Multiple Environmental Influences on the Lightning of Cold-Based Continental Cumulonimbus Clouds. Part I: Description and Validation of Model

Vaughan T. J. Phillips Department of Physical Geography, University of Lund, Lund, Sweden

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Marco Formenton Department of Physical Geography, University of Lund, Lund, Sweden

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Vijay P. Kanawade Department of Physical Geography, University of Lund, Lund, Sweden

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Linus R. Karlsson Department of Physical Geography, University of Lund, Lund, Sweden

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Sachin Patade Department of Physical Geography, University of Lund, Lund, Sweden

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Jiming Sun Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China

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Christelle Barthe Laboratoire de l’Atmosphère et des Cyclones, UMR 8105 CNRS/Météo-France/Université de La Réunion, Saint Denis, Réunion, France

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Jean-Pierre Pinty Laboratoire d’Aérologie, Université Paul Sabatier and CNRS, Toulouse, France

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Andrew G. Detwiler Department of Physics, South Dakota School of Mines and Technology, Rapid City, South Dakota

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Weitao Lyu State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences, Beijing, China

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Sarah A. Tessendorf National Center for Atmospheric Research, Boulder, Colorado

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Abstract

In this two-part paper, influences from environmental factors on lightning in a convective storm are assessed with a model. In Part I, an electrical component is described and applied in the Aerosol–Cloud model (AC). AC treats many types of secondary (e.g., breakup in ice–ice collisions, raindrop-freezing fragmentation, rime splintering) and primary (heterogeneous, homogeneous freezing) ice initiation. AC represents lightning flashes with a statistical treatment of branching from a fractal law constrained by video imagery.

The storm simulated is from the Severe Thunderstorm Electrification and Precipitation Study (STEPS; 19/20 June 2000). The simulation was validated microphysically [e.g., ice/droplet concentrations and mean sizes, liquid water content (LWC), reflectivity, surface precipitation] and dynamically (e.g., ascent) in our 2017 paper. Predicted ice concentrations (~10 L−1) agreed—to within a factor of about 2—with aircraft data at flight levels (−10° to −15°C). Here, electrical statistics of the same simulation are compared with observations. Flash rates (to within a factor of 2), triggering altitudes and polarity of flashes, and electric fields, all agree with the coincident STEPS observations.

The “normal” tripole of charge structure observed during an electrical balloon sounding is reproduced by AC. It is related to reversal of polarity of noninductive charging in ice–ice collisions seen in laboratory experiments when temperature or LWC are varied. Positively charged graupel and negatively charged snow at most midlevels, charged away from the fastest updrafts, is predicted to cause the normal tripole. Total charge separated in the simulated storm is dominated by collisions involving secondary ice from fragmentation in graupel–snow collisions.

Current affiliation: Centre for Earth, Ocean and Atmospheric Sciences, University of Hyderabad, Hyderabad, India.

© 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Vaughan T. J. Phillips, vaughan.phillips@nateko.lu.se

This article has a companion article which can be found at http://journals.ametsoc.org/doi/abs/10.1175/JAS-D-20-0234.1.

Abstract

In this two-part paper, influences from environmental factors on lightning in a convective storm are assessed with a model. In Part I, an electrical component is described and applied in the Aerosol–Cloud model (AC). AC treats many types of secondary (e.g., breakup in ice–ice collisions, raindrop-freezing fragmentation, rime splintering) and primary (heterogeneous, homogeneous freezing) ice initiation. AC represents lightning flashes with a statistical treatment of branching from a fractal law constrained by video imagery.

The storm simulated is from the Severe Thunderstorm Electrification and Precipitation Study (STEPS; 19/20 June 2000). The simulation was validated microphysically [e.g., ice/droplet concentrations and mean sizes, liquid water content (LWC), reflectivity, surface precipitation] and dynamically (e.g., ascent) in our 2017 paper. Predicted ice concentrations (~10 L−1) agreed—to within a factor of about 2—with aircraft data at flight levels (−10° to −15°C). Here, electrical statistics of the same simulation are compared with observations. Flash rates (to within a factor of 2), triggering altitudes and polarity of flashes, and electric fields, all agree with the coincident STEPS observations.

The “normal” tripole of charge structure observed during an electrical balloon sounding is reproduced by AC. It is related to reversal of polarity of noninductive charging in ice–ice collisions seen in laboratory experiments when temperature or LWC are varied. Positively charged graupel and negatively charged snow at most midlevels, charged away from the fastest updrafts, is predicted to cause the normal tripole. Total charge separated in the simulated storm is dominated by collisions involving secondary ice from fragmentation in graupel–snow collisions.

Current affiliation: Centre for Earth, Ocean and Atmospheric Sciences, University of Hyderabad, Hyderabad, India.

© 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Vaughan T. J. Phillips, vaughan.phillips@nateko.lu.se

This article has a companion article which can be found at http://journals.ametsoc.org/doi/abs/10.1175/JAS-D-20-0234.1.

1. Introduction

The first known books about weather phenomena were by Aristotle and his student, Theophrastus, in Ancient Greece around 300 BC (Brunschon and Sider 2007). In Meteorology, Theophrastus listed possible causes of lightning (Fortenbaugh and Gutas 1992). A connection between ice in clouds and lightning was hypothesized. In modern times, lightning was understood as an electrical process. In the twentieth century, various causes were proposed for charge separation in clouds [literature reviewed by Pruppacher and Klett (1997, hereafter PK97)]: 1) diffusion of ions onto inductively polarized drops, 2) convection of space charge from the environment, 3) polarized drops colliding with ice and rebounding, 4) ice breakup, and 5) “noninductive” charge separation in rebounding ice–ice collisions. Only cause 5 explained observed time scales of electrification (Helsdon et al. 2001, hereafter H01). Whereas an explanation of lightning by Theophrastus assumed collisions involving ice somehow, some modern explanations did not involve ice (causes 1 and 2) and have been discounted.

With emergence of cloud physics, it has become apparent that no physical process occurs in isolation in clouds. Lightning is no exception. Clouds consist of a myriad of interconnected physical processes, including electrical processes. The charge separation that causes lightning is known to be predominantly due to (noninductive) rebounding ice–ice collisions involving rimed ice precipitation in the presence of supercooled liquid (Reynolds et al. 1957; Takahashi 1978, 1984; Latham 1981; Jayaratne et al. 1983; Baker et al. 1987; Helsdon and Farley 1987; Latham and Dye 1989; Kumar and Saunders 1989; H01; Helsdon et al. 2002, hereafter H02; Mansell et al. 2002, 2005, 2010, hereafter M02, M05, M10, respectively). Sedimentation of heavier particles leaves a net charge aloft. Overall charge separated depends on concentrations of ice, while charge separated per collision is governed by temperature (T), liquid water content (LWC), and particle sizes. Essentially, lightning is caused by microphysical interactions.

Microphysical processes in clouds are controlled by environmental factors, such as aerosol conditions, instability, shear, and humidity. Aerosol conditions govern numbers and sizes of cloud particles (Rosenfeld and Lensky 1998; Phillips et al. 2001, 2002, 2005; Khain et al. 2004, 2005, 2008; van den Heever et al. 2006; Kudzotsa et al. 2016). There are two aerosol-sensitive mechanisms of precipitation:

Relative humidity (RH) controls the temperature of cloud base (Williams and Stanfill 2002; Khain et al. 2004; Williams et al. 2005; Zeng et al. 2009).

The vertical structure of charge characterizes thunderstorms. Most typically, a storm is “normal” (Williams 1989) with a tripole (lower positive charge beneath midlevel negative charge with upper-level positive charge) or dipole (Kuhlman et al. 2006), causing negative cloud-to-ground flashes (−CGs; negative charge to ground). Rarer storms with the opposite configuration are “inverted” (Marshall et al. 1995), with negative charge at upper levels and “+CGs” (positive charge to ground). More intense convection can be inverted with mostly +CGs and often large hail (Rust et al. 1981; Reap and MacGorman 1989; Wiens et al. 2005).

For our cloud model, representations of ice-microphysical processes were developed (Phillips et al. 2013, 2014, 2015, 2017a, 2018). Sticking efficiency for ice–ice collisions was treated with an energy-based approach (Phillips et al. 2015). Breakup in ice–ice collisions was treated for all microphysical species and predicted by to form most (95%–98%) of ice particles not from homogeneous freezing in a “cold-based” (cloud base of about 0°C) mesoscale multicellular storm (Phillips et al. 2017a,b). The storm was observed on 19/20 June 2000 in the Severe Thunderstorm Electrification and Precipitation Study (STEPS) (Lang et al. 2004). The ice-crystal process prevailed in the overall production of precipitation.

In the simulated STEPS storm, inclusion of breakup in ice–ice collisions increased the average concentration of ice by between one and two orders of magnitude from 0° to −30°C (Phillips et al. 2017b, their Figs. 5d and 8). Only by including this breakup were aircraft observations of filtered (>0.2 mm) ice concentration and LWC predicted realistically. Collisions of snow (>0.3 mm) with denser graupel/hail initiated most of the secondary fragments. Surface precipitation was modified by breakup with smaller crystals and less LWC.

In this two-part paper, to compare influences on lightning from various environmental factors, an electrical component is first developed and assessed for our Aerosol–Cloud model (AC). AC represents all empirically quantified mechanisms for initiation of drops and crystals in terms of dependencies on aerosol conditions. This electrical assessment is performed with the same cold-based cloud case from STEPS simulated and validated by Phillips et al. (2017b) against coincident observations. AC reproduced the many nonelectrical cloud-microphysical statistics observed by aircraft in that case—including ice concentration and LWC. Here the simulation is repeated including the electrical component.

STEPS occurred in the U.S. central Great Plains (CGP), combining electrical (e.g., by balloon), radar, and microphysical observations with an armored aircraft sampling fast thunderstorm updrafts (Lang et al. 2004). A Lightning Mapping Array (LMA) was deployed (Rison et al. 1999; Krehbiel et al. 2000). The storm case (19/20 June) was selected as −CGs and normal electrical structure were observed. In CGP, most storms chiefly produce negative (−CG) lightning (Orville and Huffines 2001; Boccippio et al. 2001; Fleenor et al. 2009), which is shown to be due to normal polarity charge structure (Tessendorf et al. 2007).

The aim of this two-part paper is to unravel some of the mysteries about environmental influences on lightning. First in this Part I the model and STEPS simulation are explained and validated with observations. In Phillips and Patade (2020, manuscript submitted to J. Atmos. Sci., hereafter Part II), the simulation will be analyzed with sensitivity tests to quantify the environment–lightning linkage. Focus is given in Part II to reasons for why lightning is observed more frequently over land than ocean and to how the environment controls the charge structure of storms.

2. Model description

The description by Phillips et al. (2017b) applies here in nonelectrical respects with only a few minor changes. Symbols used in this paper are summarized in appendix A.

a. AC model

AC represents clouds and aerosols with hybrid spectral bin/two-moment bulk microphysics, interactive radiation, and semiprognostic aerosol schemes. Here AC is run as a cloud-resolving model (CRM) with horizontal and vertical grid spacings of 1 and 0.5 km, and a 3D mesoscale domain 80 km wide. Mesoscale cloud systems are resolved. Microphysical species are cloud liquid, cloud ice (or “crystals”), rain, graupel/hail and snow. Seven aerosol species govern primary initiation of hydrometeors, with heterogeneous and homogeneous nucleation of ice. Three types of fragmentation are treated to form secondary ice: breakup in ice–ice collisions (Phillips et al. 2017a,b), Hallett and Mossop (1974, hereafter HM) rime splintering (cloud droplets > 24 μm) and fragmentation of freezing rain/drizzle (Phillips et al. 2018). More details are in appendix B.

b. Electrical component

The degree of complexity of the lightning scheme resembles that of Barthe et al. (2005) and is intermediate between those of H02 and M02/M05. This compromise minimizes computational expense and facilitates understanding by excluding nonessential processes.

1) Charge and its separation in ice–ice collisions

Charge on hydrometeors is represented with a “space charge mixing ratio,” ρq,x, for each xth microphysical species. It is a “bulk” quantity (i.e., for all sizes) transferred between species by microphysical conversions. Charge density in air due to ions/charged aerosols, ρq,a, is assumed to have a source from evaporation of charged drops (from prior melting of charged ice) or sublimation of charged ice, but not from diffusional growth (e.g., Barthe et al. 2005). In the laboratory, during evaporation of any charged drop, only when it has completely disappeared is charge seen to transfer to the air, and the same would be expected for ice. Hence in AC, during any evaporation/sublimation of particle size distributions (PSDs), it is assumed that there are always some hydrometeors small enough to disappear totally. No recombination of charge in air is represented, since negative or positive ions/charged aerosols are not separately resolved. No sinks of ions on cloud particles are treated.

For ice–ice collisions, the emulated bin scheme involves temporary grids of bins discretizing size distributions (section 2a) and schemes for sticking (Phillips et al. 2015) and collision (Khain et al. 2001; Pinsky et al. 2001) efficiencies. “Bulk” charge separated in collisions is from summing contributions over permutations of bin pairs. Only ρq,x is then altered. Charge per particle, q(D) = βDγ, is assumed in any species (Beard and Ochs 1986; MacGorman and Rust 1998; Barthe et al. 2005, 2012); D is particle diameter, γ is prescribed with a fixed value, and β is evaluated numerically from ρq,x. The bulk charge is distributed among all particles of a temporary grid of bins, so that larger particles have more charge per particle.

There are two main groups of schemes for noninductive charge separation from laboratory studies:

  1. Takahashi (1978, 1984);

  2. Jayaratne et al. (1983), Keith and Saunders (1989), Saunders et al. (1991), Brooks et al. (1997), and Saunders and Peck (1998).

At weak LWCs (e.g., 0.1 g m−3) Takahashi observed positive charging of the rimer at most temperatures while Saunders et al. show negative charging. Reasons for such differences are uncertain. Experiments differ in design between the two groups, 1 and 2 (e.g., Saunders et al. 2006).

We opted for group 1. This allowed −CGs and a “normal tripole” structure to be simulated, as observed in the case (sections 1 and 4). A faster impact speed in group 1 (8 m s−1) approaches convective updraft speeds (10–15 m s−1) observed here (Tessendorf et al. 2007; Phillips et al. 2017b), similar to fall speeds of graupel/hail balanced in them. Real flash rates increase with updraft speed (Williams et al. 1985; Zipser and Lutz 1994; Boccippio et al. 2001). Hail below normal thunderstorms is seen to be mostly positively charged (Kuettner 1950; Rust and Moore 1974; Magono 1977; Wahlin 1986), consistent with positive charging of graupel/hail simulated by AC (section 5b). Laboratory observations by Pereyra et al. (2000), Berdeklis and List (2001) and Takahashi and Miyawaki (2002) at weak LWCs agreed better with group 1 than group 2.

Takahashi (1978) observed the average charge, QTaka, separated per collision between a rimed rod (3 mm, representing graupel) and crystals (100 μm) from 0° to −30°C. Charging was seen to depend on T (°C) and LWC, with positive charging of the rimer for T > −10°C but only for low or very high LWCs otherwise. Charge separated per ice–ice collision, whether or not rebounding, was extrapolated with a dimensionless parameter, α:
δQ=αMIN[QTaka(T,LWC*),QTaka(T,LWC)]×[(1ξ)+ξψ]0°>T>30°C,
where QTaka is the function plotted by Takahashi (1978, Fig. 8 therein), fitting his own data. Here ξ(T > –20°C) = 0 and ξ(T < –25°C) = 1, while ψ(LWC < 0.01 g m−3) = 0 and ψ(LWC > 0.05 g m−3) = 1. Both are linearly interpolated in between. Here the actual LWC in Takahashi’s original formula has been replaced with LWC* in Eq. (1) when doing so decreases QTaka. At T > −20°C, LWC*=LWC. Takahashi counted all collisions irrespective of whether they rebound, so Eq. (1) applies to all collisions too.

For T < −24°C and LWC < 0.2 g m−3, practically no observations were made by Takahashi (1978), who extrapolated QTaka into this unobserved region assuming positive charging of the rimer. Negative charging was observed for LWC ≥ 0.2 g m−3 at T < −24°C. By videosonde in cold-based (0°C) clouds (tops near −25°C), graupel was seen to be charged negatively below and positively above the −11°C level (Takahashi et al. 2017). Peak LWC was 0.4 g m−3 so this reversal (−11°C) was warmer than for Takahashi’s (1978) lab data (see also Pereyra et al. 2008).

Consequently, at weak LWCs if T < −20°C the unobserved charging of the rimer is assumed to be negative with values from the adjacent observed region at 0.2–0.5 g m−3 (section 5b):
LWC*[g m3]={LWC×[1ξ(T)]+0.5ξ(T)0.01<LWC<0.5gm3andT<20°CLWC,otherwise.
Equation (2) improves a simulation, not shown here, of an inverted storm with +CGs observed by Wiens et al. (2005). With more prolific negative charging of the rimer at weaker LWCs, the central positive charge of the inverted storm from fallout of graupel/hail is strengthened, favoring +CGs.
Takahashi (1984) proposed that charge separated is proportional to the difference in fall speeds and surface area of the crystal. Thus, diameter (Di) and fall speeds govern α:
α=MIN[3Ξ(Di,*D0)2|VpVi|8,100],
Di,*=MIN(Di,0.3Dp).
Unrimed (cloud ice/snow) and rimed (graupel/hail or riming snow) particles are denoted by subscripts “i” and “p.” Also D0 = 100 μm. The contact area cannot be wider than some a fraction of the rimer, hence Di,*. Inspection of laboratory data by Takahashi (1987, his Fig. A3) implies it is 0.3.

As Takahashi (1978) observed collisions of only cloud-ice crystals (0.1 mm) with a rimer, our inclusion of a new factor, Ξ, in Eq. (3) treats other types of collision too. For collisions of graupel with cloud-ice crystals, Ξ = 1. For graupel–snow collisions, the charge transferred is assumed proportional to the bulk density of snow, which determines the total area of many microscopic solid contacts, not counting air spaces, during impact. Indeed, the charge transferred by collision of a 1-mm frost particle with a large rimed target (Takahashi 1987, his Fig. A3) is seen to be lower by a factor of 25 than expected by areal extrapolation from 0.1 mm (Takahashi 1978). This factor approximates the ratio of bulk densities between a 1-mm snow particle and a 0.1-mm crystal in AC (800:40) (see also Heymsfield et al. 2002, their Fig. B1). Hence we assign Ξ = ρs/800 for graupel–snow collisions and Ξ = ρs/600 for snow–crystal collisions. Here ρs is the bulk density of snow (kg m−3) while 800 and 600 kg m−3 are the estimated bulk densities of cloud-ice crystals (0.1 mm) and rime density of the target (Williams and Zhang 1996) respectively, in the laboratory experiment of Takahashi (1978).

At T < −30°C, then δQ is multiplied by fTaka(T), which is zero when colder than −40°C and unity at −30°C, being interpolated in between (fTaka = 1 − [(T + 30)/10]2 for −30° > T > −40°C). Also, QTaka(T < –30°C, LWC) = QTaka(–30°C, LWC). From Eq. (1), ±δQ is added to ρq,x per collision among different species unless the sticking efficiency is unity (Phillips et al. 2015). When the entire rimer is covered in liquid (Phillips et al. 2014) or when LWC < 0.01 g m−3, then δQ = 0 is assumed.

Equations (2) and (4) are unique here. The original factor of (Di/D0)2 for α in (3) must have somehow represented the ratio of areas of contact between the actual and observed (D0 = 100 μm) collisions. Charge separation is an interfacial phenomenon. Takahashi (1984) thresholded α to be < 10 since Marshall et al. (1978) attributed saturation of charging to a limitation on contact area. We relax the threshold to 100 as it was not directly observed and a similar limitation is present in (4). The physically plausible dependency on contact area, and similarity of morphologies of snow and crystals on the microscopic scale, suggests the validity of extrapolating beyond laboratory conditions to any crystal size and perhaps to graupel–snow collisions. Yet charging in graupel–snow collisions was never studied by Takahashi (1978). Conceivably, a slightly different dependence of charging on contact area for snow than crystals may exist in reality (Takahashi 1987). Two key morphological differences between “cloud-ice” crystals (<0.3 mm) and snow (>0.3 mm in AC) exist: bulk density drastically decreases for snow aggregates as size increases and the presence of multiple monomers per snowflake boosts the sticking efficiency (Phillips et al. 2015).

In summary, Eqs. (1)(4) are applied for charging in collisions between graupel/hail and crystals, graupel/hail and snow, and riming snow and crystals. Only charging in graupel–crystal collisions was observed in the laboratory by Takahashi, however.

2) Electric field

AC uses two domains, a “dynamics domain” for prognostic variables inside an extended finer “potential domain” for electrical quantities. The potential domain (120 km × 80 km × 30 km; a 3D cubic grid of 0.5-km resolution) is much wider (by 50%) and higher than the dynamics domain for open lateral (eastern and western) and upper boundaries, following M05. Northern and southern lateral boundaries coincide and are periodic for both domains. The electric field, E = −∇ϕ, is calculated on the potential domain with potential, ϕ, from solving the Poisson equation for net space charge density, ρq (Adams 1989). For the potential domain, horizontal components of electric field are zero on lateral open boundaries (M05, M10) far from the dynamics domain, while upper and lower boundaries are prescribed with the background potential and zero volts, respectively. In clear-sky conditions the fair-weather electric field is reproduced (e.g., about −50 and −5 V m−1 at 1.6 and 10 km MSL; e.g., PK97). The upper boundary (30 km above ground) is so high that fixing its potential has little influence on the storm (cloud tops about 14 km above ground).

Only after each flash and its partial neutralization of charge is ϕ evaluated. This is less expensive than frequent updates of electric field in more explicit models (e.g., M02; Fierro et al. 2006). Though electric fields may influence coagulation (reviewed by PK97), such effects are neglected here.

3) Lightning

Lightning is simulated partly following MacGorman et al. (2001) and Barthe et al. (2005) with some modifications. The discharge is triggered where E = |E| reaches a threshold (Marshall et al. 1995; Riousset et al. 2007; Krehbiel et al. 2008) in V m−1 of
Einit=1.8×105ρa(z).
The plasma channel is modeled as two leaders with opposite polarities of charge propagating from the trigger in opposite directions. The positive and negative leaders propagate toward negative and positive ambient charge, respectively, if preflash fields exceed a fraction, fprop = 5%, of Einit; Winn et al. (1978) observed 15 kV m−1 fields below a thunderstorm. Propagation stops if the channel doubles back. Each leader is traced exactly parallel or antiparallel to the preflash electric field vector irrespective of gridpoint locations. The same lateral boundary conditions are applied to leaders and their branches as for other predicted quantities. Any leader crossing a periodic boundary simply reenters on the other side.

A −CG (+CG) occurs if a leader goes below 1.5 km (3 km) above ground. This threshold is from observations by a lightning positioning system at Guangzhou (Lyu et al. 2014, 2016; Fan et al. 2018) and in STEPS (Wiens et al. 2005). Also the potential of the trigger point, ϕ0, must satisfy ϕ0 × ρch ≥ 0 where ρch is charge density in the leader approaching ground (positive for +CGs, negative for −CGs) and |ϕ0| > 20 M V (Tan et al. 2014, their Fig. 3). If both criteria are met, the leader is sent vertically to ground.

Branches are treated statistically, without tracing channels, when |ρq| > ρcrit = 0.2 nC kg−1 and ambient ϕ is lower (higher) than the positive (negative) leader’s ϕ0. A grid box must satisfy both conditions and be adjacent to one satisfying them so as to be added to the branch cluster of a leader. The maximum number of branching grid boxes is (Barthe et al. 2005)
N=(rLχ)χ.
In this fractal law N is the number of junction points of branches > 0.5 km in the sphere of radius r from the preflash trigger point while Lχ is a length scale. Figure 1 conveys the geometry of fractal branching schematically. The number of branch junction points of a polarity in the jth hemispherical shell (δr) of radius r from the preflash trigger point is
δNδr2dNdr=χ2(1Lχ)χrχ1δL¯=χ2(δL¯Lχ)χjχ1,
Ngrid(j)=κ×δN,
where Ngrid is the maximum number of grid boxes with branches in the shell; κ is the number of grid boxes (0.5 km) per junction point of branches crossing them; δr=δL¯, where δL¯ is the diagonal gridbox width and r=jδL¯. Equations (7) and (8) are unique. Appendix C implies κ7(Lχ/δL¯). By cycling over all j from the trigger, branched grid boxes of the leader are amassed.
Fig. 1.
Fig. 1.

Schematic diagram of the flashes branching algorithm as applied to a single flash. Two leaders propagate upward and downward from the trigger point. The branching volume around each leader is divided into many concentric hemispheric shells. Each hemispheric shell depicted here has a certain number of branches according to the number of its junction points.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-19-0200.1

For constants in (7), three composite images of lightning were taken from the Tall-Object Lightning Observatory in Guangzhou (TOLOG) in China with a high-speed video camera (Fig. 2). These were three downward −CGs both in and out of cloud, with upward leaders from tall structures. Junction points were counted for branches >0.3 km in projected length (plane normal to view), corresponding on average to branches > 0.5 km in 3D. Trigger points aloft were assumed near −10°C as typically observed there (Zheng et al. 2019). The mean for all three photos implies Lχ ≈ 1.4 ± 0.2 km (90% confidence interval, t statistics).

Fig. 2.
Fig. 2.

Composite visible images of flashes to ground taken with a high-speed video camera from Guangzhou City in southern China, from (a) 22 Jul and (b) 7 and (c) 24 Sep 2012. The resolution is 3 and 4.7 m per pixel and the distance to the striking point is 2.1 and 3.3 km, for (a)/(b) and (c), respectively. These were all downward CGs, involving upward leaders from tall structures.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-19-0200.1

4) Neutralization of ambient charge by lightning

Ambient charge, on hydrometeors and air, is neutralized in each grid box of the flash as follows. The charge in the flash is ρch = ζ|ρqρcrit| where ζ = −ρq/|ρq| is the polarity of plasma with opposite sign to the net ambient space charge density, ρq=ρq,a+xρq,x. Ambient space charge densities in air and in each xth species, ρq,a and ρq,x, are incremented by δρq,a = ζMIN(|ρch|, |ρq,a|) and δρq,x = ζMIN(|ρchδρq,a|, |ρq,x|χx), while χx is its fractional contribution to total hydrometeor surface area. Neutralization is incomplete in nature (Williams et al. 1985), as treated by ρcrit. For ICs, δρq,a and δρq,x are normalized to have a total charge of zero over the flash, before altering ρq,a and ρq,x. The normalization is not done for CGs as net charge in the flash after neutralization flows to ground.

3. Description of observed case and model setup

STEPS in summer 2000 observed convective storms by aircraft, balloons, radar, ground-based measurements, and satellite (Lang et al. 2004). The storm on 19/20 June had high cold cloud bases near 0°C (4.4 km MSL) at 3 km above ground (1.3 km MSL). The case is representative of continental multicell storms in U.S. CGP (section 1) where cloud bases are usually colder than further south. It was a multicell system of convection 50–100 km wide.

The storm began over Colorado at about 2200 UTC (1600 local time) and moved almost eastward (about 70° from north). Graupel (<0.5 cm) and snow were ubiquitous in aircraft observations. During flights, small hail (>0.5 cm) was detected, especially on flanks of convective updrafts (Goehring 2005; Phillips et al. 2017b). Convective cells (reflectivities up to 55 dBZ aloft) coexisted with a lightly precipitating stratiform cloud deck (about 20 dBZ).

The multicellular storm is simulated in a 3D domain (80 km × 80 km × 16 km), approximating it as a convective line with cells initiated at x = 30 km. Translation of the domain keeps them in it. The horizontal x axis points 70° from north. Phillips et al. (2017b) elaborate further.

4. Results from model validation

Regarding nonelectrical quantities, Phillips et al. (2017b, their Figs. 5 and 6) showed agreement between the AC simulation of the STEPS case (1145 UTC 19 June–0215 UTC 20 June 2000; section 3) and coincident observations by aircraft, satellite, and ground-based instruments for many quantities. Vertical profiles of mean diameter and concentration of droplets, LWC, radar reflectivity, filtered ice concentration, and PSDs were among quantities predicted accurately. Predicted and observed concentrations of ice particles (>0.2 mm) identically averaged in convective updrafts were on the order of 10 L−1 at flight levels. Differences between prediction and observations were less than the spread of observations.

With electrification represented, Fig. 3 compares predicted and observed flash rates (Tessendorf et al. 2007) and ascent statistics at flight levels of 6–7 km MSL. The flash rate is simulated with errors mostly less than a factor of 2. Aircraft data of ascent are more accurate than the radar data and agree with the predicted ascent at all cumulative frequencies of vertical velocity. Dual-Doppler ascent retrievals have wide biases (Dahl et al. 2019).

Fig. 3.
Fig. 3.

(a) The total flash rate in the simulated domain, predicted (closed circles) and observed (thin line) for the STEPS case (2345 UTC 19 Jun–0215 UTC 20 Jun 2000). Lightning data observed by LMA were computed on a moving grid that followed the model domain (Phillips et al. 2017b). Also shown is (b) the cumulative distribution of vertical velocity between 5 and 6.5 km MSL in fast convective updrafts >5 m s−1, comparing the prediction (thick line) with observations by aircraft and Doppler radar (lines with symbols). Transient fluctuations of up to about 3 m s−1 arose from flight maneuvers, so the relative error in aircraft data of ascent is assumed to be ±30%.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-19-0200.1

Figure 4 depicts numbers of all flashes. Most (>99%) are IC, the rest (1%) −CG. Model and observations agree, differing by about 10%. A few (4%) observed CGs were apparently +CG as predicted, though misclassification of ICs as +CGs is possible (Cummins et al. 1998; Leal et al. 2019). The IC/CG ratio of 102 is large (Lang et al. 2000; Williams 2001; Boccippio et al. 2001). A high cloud base (3 km above ground) from a dry lower troposphere inhibited charged surface precipitation and hence CGs, for reasons noted below.

Fig. 4.
Fig. 4.

Observed and simulated (AC) total numbers of flashes for the three types of lightning in the STEPS case (2345–0215 UTC), namely, intracloud (IC; black), negative cloud-to-ground (−CG; gray), and positive cloud-to-ground (+CG; white) lightning. For observed flashes, only those in the 3D domain simulated (Phillips et al. 2017b, their Fig. 1a) are counted.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-19-0200.1

Figure 5 shows −CGs and estimated altitudes of trigger points. Observations and predictions agree in timing and frequency. Most −CGs were initiated in convective cloud near 6–7 km MSL (−10° to −17°C), in the lower half of the central negatively charged region (5.5–9 km MSL) of the large-scale tripole. Heights of triggering for all flashes are predicted adequately, but with a peak 2 km too high (Fig. 5b). Observed and predicted ICs were mostly initiated at 6–9 MSL altitude (about −10° to −30°C) with a peak near 8 km MSL (about −25°C). They were predicted to arise often from intense charge of transient graupel/hail fall shafts in convective cells, at levels in the midst of the central negative region of the large-scale tripole. A few flashes (10%) were observed to trigger at 3–5 km MSL, especially ICs at 4.4–5 km MSL (0° to −4°C; 0110 to 0200 UTC), levels where none are predicted (section 6).

Fig. 5.
Fig. 5.

Negative cloud-to-ground flashes for the STEPS case (2345–0215 UTC), observed (open symbols) and predicted by AC (closed symbols). (a) Their time evolution in terms of numbers of −CGs every minute (black and blue lines). (b) The vertical profile of relative frequencies of their estimated trigger-point altitude. Also shown in (b) is the corresponding vertical profile for all flashes, the vast majority of which are IC. Finally, superimposed in (a) is the time evolution of accumulated absolute magnitude of charge at the ground (red lines) from precipitation (positive) and CGs (negative).

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-19-0200.1

Timing of −CGs is explicable in terms of surface precipitation (Fig. 5a). Predicted accumulation of charge in surface precipitation from similarly charged graupel causes that of opposite charge to ground from CGs (section 5b), as with all simulations in Part II (not shown), leading it by 10–20 mins, because removal of charge in precipitation creates the total storm charge aloft. In normal (inverted) storms, AC predicts net transfer of positive (negative) charge to the surface in precipitation and then −CGs (+CGs) as a lagged response in all simulations. As most −CGs or +CGs originate from midlevels and conduct negative or positive charge toward the higher or lower potential of the ground (always zero volts), respectively, they respond to the total of all net charge and average potential of the entire convective core caused by fallout of oppositely charged precipitation to ground. This is why arrival of surface precipitation in the second half of the simulation (Phillips et al. 2017b, their Fig. 5) coincides with −CGs. Generally, onsets of CGs and surface precipitation are observed to coincide to within a few mins (Gungle and Krider 2006) with CG number controlled by rainfall volume (Battan 1965; Kinzer 1974; Piepgrass et al. 1982). Here, to summarize, positive charge of surface precipitation causes negative average charge on the simulated normal storm and −CGs as a response. Corona discharge may complicate this picture (section 6). Equally, charged precipitation shafts below cloud promote propagation of CGs downward.

Number density of LMA sources (VHF) was observed in STEPS. They have no polarity, but the negative end of a flash produces more sources (e.g., Dwyer et al. 2004, 2005) than the other end, so polarity of ambient volume can be inferred. For comparison, this source density is diagnosed from AC output by assuming proportionality with charge neutralized in each grid box, a novel method. The constant of proportionality gives observed numbers (~103) of sources per flash for storms generally (Wiens et al. 2005). The constant (200 or 50 nC−1) is 4 times higher for “positive LMA” sources (negative breakdown through ambient positive charge) than negative sources (vice versa), as seen in storms by Wiens et al. (2005), Rison et al. (1999) and the case here.

Figure 6 shows agreement between predicted and observed profiles of total, positive and negative LMA sources for cloudy levels (4–13 km MSL). There are deep broad maxima for all and positive sources near 25 and 23 dB, respectively, over 5–10 km MSL (about −5° to −40°C) and a narrower peak of negative sources of about 20 dB. Predicted negative LMA sources are about 1 km too high, consistent with ICs being triggered at levels too high also.

Fig. 6.
Fig. 6.

The density of (a) all LMA sources expressed in decibels (dB; 10log10[number of average sources per min per volume element (min−1)]) for the AC simulation (2345–0215 UTC) (thick full line) and STEPS observations from the region (80 km × 80 km) simulated (dashed line). In both observations and simulation, the 3D domain is divided into volume elements 0.5 km deep and 10 km wide in one horizontal direction, spanning the domain in the other, with averaging over all elements at each level (Wiens et al. 2005). The source profiles in dB for every minute are then averaged over the simulated period. Corresponding contributions from (b) positive (negative breakdown through ambient positive charge) and (c) negative (vice versa) LMA sources are compared similarly. Errors of the prediction arise from choice of average value of LMA sources per flash (about 1000), which may vary by about half an order of magnitude (e.g., Wiens et al. 2005). The error shown for the observed profile is the standard deviation for the variability over time. For simplicity, the observations were analyzed over a fixed domain of the same size as the model.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-19-0200.1

Figure 7 shows horizontal distance of predicted trigger points from maximum ascent (13 ± 4 m s−1) in the nearest convective core. Most (97%) are in the core. About 3% are in stratiform/cirriform cloud, being triggered “remotely,” 10–20 km from the core’s maximum ascent, with a few (0.02%) 18–20 km away. A compass plot shows simulated trigger points in the horizontal plane relative to the nearest core (Fig. 7b). Favored locations of remote trigger points (>10 km) are between adjacent cells where electric fields superpose. Triggering here slightly prefers sides of any cell that are upshear and to the left of the system’s propagation direction, because the environmental vertical shear is not unidirectional.

Fig. 7.
Fig. 7.

From the AC simulation are shown (a) relative frequency of the horizontal distance of the trigger points from the center of the nearest convective core for all flashes of the STEPS case (2345–0215 UTC) and (b) their relative positions compared to this composite core in terms of flash density (logarithm of number per km2) in the horizontal plane. The vast majority are IC. In (b), the origin of the compass plot is the axis of the strong convective updraft nearest to each trigger and the blue arrow delineates the direction of storm propagation. Angles in (b) are directions defined clockwise from north while distances are in km.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-19-0200.1

Most charging of graupel occurs in convective ascent (section 5b). Yet broad continua of sizes and fall speeds of graupel in any cloudy volume create vertical charge structure in the layer cloud from outflow, causing a continuous distribution of horizontal distances of triggering from the core (by up to 20 km away here). The prediction (Fig. 7) accords with observations of most lightning being triggered in cores and with a few flashes triggered tens of kilometers away (Proctor 1991; Wiens et al. 2005; Dye and Willett 2007; Weiss et al. 2012; Kuhlman et al. 2009). Supercooled cloud liquid must be present for any charge separation to occur in AC.

In the STEPS case, electrical properties of the storm were measured by balloon. Figure 8 shows the balloon trajectory, initially 20 km downshear of convection (<50 dBZ) in weak reflectivity (about 0 dBZ). Struck by lightning at 9 km MSL, all other flashes around it were triggered at least 5 km away (Fig. 8). The balloon rose through layer cloud toward the downshear anvil of approaching cells, missing them (Goehring 2005, Fig. 26 therein).

Fig. 8.
Fig. 8.

Plan view of balloon trajectory (large black plus symbols) superimposed on a 0.48° elevation angle scan of equivalent reflectivity factor (dBZ) near the ground from the Goodland radar at the moment of release (0059 UTC 20 Jun 2000). The balloon is almost 20 km to the east of a reflectivity maximum of about 40 dBZ when launched and then drifts several kilometers northeastward. Also shown are the IC flash trigger points (magenta tiny plus symbols) during ascent of the balloon until it was struck by lightning at 9 km MSL altitude. The balloon was launched by the National Severe Storms Laboratory (NSSL) from Goodland airport.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-19-0200.1

Figure 9 compares quantities measured by balloon and simulated for an ensemble (30) of virtual balloon trajectories. The AC simulation is idealized and cannot reproduce positions of real clouds. From AC output, each trajectory was traced, ascending 7 m s−1 faster than the evolving wind to match observed altitudes. Trajectories were initiated randomly from a square (10 km) about 10 km east of a cell with a similar reflectivity to that near the real balloon. The predicted mean electric field (Fig. 9a) agrees with the observations. The maximum measured was 30 kV m−1 (8 km MSL), much weaker than the breakdown threshold (90 kV m−1). The balloon sampled large-scale conditions ahead of the convective line.

Fig. 9.
Fig. 9.

(a) The vertical component of electric field (Ez) both observed (black open symbols) and predicted by AC (red closed symbols). Measurements were from an electric field meter on the balloon plotted in Fig. 8. The full line of the model is the mean of an ensemble of many possible simulated trajectories of the balloon. Positive values indicate an upward electric field. (b) A vertical profile of the inferred net charge density from the same balloon observations (black open symbols) compared with the average for the ensemble of simulated trajectories (red closed symbols). Both model and observations depict the large-scale normal tripole of charge structure. Errors plotted in (a) and (b) are standard deviations (thin dotted lines and error bars), to depict the spatial variability. Observational points (1 s) of electric field were binned in layers, with the standard deviation shown for each bin in (a) and determining that of inferred charge density for error bars in (b).

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-19-0200.1

Total space charge density, ρq, is from the balloon sounding using dEz/dzρq/ϵ, where Ez is the vertical component of E. Figure 9b compares it with AC’s ensemble. A normal tripolar charge structure (e.g., Williams 1989) is predicted and observed:

  • weak positive charge below the −10°C (6 km MSL) level;

  • strong negative charge at 6.5–8 km MSL (about −15° to −25°C);

  • moderate positive charge at 8–10 km MSL (about −25° to −38°C) aloft.

Normal tripoles are associated with predominance of −CGs among strikes to ground, as in the simulation. Most−CGs were triggered at lower levels of this central negative region, where negatively charged snow remains as graupel/hail falls out. The normal tripole is here predicted for reasons (central negative charge due to snow/crystals, rather than graupel) differing from the traditional explanation of normal tripoles (Williams 1989) (section 5a).

Figure 10 compares predicted charge to ground in −CGs with that inferred from peak currents (Rakov and Uman 2003; Schoene et al. 2010) observed in STEPS. They agree in terms of both the median charge per flash and the statistical distribution among all flashes to ground.

Fig. 10.
Fig. 10.

Observed (“NLDN”) and predicted (“AC”) amounts of negative charge in any flash to ground among all −CGs in the simulated region of STEPS. The logarithm of the absolute magnitude of the charge transferred to ground is shown. The −CG flashes are the same set shown in Fig. 4. Charge was inferred from NLDN observations of peak current by assuming proportionality to the square of the peak current in view of empirical relations for rocket-induced artificial lightning by Schoene et al. (2010, their Fig. 4). The constant of proportionality was constrained by general known characteristics of −CGs from Rakov and Uman (2003, their Table 1.1) showing about 30 C (30% error) typically transferred for a peak current of 30 kA. The error in observed charge is about half an order of magnitude, partly due to variability of the exponent (2) among the empirical relations.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-19-0200.1

5. Results for other electrical quantities of STEPS case

a. Spatial distribution of predicted charge and electric field

Figure 11 shows space charge density averaged over the domain. Features seen in the balloon sounding (Fig. 9) are evident. The most intense charges are on graupel (positive) and snow (negative). The net charge in the storm is from differential sedimentation of graupel versus snow/crystals. Most charge on graupel is from rebounding collisions with (cloud-ice) crystals, the process observed by Takahashi (1978) (section 2b). Once charged, many crystals grow to become charged snow. Fluctuations of LWC below average are the cause of graupel mostly charging positively (section 5b), as noted below.

Fig. 11.
Fig. 11.

Unconditional averages from the STEPS simulation by AC over the entire domain of total space charge density (thick full line) and its components from cloud ice, snow, and graupel (thin lines with symbols). Charge densities in air and on rain are shown (thin dotted and dot–dashed lines). The net charge on cloud liquid is negligible.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-19-0200.1

The normal tripole in Fig. 11 is explicable as follows. First, at midlevels below 10 km MSL (about −38°C), the averaging includes regions of negative and positive charge on graupel in extreme (rich LWC) and moderate/weak (low LWC) convective ascent, respectively. Weaker ascent is wider and prevails, so graupel charges mostly positively. Above 8–9 km MSL (near −30°C) the polarity of both oppositely charged ice species dominating charging (graupel and cloud-ice crystals) in Fig. 11 reverses, as in the laboratory data of charging (Takahashi 1978, his Fig. 8). In the laboratory the rimer was seen to charge negatively at LWCs > 0.1 g m−3 if T < −30°C, but always positively if T > −10°C (section 5b). Graupel in the fastest ascent (LWC ~ 1 g m−3) becomes negatively charged when T < −15°C, being upwelled into cirriform outflow aloft. Conversely, graupel in moderate/weak convective ascent charges positively and likely remains at lower levels, detraining into stratiform cloud.

Net charge in narrow graupel/hail shafts (6–8 km MSL; −10° to −25°C) is predicted to be mostly positive. This agrees with LMA observations of net charge being positive at midlevels (6–8 km MSL) in a graupel/hail shaft (45 dBZ) next to an updraft (>5 m s−1) from Tessendorf et al. (2007, their Figs. 12c,d, their x = −68 km) in this STEPS case at 0019 UTC. In stratiform outflow (30–40 dBZ) from this updraft (0019 UTC), observed LMA sources are negative at midlevels. Positivity of graupel/hail shafts is observed to coexist with average negativity on the large scale (>5 km), as in the simulation.

In Fig. 11, at upper levels predicted net charge (positive above 9 km MSL) is caused by cloud-ice crystals, which are spread by their slow fall speeds coupled with large storm-relative flow over wide areas. Positivity of cloud ice and negativity of graupel are due to charge reversal in the laboratory data, as noted above. At lower levels (5 km MSL), weak positive net charge is from positively charged graupel falling out.

Comparing Figs. 5, 6, and 11 reveals that −CGs, simulated and observed, originate from intense electric fields between the strong central negative and low-level positive centers of the large-scale tripole. ICs originate from most subzero levels, but especially in the upper half (−20° to −30°C) of the central negative region of the large-scale tripole. ICs are often triggered near transient narrow shafts of intensely charged graupel/hail (see snapshots below).

Figures 12 and 13 show snapshots of electrical and microphysical quantities in vertical sections where lightning was triggered in a cell at 0055 UTC. Peak updraft speed is 15 m s−1 with LWC < 2 g m−3. Supercooled liquid is confined to the convective updraft (5 km wide) below the −20°C level, while cirriform and stratiform cloud is ice-only (Figs. 13a,b) due to weakness of ascent (Korolev 2007). Both charging and graupel production (Figs. 12a,d and 13b,e,f) coincide in the cell. Graupel forms near updraft edges from riming of snow (Figs. 13d,e). Collisions between graupel and secondary cloud-ice crystals cause the charging.

Fig. 12.
Fig. 12.

Snapshots at 0055 UTC from the control run by AC of the STEPS case showing: densities of charge (nC m−3, shading) in (a) total, and on (b) cloud-ice particles, (c) snow, and (d) graupel; (e) electric field strength (kV m−1, shading) and electric field vectors (arrows); and (f) charge acquired by cloud from flashes after neutralization (nC m−3, shading). For each quantity, a vertical slice at y = 29 km through the trigger points of three flashes triggered near 8 and 11 km MSL is shown. On all snapshots are marked the bidirectional leaders (magenta lines) traced from the trigger points (magenta filled stars) for the flashes at 0055 UTC. Vertical velocity (w) of convective ascent (thick black dashed contours, spaced by 5 m s−1) and isotherms (thin cyan contours, spaced by 10 K) are superimposed on each.

Citation: Journal of the Atmospheric Sciences 77, 12; 10.1175/JAS-D-19-0200.1

Fig. 13.