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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 460 JOURNAL OF APPLIED METEOROLOGY 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 VOLUME 40 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, MARCH 2001 MACGORMAN ET AL. 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- 462 JOURNAL OF APPLIED METEOROLOGY 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. VOLUME 40 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- MARCH 2001 MACGORMAN ET AL. 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. 464 JOURNAL OF APPLIED METEOROLOGY 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 MARCH 2001 MACGORMAN ET AL. 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 466 JOURNAL OF APPLIED METEOROLOGY VOLUME 40 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 470 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 472 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. 474 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. 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