure.
Instead, at each end of the flash inside storm charge,
the parameterization uses a ‘‘wildfire’’ technique to expand the flash from the end of the channel to all contiguous grid points satisfying two conditions: |r(i, j, k)|
$ rchan and |f (i, j, k)| $ |f end |, where r(i, j, k) and
f (i, j, k) are, respectively, the net charge density and
the ambient electric potential at a grid point, and f end
is the ambient electric potential at the end of the channel
before extension. During the wildfire expansion, each
new point that is added must adjoin a point that satisfied
these conditions previously as the flash developed. As
can be inferred from Fig. 5, the condition on |f (i, j, k)|
keeps further flash development within the volume
bounded by the equipotential surface that passes through
the channel at the point where the initial stage of pa-
MARCH 2001
MACGORMAN ET AL.
rameterized channel propagation stops. Equivalently, it
restricts further flash development to a region in which
the ambient electric field is small and prevents the flash
from extending far into regions in which the ambient
electric field would begin to retard propagation.
When Estop is as small as 10–15 kV m21 , which is a
reasonable value for allowing propagation to ground
beneath the cloud, the initial stage of flash development
produced by some configurations of thunderstorm
charge can stop inside the cloud in a region of charge
density too small to support further propagation. This
sometimes has happened after several flashes eliminated
much of the net charge density near the middle of a
region that had been filled with charge. Often within
2–3 grid points (1–1.5 km) of stopping in such cases,
the lightning channel passed a region of net charge density greater than rchan , where the conceptual model suggests branching might occur. Thus, if the charge density
is too small at the end of the first stage of the flash, the
parameterization searches back along the channel to find
if there is a large enough charge density somewhere on
the channel. If so, the part of the channel at which a
large enough charge density is found is treated as the
end of the channel for the rest of flash development. If
not, that end of the flash simply stops at the grid point
determined by Estop , though the other end of the flash
can still expand through charge.
Because realistic storm charge distributions can cause
the initial stage to stop beyond the center of a charge
region when the stopping criterion is simply to reach a
point with a small electric field magnitude, an option
(not used for the examples in section 2d) is for the
parameterization to use more complex stopping criteria
for the first stage of flash development. If a sufficiently
large charge density is encountered at a point with an
ambient electric field magnitude below some threshold
that is significantly smaller than Einit but larger than Estop ,
the second stage of flash development is begun at that
point. Otherwise, the channel continues lengthening until it reaches a point with an ambient electric field magnitude smaller than Estop .
If one end of the flash exits the storm before reaching
Estop , it cannot be extended beyond the point where
|E(i, j, k)| # Estop , under the above rules for propagation
in charged regions. If the flash has reached ground, it
obviously continues no farther downward. If an end
outside the cloud is near the ground (z # zcg ), it is
extended vertically down to the ground. This is done
because our parameterization does not determine the
contribution of the channel to the electric field at the
tip, and this contribution for a channel nearing ground
would typically be enough to cause the flash to continue
propagating to ground. If the end is outside the cloud,
but farther from ground (z . zcg ), it simply stops where
the ambient electric field becomes small enough.
Though at the ground or outside the storm the end of
the flash cannot be extended into a region of thunderstorm charge, the other end of the flash inside the storm
469
still is extended into charge. When flash growth stops
at both ends, the flash is categorized as a cloud flash,
ground flash, or air discharge on the basis of the location
of each end. Cloud flashes have both ends in the storm,
ground flashes have one end at the ground, and air discharges have one end outside the cloud and precipitation, but above ground.
If neither end of a flash can expand through any
charge (this has happened in no simulation so far), the
flash is discarded and then initiated again at another grid
point. If a preset number of successive flash attempts
are discarded in a single time step, the model moves
forward one time step (typically 5 s) before trying again.
Such flashes conceivably could occur in nature, but
when using the estimation scheme for lightning charge
in the next section, they would have little or no affect
on the thunderstorm charge distribution. Treating the
charge of these flashes as described in the next section
would waste computer resources, and so is not done.
These cases are tabulated, however, so they can be included in flash counts if desired.
One more test is applied to a cloud flash before accepting it as a flash for the model. The polarity of the
ambient charge on the end of the flash that propagated
parallel to the electric field must be opposite to that of
the ambient charge on the end that propagated antiparallel. In simulations of some large storms, when a flash
reached a grid point with E(i, j, k) # Estop , it had, on a
few, rare occasions, propagated a long distance horizontally to a region with a small charge density of the
same polarity as the other end. This situation could not
be treated as described in the next section. Several remedies seemed reasonable to us. The one chosen for simplicity was to discard the flash and initiate it again at
another grid point. Thus far, the second flash has never
produced the same problem, but if the problem remains
after a preset number of attempts, the model moves
forward one time step before trying again.
c. Charge neutralization
Charge estimation and neutralization were parameterized by using the technique developed by Ziegler and
MacGorman (1994), except that Ziegler and MacGorman neutralized charge at all grid points having
|r(i, j, k)| $ rchan throughout the storm, but the new parameterization neutralizes charge only at such grid
points within a single localized flash. Other techniques
could be used to estimate the charge involved in simulated lightning whose structure is determined as described in previous sections, but it was simplest to adapt
the technique of Ziegler and MacGorman to provide
charge estimates consistent with observed values. For
later convenience, the nomenclature used here departs
from that used by Ziegler and MacGorman by treating
each end of the flash separately. With the procedure
described in the previous section, the end that propagates parallel to the field stops in negative ambient
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JOURNAL OF APPLIED METEOROLOGY
charge, if it reaches a significant amount of charge. Because this end releases positive charge to neutralize the
thunderstorm charge in which it is embedded, this end
will be labeled the positive end. Similarly, the end that
propagates antiparallel will be labeled the negative end
because it releases negative charge. The magnitude of
charge neutralized by lightning is computed in a few
steps.
1) The model counts the number of grid points in the
flash at which the magnitude of net ambient charge
density r(i, j, k) exceeds rneut . A preset fraction f r
of the charge density excess is summed over all grid
points separately for each polarity to provide a first
estimate of the positive and negative charge involved
in the lightning flash. If r1 (i, j, k) is the ambient
charge density of a grid point at the positive end of
the lightning, and r2 (i, j, k) is the ambient charge
density of a grid point at the negative end, then the
first estimate of the corresponding excess charge
density dr1 (i, j, k) or dr2 (i, j, k) to be added at the
grid point is given by
0
for rchan # | r6 (i, j, k)| , rneut

dr6 (i, j, k) 5 6 f r (| r6 (i, j, k)| 2 rneut )

for | r6 (i, j, k)| $ rneut .

(1)
Note that rneut may be either larger or smaller than
rchan ; the first line of Eq. 1 applies only if rneut is
larger. Also, 6|r6 (i, j, k)| 5 2r6 (i, j, k), because the
subscript indicates the polarity of lightning charge,
and the polarity of ambient charge is opposite to that
of charge on a lightning channel at the same location.
2) For cloud flashes, a correction drcor is computed for
all grid points involved in the flash to ensure that it
neutralizes equal amounts of positive and negative
charge:
drcor 5
O dr (i, j, k) 1 O dr (i, j, k) ,
1
2
N1dis 1 N2dis
(2)
where N1dis is the number of grid points in the flash
at the positive end, N2dis is the number of grid points
in the flash at the negative end, and each summation
is over all points in the corresponding end of the
flash. If the flash is not a cloud flash, drcor 5 0.
3) At each grid point in the flash, ambient charge is
neutralized by distributing the lightning charge over
all hydrometeors at that location. The total lightning
charge density (dr6final ) actually added at a grid point
(i, j, k) involved in the flash is given by
dr6final (i, j, k) 5 dr6 (i, j, k) 2 drcor .
(3)
For ground flashes and air discharges, charge is neutralized by the parameterization only in regions having cloud or precipitation particles. The charge that
lightning transports to ground or deposits outside
cloud and precipitation is assumed to be lost to the
VOLUME 40
storm, because our storm model does not treat free
ions and is concerned only with the charge on hydrometeors.
For cloud flashes, the new parameterization adds a
condition not used by Ziegler and MacGorman (1994),
to make sure that the volume of grid points for each
polarity is sufficient to hold the neutralized charge without causing an electric field magnitude larger than Einit
at the outer boundary of an equivalent sphere. When
one end encompassed much more charge than the other
before the correction for charge was computed, the correction occasionally added enough charge at the smaller
end to create a new flash at the outer boundary of the
original flash volume. In this situation, a real lightning
flash would be expected to keep propagating. Therefore,
our parameterization expands the volume of the flash at
the end at which the uncorrected total charge magnitude
was smaller, if the volume is too small to hold the corrected charge without causing breakdown. The condition on the flash volume means that N1dis and N2dis each
must be greater than Ncrit , where Ncrit is given by
Ncrit 5
1
2
4p
Q dis
3dy 4pe E init
3/2
,
(4)
and dy 5 dx dy dz is the volume represented by each
grid point, e is the permittivity of air, and Qdis 5 dy
S dr1final (i, j, k) 5 dy S |dr2final (i, j, k)|. If N1dis or N2dis
is #Ncrit , the parameterization makes the number of grid
points at that end equal to Ncrit 1 1 by adding adjoining
points, and the magnitude of charge density added to
each grid point at that end becomes
dr6final (i, j, k) 5 dr6 (i, j, k) 2
N6dis
dr ,
Ncrit 1 1 cor
(5)
where dr6 (i, j, k) 5 0 at the new grid points.
The charge density added at a grid point is apportioned to each hydrometeor category according to its
relative surface area, because a hydrometeor’s cross section for ion capture tends to increase in proportion to
its surface area. Thus, the charge density dr m (i, j, k)
deposited by lightning on hydrometeors of the mth category at a grid point is simply
drm (i, j, k) 5
O
Sm (i, j, k)
dr6final (i, j, k),
Sn (i, j, k)
(6)
n
where S m (i, j, k) is the total surface area of hydrometeors
in the mth hydrometeor category in the volume represented by (i, j, k) and the summation is over all hydrometeor categories in that volume. This apportionment
ignores the effect of preexisting hydrometeor charge.
Though such an effect could be included, this refinement
is unwarranted for use with our very rough technique
for estimating the lightning charge at a grid point, Furthermore, even if the estimate of lightning charge were
accurate, the effect of preexisting charge would likely
MARCH 2001
MACGORMAN ET AL.
be noticeable only for precipitation particles, because
the mean charge per cloud particle is generally very
small.
So far, the charge on the channel of the initial stage
of a flash has been ignored. However, it would be possible to include this charge by interpolating the ambient
charge density from grid points adjoining the initial
channel, using the above rules to estimate the lightning
charge, and assigning a corresponding portion of lightning charge to each adjoining grid point. Because the
number of grid points and the ambient charge density
along the channel of the initial stage both tend to be
smaller than those of the extension of the channel
through regions of charge, the contribution to lightning
charge from the initial channel would tend to be small
compared with that from lightning structure through
charge. This tendency would also be expected if the
parameterized charge were proportional to the ambient
potential difference between the channel segment and
the initiation point, as computed by Helsdon et al.
(1992). It also is consistent with the observations of
Krehbiel (1981) showing neutralized charge mainly
along the horizontally extensive part of flash structure.
In simulations thus far, rneut has been set equal to rchan
to take advantage of sensitivity studies by Ziegler and
MacGorman (1994) to set parameter values for preliminary simulations with our lightning parameterization.
Ziegler and MacGorman found that estimated flash rates
were comparable to observed rates and the maximum
electric field remained bounded throughout simulations,
when their parameterization used rneut 5 0.5 nC m23
and f r 5 0.3. These values were used with the new
parameterization to produce the results in the next section. A comparable effect would be obtained if one set
rneut 5 0 and neutralized a smaller fraction ( f r ) of the
charge at grid points at which r(i, j, k) $ rchan , but this
could lead to numerical problems by introducing larger
gradients in r(i, j, k). In their sensitivity tests, Ziegler
and MacGorman found that using f r 5 0.1 and rneut 5
rchan 5 0.5 nC m23 removed too little charge to prevent
the electric field magnitude from growing larger than
needed to create a spark in clear air.
d. Examples of parameterized lightning flashes
An exhaustive study of our parameterization’s sensitivities to different parameter choices and different
storm morphologies would be a major study in its own
right and is beyond the scope of this paper. Our intent
in this section is simply to provide a few examples to
demonstrate that, by using a physically realistic conceptual model firmly based on observations, our new
parameterization produces lightning whose structure is
significantly more realistic than that produced by previous parameterizations. To provide examples of flashes
produced by our parameterization, a supercell storm was
simulated.
The three-dimensional cloud model used to provide
471
these examples incorporates a comprehensive microphysics package (e.g., Straka and Anderson 1993;
Straka and Rasmussen 1997) that includes prognostic
equations for hydrometeor mixing ratio, concentration,
temperature, charge, and other microphysical characteristics. Hydrometeor species were treated by bulk microphysics and included cloud droplets, drizzle, rain,
three ice crystal habits (plates, columns, and rimed crystals), snow aggregates, three graupel density habits, frozen drops, small hail, and large hail (diameter . 20
mm). Hydrometeors were charged by using the noninductive and inductive parameterizations developed by
Ziegler et al. (1991). The charge and electric force had
no effect on storm dynamics or on other hydrometeor
properties. The model domain was 40 km 3 40 km 3
20 km in x, y, and z, respectively, with a grid spacing
of 1 km 3 1 km 3 0.5 km.
The atmosphere was assumed initially to be horizontally homogeneous. The atmospheric sounding had convective available potential energy of approximately
2200 J kg21 and a half-circle hodograph in which the
wind speed changed from 0 m s21 at the ground to 50
m s21 at a height of 6 km and above. Convection was
initiated with a warm, moist spheroid (10.0 km 3 10.0
km 3 1.4 km) centered 1.4 km above ground in the
middle of the domain, as done by Weisman and Klemp
(1984).
The model simulation was run for a total of 40 min
after initiation, with a time step of 5 s. After the kinematic, thermodynamic, and microphysical fields (including charge density) were adjusted at the end of each
time step, the electric field was evaluated to check
whether the condition for flash initiation (|E(i, j, k)| $
Einit ) was satisfied anywhere in the model domain. If the
condition was satisfied, the resulting lightning flash was
treated as an instantaneous event between time steps;
the charge from lightning was added to the charge of
the water substance categories after each flash, but before the next time step, as described in section 2c. Multiple flashes were allowed after any time step, if somewhere in the model domain the criterion |E(i, j, k)| $
Einit was met after the charge from the previous flash
was added.
To illustrate the model simulation, various output
fields are presented at the time of the first simulated
lightning flash. Water substance fields, charge density,
electric potential, and electric field are shown in Figs.
8 and 9 for one vertical cross section of the storm. Note
the two broadly horizontal regions of net charge in Fig.
8b. This geometry is important in producing the predominately horizontal lightning structure through each
of these two regions, particularly in the second example
to be discussed shortly. The descending altitude of net
charge with distance from the updraft is caused by the
sedimentation of charged snow and graupel (Figs. 8f
and h) lofted by the strong updraft (Fig. 8a). The regions
of positive snow and ice overlap with that of negative
graupel, thereby creating a mixture having little net
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JOURNAL OF APPLIED METEOROLOGY
VOLUME 40
FIG. 8. A vertical cross section of model fields at the time of the first lightning flash in a simulation of a
supercell storm. The heavy contour that is identical in all panels indicates the cloud boundary. The approximately horizontal solid and dashed lines depict the 08 and 2408C isotherms, respectively. Unless otherwise
indicated, solid contours indicate values $ 0; dotted contours indicate , 0. (a) Vertical wind (contour interval
5 4 m s21 ), (b) net charge density (interval 5 0.25 nC m23 , centered on zero), (c) cloud water (solid contours,
interval 5 1 g kg21 ) and cloud ice (dotted contours, interval 5 0.5 g kg21 ) mixing ratio, (d) charge density
on cloud ice (interval 5 0.4 nC m23 ) (e) snow aggregate (solid contours, 0.005, 0.01, and 0.05 g kg 21 and
then interval 5 0.1 g kg21 ) and rain mixing ratio (dotted contours, interval 5 0.01 g kg21 ), (f ) charge density
on snow aggregates (interval 5 0.5 nC m23 ), (g) graupel/hail mixing ratio (interval 5 2 g kg21 from 0.1 g
kg21 ), (h) charge density on graupel/hail (interval 5 0.5 nC m23 ).
charge between the two main regions of net charge. This
mixture provides a reservoir from which subsequent
sedimentation can help replenish the charge neutralized
by lightning. Note also that the region of large electric
field magnitude (Fig. 9a), where lightning initiation is
most likely, is between the main net positive charge and
the main net negative charge. The few dots above the
main positive charge in Figs. 8b and 8d depict negative
charge (the line is too short to be dashed), indicative of
a growing screening layer on cloud ice near the upper
cloud boundary (e.g., Ziegler et al. 1991).
Values used for the parameters of the lightning parameterization are shown in Table 1. Threshold value
Einit was constant over the entire model domain and was
set to 150 kV m21 , because in situ observations of the
magnitude of the electric field have rarely exceeded this
value. Though the specific value chosen is arbitrary, the
simulated lightning structure is not particularly sensitive
MARCH 2001
MACGORMAN ET AL.
473
FIG. 9. A vertical cross section of (a) the electric field magnitude
and (b) the electric potential at the time of the first lightning flash in
a simulation of a supercell storm. Solid contours indicate values $
0; dotted contours indicate , 0. The contour interval for the electric
field magnitude is 25 kV m21 and for the electric potential is 5 3
10 3 V.
to the magnitude of Einit , but the timing of the first flash
and other electrical properties of the storm are affected,
as discussed previously.
The first flash produced by the simulation is shown
in Fig. 10, which presents various vertical cross sections
of the lightning structure. Note that the vertical channel
at x 5 23 km in Fig. 10d spans the region of large
electric field between the two main charge regions (compare with Figs. 8 and 9), and the horizontal structure is
given by the intersection of regions of sufficient charge
density with regions in which the magnitude of the electric potential is sufficiently large. The seventh flash,
which occurred two minutes after the first, is shown in
Fig. 11. Note that each of these two examples has two
layers of flash structure, much like each of the observed
flashes shown in Figs. 1b–3. Furthermore, the upper
layer coincides with positive thunderstorm charge, and
the lower layer, with negative charge, much as was inferred for the observed flash shown in Fig. 4. The parameterization of Helsdon et al. (1992), without the extensions described in this paper, would not have produced flash structure having two layers, but would have
produced only an essentially vertical channel (similar
to the one between the two layers of each flash in Figs.
10 and 11).
The increase in the horizontal extent of lightning
structure from the first flash (Fig. 10) to the latter flash
FIG. 10. The first flash produced by the storm simulation shown in
Figs. 8 and 9. Each cross section through the flash shows the charge
density contour for 0.5 nC m23 (lighter contours, solid for positive values,
dotted for negative) and its intersection with the volume within the equipotential contour for f end (shaded area). The spatial structure of each end
of the lightning flash is given by the region of intersection. (a)–(c) Vertical
cross sections through part of the lightning flash, spaced 1 km apart in
the y direction, and (d) the footprint of the flash in the x–z plane, obtained
by plotting the outermost boundary of the flash at any value of y. The
solid line connecting the two vertically separated areas depicts the channel
from the initial stage of the parameterization’s flash development, when
the channel traced the electric field.
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JOURNAL OF APPLIED METEOROLOGY
FIG. 11. The seventh flash in the supercell storm simulation. It
occurred 140 s after the first flash, shown in Fig. 10. See the caption
for Fig. 10. (a)–(c) Vertical cross sections through part of the lightning
flash, spaced 1 km apart in the y direction, and (d) the total footprint
of the flash on this vertical cross section.
VOLUME 40
(Fig. 11) is consistent with observations by several investigators (e.g., Krehbiel 1986; Scott et al. 1995; Maier
et al. 1995). These investigators attributed the growth
of the horizontal extent of lightning to a corresponding
horizontal growth of regions containing charged hydrometeors. A similar relationship explains the simulated
behavior. Horizontal growth of regions of larger charge
density caused horizontal growth of lightning structure,
and both were caused by horizontal growth of the regions containing the hydrometeor carriers of charge.
The tendency for flashes to occur in layers, each associated with a charge region, is reasonably independent
of the values chosen for the parameters in Table 1; it is
inherent in the parameterization. The tendency for the
horizontal extent of flashes to grow early in the storm
also would be expected over a fairly broad range of
parameter values. However, the properties of flashes do
depend to some extent on the values chosen for the
controlling parameters. For example, Einit affects the
timing of the first flash and the amount of charge available to flashes in the thunderstorm. The functional dependence of Einit on altitude affects the height of flashes
and thereby can affect flash type, the maximum ambient
electric field at a given grid level, and the vertical distribution of thunderstorm charge. The effect of rchan depends to some extent on Einit , but for a given value of
Einit , a larger value of rchan causes flashes to become
smaller, because the size of the region in which r(x, y, z)
$ rchan becomes smaller. The value of rchan tends to
affect the minimum and maximum height of a flash
much less than the horizontal extent, because the vertical
extent of each of the larger charge regions in a storm
tends to be less than the horizontal extent. One can make
rchan large enough to make it impossible for most flashes
to expand in the second stage of our parameterization.
However, this problem can be avoided altogether by
choosing a fairly small value of rchan , one just large
enough to avoid regions with r(x, y, z) near zero.
The charge neutralized by a given flash can vary considerably for different choices of rneut and f r , but the
parameter values can be readily adjusted to give reasonable values in an ensemble of flashes. The charge
neutralized in the thunderstorm by each of the first 11
flashes in our simulated storm is shown in Table 2. The
amount of charge neutralized by individual flashes tended to increase during the first few minutes after the storm
began producing lightning. With the same values for the
lightning parameterization, a simulated flash toward the
end of a severe storm having a large anvil sometimes
has neutralized more than 100 C. The storm simulation
used for this paper did not continue long enough to allow
this to happen. The tendency for the charge neutralized
by flashes to increase early in the storm occurs over a
broad range of parameter values. This tendency and the
charge values for individual flashes in this simulation
are both consistent with observations of thunderstorms
(e.g., Uman 1969; Krehbiel 1986).
MARCH 2001
475
MACGORMAN ET AL.
TABLE 2. Charge neutralized by each of the first 11 flashes in a
supercell storm simulation. All were cloud flashes, each neutralizing
two equal, but opposite, charges having the listed magnitude.
Sequence
No.
Model time
(s)
Charge
(C)
1
2
3
4
5
6
7
8
9
10
11
1950
1980
1990
2010
2025
2045
2090
2125
2155
2160
2190
3.1
2.7
7.9
5.1
9.6
34.6
37.0
39.6
10.4
29.7
25.7
3. Concluding remarks
Differences between the new parameterization and
the parameterization of Helsdon et al. (1992) are summarized in Table 3. The new contributions of our parameterization are the features that affect lightning initiation and structure, not the scheme for estimating the
charge of lightning. A different lightning charge scheme
could be used without affecting the lightning structure
produced by the parameterization. The charge scheme
of Ziegler and MacGorman (1994) was chosen because
it was simple to adapt it to the new parameterization to
give reasonable results. In this scheme, the choice of
values for rneut and f r obviously have a large effect on
the amount of charge involved in a flash and, hence, on
flash rates. However, the values should be adjusted to
force individual flashes in the ensemble produced by
simulated storms to neutralize amounts of charge typical
of observed lightning. The simulated flash rates will
then be scaled accordingly.
With additional time and effort for development, it
would be possible to replace the present charge scheme
with a version of the charge estimation technique developed by Helsdon et al. (1992), who based their estimate of a flash’s charge distribution on the difference
in ambient electric potential between a given point on
the channel and the initiation point. Because the flash
structure produced by our parameterization can be much
more complex than the single channel used by Helsdon
et al.’s parameterization, however, it probably would be
necessary to treat the effect of channels at surrounding
grid points on the charge induced at a given grid point
embedded in the flash. Such a treatment may require
simplifying assumptions and an iterative procedure to
compute the induced lightning charge.
One could also develop a completely different technique to estimate lightning charge. For example, one
might use the boundary condition that the surface of the
lightning volume is equipotential and solve iteratively
for electric potential throughout the thunderstorm grid
with this boundary condition satisfied. The charge density at grid points in the lightning volume could then
be calculated from the electric potential by using Poisson’s equation.
Once the charge involved in a lightning flash is estimated at a grid point, the way that the charge is handled
is governed by the capabilities of the thunderstorm model being used. If one uses the model described by Helsdon and Farley (1987), the lightning charge at a grid
point is converted to singly charged ions and added to
the ion population treated by the model, which also
explicitly treats the subsequent capture of ions by various types of hydrometeors. In the model used for this
paper, ions are not treated explicitly. Instead, it was
assumed that the charge on lightning in the thunderstorm
was captured by hydrometeors within a model time step
of 5 s, so the charge was distributed over hydrometeor
categories in proportion to their cross section. The errors
from assuming all lightning charge is captured within
a time step are expected to be much smaller than those
caused by uncertainties in our knowledge of the amount
and distribution of charge on lightning channels.
The main contribution of the new parameterization is
that it modifies the occurrence and structure of simulated
lightning by adding new options for flash initiation and
by continuing flash development into regions having a
weak electric field and a substantial charge density. Both
of these extensions are based on conceptual models of
the physics of lightning, as described in a previous section, and can improve the realism of simulated lightning.
Though the choice of values for Estop and rchan can affect
the dimensions of the flash and details of the flash struc-
TABLE 3. Comparison between the parameterization of Helsdon et al. (1992) and the parameterization described in this paper.
Characteristic
Helsdon et al. (1992)
Threshold |E| for initiation
Einit uniform in model domain
Starting grid point
Point having |E|max
Channel propagation
Grid point to grid point, roughly parallel
and antiparallel to ambient E
Four grid points beyond the point at which
ambient E , Estop
Termination of flash
Lightning charge
Estimates charge induced on channel by
ambient E
New
Einit either uniform or decreasing with
height
Chosen randomly from points having |E| .
Einit 2 dEinit
In small steps parallel and antiparallel to
interpolated ambient E
Initially stops where ambient E , Estop, then
extends to all adjoining grid points at
which |r| $ rchan and |f| $ fend
Assumes charge is approximately
fr [r(i, j, k) 2 rneut ]
476
JOURNAL OF APPLIED METEOROLOGY
ture, the qualitative features of flash structure are similar
for different choices of values as long as Estop is small
enough that the parameterization’s initial stage of flash
propagation tends to end within regions of substantial
charge density and rchan is less than charge density magnitudes produced over reasonably large regions of the
simulation. (The magnitude of the larger charge densities typical of a simulation is a strong function of the
value of Einit when using any lightning parameterization
capable of preventing the electric field magnitude from
increasing indefinitely.)
The choice of magnitude for Einit and its functional
form, if a function of height is used, should conform to
some hypothesis, such as those discussed in the previous
section. The choice has some obvious repercussions,
affecting, for example, the maximum electric field allowed in the simulation, the maximum charge density,
the timing of the first flash, the evolution of flash rates,
and possibly the location of flash initiation. These effects, in turn, may have more subtle repercussions.
These effects will be investigated and reported more
thoroughly in future studies.
Note that this paper presents only a few initial examples showing that the realism of simulated flash structure can be improved by using the conceptual model of
MacGorman et al. (1981) and Williams et al. (1985) to
extend the lightning parameterization of Helsdon et al.
(1992). This paper neither explores all the types of lightning that can be produced by the parameterization nor
attempts to demonstrate that all types of lightning structure can be produced by the parameterization in its present form. Doing so would be an enormous undertaking,
because flash structure depends intimately on the thunderstorm charge distribution, which varies from storm
to storm and during the life cycle of a given storm. For
example, all simulated flashes thus far have been initiated between two oppositely charged regions. Though
not shown in this paper, the parameterization has produced positive and negative cloud-to-ground flashes in
some storm simulations, but only when the storm’s
charge distribution has had specific characteristics. Initiation must occur on an electric field line that connects
with the ground or that terminates in a region of charge
that reaches near ground. For negative cloud-to-ground
flashes, this condition requires a significant lower positive charge. In many simulations, the parameterized
noninductive charging mechanism never produced
enough lower positive charge to cause negative cloudto-ground flashes.
Thus, to evaluate the parameterization’s ability to produce specific types of lightning, one might have to simulate many different storms over most of their life cycle.
Furthermore, if some observed types of lightning were
not produced by these simulations, it would be necessary to determine whether the lack of observed types
was caused by flaws in the lightning parameterization
or by systematic differences between observed thunderstorm charge distributions and the charge distribu-
VOLUME 40
tions produced by the parameterized electrification
mechanisms. Though it is possible to conceive of idealized storm charge distributions in which our lightning
parameterization would produce all the commonly observed types of lightning, extensive sensitivity studies
and comparisons with observations obviously are needed before one can thoroughly understand how the lightning parameterization behaves and can have much confidence that it will produce the correct lightning flash
types in a given situation. Such studies of the lightning
parameterization will be included in our future modeling
studies of storm electrification.
By permitting extensive development of flashes in
regions of substantial charge density and weak ambient
electric field, our new parameterization produces flash
structure much more like that of observed flashes, as
would be expected from the observed correlation between horizontal lightning structure and thunderstorm
charge. However, like most parameterizations, our lightning parameterization incorporates several approximations and arbitrary choices and should not be extended
beyond the limits of its design. For example, our parameterization does not include detailed lightning physics, and so is unsuitable for studying the details of flash
propagation. In fact, our parameterization could benefit
from comparisons with models that include more physics of channel propagation, to aid in choosing its parameter values and to refine some aspects of the parameterization itself. The examples shown in this paper were
produced by using particular parameter values, but it is
by no means certain that these are the best values. Furthermore, it may be that some parameters need to vary
randomly within some range or to depend on storm
conditions. On the other hand, models with more detailed lightning physics require too much computing for
them to be used routinely at present in numerical simulations focusing on the overall electrical properties of
storms. The primary benefit of the new parameterization
is that, by making simulated flash structure substantially
more realistic while keeping calculations tractable, it
improves estimates of the regions of storms affected by
lightning for studies using numerical cloud models.
Acknowledgments. Partial funding of this research
was provided by the National Science Foundation,
Grants ATM 9807179, ATM 9613718, and ATM
9311911. The Cooperative Institute for Mesoscale Meteorological Studies provided an office and office support for the first author during the reported research and
manuscript preparation. We thank Joan O’Bannon for
drafting some of the figures. We also thank Vlad Mazur,
John Helsdon, and anonymous reviewers for their helpful suggestions concerning earlier versions of this paper,
though the authors bear sole responsibility for all statements in this paper.
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