A stochastic lightning-flash scheme for 3D explicitly resolving cloud ...

QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY Q. J. R. Meteorol. Soc. 135: 113–124 (2009) Published online 21 January 2009 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/qj.364

A stochastic lightning-flash scheme for 3D explicitly resolving cloud models G. Molini´ea , J. Escobarb and D. Gazenb a

Laboratoire de Transfert en Hydrologie et Environnement, Universit´e de Grenoble (CNRS, INPG, IRD, UJF), France b Laboratoire d’A´erologie, UMR 5560, Toulouse, France

ABSTRACT: The paper presents a stochastic lightning-flash scheme designed for mesoscale cloud-resolving models (MCRMs). The lightning-flash scheme is implemented on-line in an MCRM. It is fully parallellized and vectorized. A lightning flash is schematized as two single conducting channels (single tracks) propagating in opposite directions from the lightning ignition point, and of branch patterns propagating from the single tracks. On the base of scale similarities between lightning flashes and discharges in dielectrics at centimetre scales, a stochastic scheme has been designed to compute branch trajectories. A fractal relationship is used to limit the branch number. Charge neutralization operates along the single tracks and branch trajectories to threshold the cloud charge density. The scheme has been implemented in the French meteorological community model M´esoNH. Two kinds of tests were designed to assess the scheme’s capabilities. A first set consists of single-lightning simulations, which demonstrate that, thanks to branches, the simulated lightning flashes are (i) able to reach sparse electric charges and (ii) are fractal objects. The second set consists of comprehensive 3D-thundercloud life-cycle simulations. A simple non-inductive charging process is activated in order to assess the sensitivity of thundercloud electrical behaviour to lightning patterns. It is shown that, paradoxically, lightning flashes with quasi-plane branch propagations (i.e. fractal dimension close to 2) lead to more steady c 2009 Royal electrical behaviour than those completely filling volumes (i.e. fractal dimension close to 3). Copyright
Meteorological Society KEY WORDS

atmospheric electricity; fractal lightning

Received 14 March 2008; Revised 14 November 2008; Accepted 26 November 2008

1.

Introduction

Electrical activity in storms results from a competition between electrical charge generation and neutralization processes. Lightning flashes are among the most efficient mechanisms in rearranging thundercloud electrical charges (neutralization and/or deposition) (Ziegler and MacGorman, 1994; MacGorman and Rust, 1998, pp. 53–54; Marshall and Stolzenburg, 2001, 2002; Coleman et al., 2003; Jacobson and Heavner, 2005; Riousset et al., 2007). We are interested in thunderstorm mechanisms at the γ -mesoscale (∼100 km2 , the thundercloud scale). At this scale, lightning features branch. This is obvious to the naked eye and on lightning pictures, and over recent decades has been widely observed in the VHF–UHF frequency range, either using interferometry (Richard et al., 1986; Mazur, 1989; Rhodes et al., 1994; Shao and Krehbiel, 1996; Dotzek et al., 2001; Mardiana et al., 2002) or the time-of-arrival (TOA) technique (Rison et al., 1999; Thomas et al., 2000; Krehbiel et al., 2000). In the LF–VLF frequency range, the spatial dispersion of radiation sources (Betz et al., 2004, 2007) could be ∗ Correspondence to: G. Molini´e, LTHE, 1025, Rue de la Piscine, 38130 Grenoble, France. E-mail: [email protected]

c 2009 Royal Meteorological Society Copyright


partly due to lightning-flash branches. Lightning branches carry electrical charges and mainly propagate through cloud charged regions (Krehbiel et al., 2000; Shao and Krehbiel, 1996; Rison et al., 1999; Thomas et al., 2000; Coleman et al., 2003; Carey et al., 2005; Wiens et al., 2005). Thus, knowledge of lightning-flash patterns is essential in determining the thunderstorm electrical charge rearrangement by lightning flashes. It is also essential in determining other lightning-flash features such as lightning-emitted nitrogen oxides (Choi et al., 2005; DeCaria et al., 2000). Helsdon et al. (1992) and Solomon and Baker (1996) were among the first to run simulations of comprehensive thunderstorm life cycles including lightning-flash parametrizations. Their work constitutes the basis of the current study. Basically, the lightning-flash parametrization of Solomon and Baker (1996) (implemented in a 1.5D model) and that of Helsdon et al. (1992) (in a 2D model) simulated lightning flashes as bi-directional leader discharges, initiated at the location of the maximum electric field. From the initiation point, simulated lightning flashes were composed of two conducting channels propagating in opposite directions along electric field lines, as long as an electric field magnitude criterion is valid. The anisotropy in thundercloud charge neutralization due to branch patterns was not taken into account.

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MacGorman et al. (2001, hereafter McG01) attempted to simulate the gross effect of lightning-flash branches. Numbers of simulated lightning flashes are composed of two quasi-horizontal layers joined by a vertical bridge. (Such lightning-flash patterns have been observed by Mazur, 2002.) The two quasi-horizontal lightning layers simulate the gross branch patterns propagating into charge pockets even though the electric field is low. The bridge channel is driven by the electric field between two charge pockets. Then, the idea of branch simulation in mesoscale cloud-resolving models (MCRMs) was raised. The physical mechanisms leading to the branch tortuosity have been revealed by studies of the microscopic processes involved in lightning flashes. In an extreme simplification (Mazur, 2002; Rakov and Uman, 2003, provide more details), lightning flashes consist of several steps during which centimetre electric discharges amplify in macroscopic discharges. At the centimetre scale, the sketch of the discharge process is illustrated in Pietronero and Wiesmann (1988). An ionisation wave (streamer) creates a plasma in which an electron avalanche (leader) propagates, drawing a luminous filament (the streamer–leader system). Step-by-step, these two processes build an ionized channel of several hundred metres in length. A lightning flash consists of a positive and negative streamer–leader system propagating in opposite directions. Depending on their polarities, streamer–leader systems propagate within one to a few tens of milliseconds. Following a streamer–leader system, several intense electrical discharges (return strokes in the case of cloud-to-ground lightning flashes or a recoil leader in the case of intra-cloud ones) borrow the streamer–leader channel. The total lightningflash duration can reach several hundreds of milliseconds. The lightning-flash tortuosity is thus due to the streamer–leader propagation. The behaviour of the positive and negative streamer–leader propagation is obviously different (Williams, 2006). However for both systems, the electron avalanche propagation velocity is proportional to a power of the ratio of the local to the surrounding electric field magnitudes. A competitive branch growth takes place because the fastest growing branches shield the electric field in their surroundings (the Faraday screening effect) preventing the propagation of the closest branches. Thus, the growth probability of a branch in a given direction is related to a power of the local electric field magnitude (Pietronero and Wiesmann, 1988). At the centimetre scale, this kind of growing phenomenon is known to lead to fractal objects, as shown in an explicit dielectric breakdown model (DBM; Niemeyer et al., 1984; Wiesmann and Zeller, 1986; Barclay et al., 1990. Lightning-flash structure properties have also been deduced from experimental dielectric discharges. Williams et al. (1985) studied electric discharge propagations in a dielectric slab of some centimetres in size. Assuming scale similarities, they concluded that these results could be extrapolated to lightning flashes. On the basis of scale similarities, Mansell et al. (2002) and Riousset et al. (2007) applied an adapted DBM c 2009 Royal Meteorological Society Copyright


at the mesoscale (Niemeyer et al., 1984; Wiesmann and Zeller, 1986). Running their stochastic lightning model (SLM), Mansell et al. (2002) get very realistic branch simulations. The SLM consists of propagating lightning leaders with multiple branches. A given branch propagates stochastically. The branching probability is a function of the electric field at branch tips. The approach of Riousset et al. (2007) is similar, except that the authors pay particular attention to solving the electrical potential in the simulation domain. The realism of simulated branches and the low number of the scheme parameters (two electric field thresholds) are the main advantages of the SLM. Tan et al. (2006) implemented a 2D version of the SLM of Mansell et al. (2002). They show that solving the explicit equations of electrical breakdowns (as those solved in the SLM) at scales of few hundred metres does not provide realistic lightning branch patterns. Therefore, to be coupled to a MCRM, the SLM presents two major drawbacks. First, it necessitates a Cartesian discretization grid finer than the MCRM one. Usually MCRM discretization grids are curvilinear to fit terrain elevation (e.g. Laprise, 1992; Lafore et al., 1998). Second, the computation of an electric potential or an electric field necessitate solving the Poisson equation for the electrical potential. This has a high computational cost and complicates vectorization and parallelization of the scheme. In this paper, we propose a simple stochastic scheme to simulate lightning-flash patterns in an MCRM. The scheme is designed to be integrated into the M´esoNH MCRM (Lafore et al., 1998; http://mesonh.aero.obsmip.fr/mesonh/) and then it is fully vectorized and parallellized. Section 2 describes the lightning-flash scheme. The scheme capabilities are demonstrated in section 3 for 2D and 3D static configurations and for comprehensive thunderstorm life-cycle simulations. In section 4, the behaviour of the thundercloud electrical state is discussed as a function of the lightning morphology features (fractal dimensions). 2.

Conceptual scheme

2.1. The model The lightning model is designed to resolve the overall effect of lightning flashes on the cloud charge neutralization at the cloud scale. To explicitly simulate the microphysics and dynamics of the thundercloud, MCRMs must be run with spatial and temporal resolutions of a few hundred metres (typically 1 km in the horizontal and 250 m in the vertical) and few seconds, respectively. Therefore, the successive stages of a single lightning flash (described in the introduction) are not explicitly resolved. The electric field is computed at each time step (Molini´e et al., 2002, provides details on the electric field computation), before and after a lightning-flash generation but not during the successive lightning stages. Some parts of this model are directly inspired from the Helsdon et al. (1992) and McG01 schemes. Relative to these previous works, the lightning branch design Q. J. R. Meteorol. Soc. 135: 113–124 (2009) DOI: 10.1002/qj

A 3D STOCHASTIC LIGHTNING-FLASH MODEL

constitutes the main improvement. Lightning-flash patterns are made up of two kinds of objects: single tracks and branches. Driven by the electric field, one single track propagates in the electric field direction from the initiation point, and the other one propagates in the opposite direction. Branches are attached to the single tracks and propagate towards and inside high (in absolute values) charge-density regions. Once the lightning-flash pattern is established, the cloud charge neutralization is applied along single tracks and branches. The lightningflash paths are discretized on the 3D curvilinear MCRM grid. At each grid node, the lightning single tracks and branches are able to propagate toward the closest grid node including diagonals. In consequence, there are 7 available directions in the 2D geometry and 26 in 3D. 2.1.1. The initiation point Lightning flashes are initiated because the electric field overwhelms the dielectric strength of air around the initiation spot. Nevertheless, the maxima of the electric field magnitudes (measured in thunderclouds during various studies) are much less than the dielectric strength required to disrupt clear air (around 1.6 MV m−1 at 6 km height for example; Rakov and Uman, 2003, p. 84, provide a review). Therefore, several different processes such as runaway electrons (Gurevich et al., 1992) or hydrometeor distortions (Cocquillat and Chauzy, 1993) locally enhance the electric field to allow lightning-flash initiation (Petersen et al., 2008). The electric field in grid spacings of hundred of metres cannot reflect local enhancements required for the initial spark. Thus, following McG01, lightning flashes are initiated at grid points randomly selected from those in which the electric field is higher than 90% of a given threshold Etrig . Independently of local enhancements of the electric field, the pressure reduction with altitude reduces the dielectric strength of the air. Even though the dielectric strength reduction is not only due to the break-even process, electric field magnitudes in thunderclouds are grossly bounded by the break-even electric field calculated as Equation (1) in Marshall et al. (1995). Therefore, we parametrize the Etrig altitude dependence as
z  , (1) Etrig = Etrig0 exp − 8.4 where z is the altitude in kilometres and Etrig0 ∼ 200 kV m−1 , the electric field threshold at ground level. 2.1.2. The single tracks The aim of single tracks is to connect two regions of opposite electric charge density in which lightning branches can propagate. It consists of segments, the positive ones propagating along the electric field lines and the negative ones in the opposite direction. The propagation of each single track stops independently of the other when the local absolute value of the ambient electric field at the single-track tip falls below a given threshold Eprop . Then, the lightning-flash branch computation is activated. c 2009 Royal Meteorological Society Copyright


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2.1.3. The branches Our intent is to determine the physical parameters at MCRM space- and time-scales which allow simulation of realistic branch patterns. Our intent is not to solve at such scales the explicit equations of electrical discharge propagation, as done in the DBM of Wiesmann and Zeller (1986) and Pietronero and Wiesmann (1988) and following the SLMs of Mansell et al. (2002) and Riousset et al. (2007). Lightning-flash branch simulations must satisfy different features. They derive from stochastic processes (Pietronero and Wiesmann, 1988; Wiesmann and Zeller, 1986), and they propagate from single tracks towards and into regions of high charge density (Coleman et al., 2003). The regions where branches are allowed to propagate are demarcated using a charge-density threshold (Qxc ≥ 0). All contiguous grid points (including at least one on the single track) where the electric charge-density absolute value, |Q|, is greater or equal than Qxc are potential candidates for a branch bond. In those demarcated regions (|Q| ≥ Qxc ), the model selects the discrete paths of lightning branches. It proceeds sequentially, each step adding new bonds to the branch pattern. An example is shown in Figure 1 in which, during step 1, index-1 bonds were added to the lightning-flash pattern and, during step 2, index-2 bonds were added, etc. At each step, potential new bonds correspond to lattice nodes neighbouring the existing lightning pattern. A branching likelihood is attributed to the potential new bonds as in the DBM of Wiesmann and Zeller (1986), except that the probability is a four-term product including: 1. The likelihood for an electrical discharge to propagate in a specific direction depends on the electric field magnitude in that direction. The fastest propagating discharge shields the electric field in its surroundings, slowing down and even preventing close branches developing (Pietronero and Wiesmann, 1988). Therefore, the first term of the product is the inverse of the number of the closest-neighbour grid points yielding a branch or being a potential new bond. 2. To favour the discharge expansion, the second term is the distance from the initiation point normalized by the maximum distance from the initiation point. 3. To draw propagating branches toward the highest charge-density regions, the third probability term is the absolute value of the electrical potential. This is in agreement with observations of Coleman et al. (2003) and Tan et al. (2006) that, at the cloud scale, highest branch numbers associate with the highest electrical potential values. In laboratory experiments, Williams et al. (1985) and Temnikov et al. (2005) show that electrical discharges similar to lightning flashes deeply penetrate into the high charge-density layer. 4. To favour branch propagation into highest chargedensity regions, the fourth term is the normalized ambient electric charge density. Q. J. R. Meteorol. Soc. 135: 113–124 (2009) DOI: 10.1002/qj

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Lmax step3 3 3

  Q(r) × Q(r) − Qxc if Q(r) > Qxc , Q(r) δQ(r) = 0 otherwise, (3)

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where Q(r) is the ambient electric charge density at bond r. If the lightning flash does not reach the ground, δQ(r) is adjusted to ensure the neutrality of the lightning flash. A fraction of δQ(r) neutralizes ions, and its complement removes solid and liquid particle charges proportionally to their equivalent efficient cross-section.

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grid location), the neutralized charge density δQ(r) is therefore:

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3.

Implementation of the stochastic lightning scheme

Figure 1. Schematic representation of a lightning single track (thick line segments) and branches (dashed line segments). The I sign indicates the lightning initiation point and the mesh is a rectangular model grid. Branch bond indices are enclosed in white circles and black circles identify index-4 potential new bonds. For legibility, the grid is a Cartesian one, a specific case of curvilinear grids.

As the stochastic lightning scheme is designed to run within MCRMs, the following implementations are done with grid cells and time steps as large as possible to save computing resources, but small enough to allow explicit simulations of thundercloud microphysics and dynamics.

Then the stochastic selection of new effective lightning bonds is applied based on the branching probabilities. Moreover, assuming the fractal nature of lightning flashes, the bond number must satisfy the relation (Mandelbrot, 1975; Barnsley et al., 1988):

3.1. Static tests

N = Lmax χ × λ1−χ ,

(2)

In this section, we focus only on the simulated lightning trajectories; no charge neutralization is considered. The sensitivity tests are performed with a horizontal and a vertical grid spacing of 250 m and 125 m, respectively. The domain size is of 12 × 12 km2 . Two kinds of static test of the lightning stochastic scheme are presented. The first set of tests involves 2D-unidirectional discharges. It is devoted to assess the scheme’s ability to preferentially propagate discharges towards highest (in absolute values) chargedensity regions (subsection 3.1.1). The second set of runs explores the sensitivity of the simulated 2D lightning structures to the scheme parameters (subsection 3.1.2).

where Lmax is the diameter of a virtual circle or sphere (depending on the model geometry) in which the lightning flash is entirely included at a given building stage (set to the domain size), λ is the radius of a virtual sampling circle or sphere and χ the fractal (or self-similarity) dimension of the lightning flash. λ is constant along the lightning-flash simulation. Equation (2) is used in the scheme not to enforce but to limit the branch number of the positive and negative parts of a flash. The maximum branch number can be adjusted by tuning the parameters 3.1.1. Unidirectional discharges χ and λ. Unidirectional electrical discharges are simulated in uniformly charged domains, except in a spot filled with an 2.1.4. The charge neutralization excess of electric charge (Figure 2(a)) or depleted of elecThe charge neutralization is the result of recombina- tric charge (Figure 2(b)). In these configurations, only tions of ions supplied along the lightning channel. The the negative segments of the single track are simulated lightning-emitted ions are captured by cloud ions, cloud and they are compelled to propagate vertically across ten water (non-precipitating particles) and hydrometeors. grid meshes from the upper domain boundary. Branch Because of several factors (both dynamic and electrically tracks are allowed anywhere in the domain (Qxc = 0). related motions), lightning neutralizes cloud charges in Figure 2(a) displays a simulated discharge propagating some volume around the lightning channel. Following toward the highest charge-density region when triggered Ziegler and MacGorman (1994) and McG01, we applied in the lowest one. Figure 2(b) shows the discharge avoida charge neutralization parametrization which consists of ing the lowest charge-density region when triggered in removing electric charges in excess of a given thresh- the highest one. This is the kind of lightning behaviour old Qxc . The charge is neutralized at grid points crossed reported in McG01. Such a discharge should be able to by the lightning. If r denotes a bond location (and a neutralize an unusually charged spot. c 2009 Royal Meteorological Society Copyright


Q. J. R. Meteorol. Soc. 135: 113–124 (2009) DOI: 10.1002/qj

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Figure 3. The academic charge distribution used in the 2D simulations. It includes a upper positive charge (A: 1 nC m−3 ), in which two zones of higher positive charge (A1 and A2: 1.5 nC m−3 ) are embedded. In the lower part of the figure is a negative charge (B: 1.5 nC m−3 ) imposed around a 1 nC m−3 charge-density area (B1).

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Figure 2. Simulations of 2D unidirectional discharges. The black ellipse marks the initiation point, the thick line shows the positive single track and the thin ones the lightning-flash branches. (a) shows a domain uniformly filled with a positive electrical charge of 0.1 nC m−3 , except the hatched square which is filled with a charge density of 10 nC m−3 . (b) is the opposite of (a), with the hatched area charged at 0.1 nC m−3 .

3.1.2. 2D lightning-flash structure On the one hand, lightning-flash patterns result from stochastic proccesses. On the other hand, they are constrained by the charge-density distribution (through the branching probability) and by the limit on the number of branches (Equation (2)). The purpose of this section is to explore the lightning pattern sensitivity to the parameters χ and λ which limit the branch number at a given distance from the initiation point (section 2.1.3). M´esoNH is processed at one time step during which only the electric field computation and the lightning-flash scheme are activated. The initial electric charge configuration is illustrated in Figure 3. The single-track trajectories are forced to the same configuration for all the runs. The trajectories run from the initiation point (coordinates 12.5 km, c 2009 Royal Meteorological Society Copyright


12.5 km) toward the A and the B region along the maximum electric-field line. To constrain the lightning-branch propagations in regions A and B, Qxc has to be positive and lower than 1 nC m−3 (the absolute value of the background charge density in regions A and B is 1 nC m−3 ). It is set Qxc = 0.9 nC m−3 . Figure 4 displays branched lightning flashes reaching sparse charge-density regions. The simulated branch patterns look like those displayed in DBM-related papers (Pietronero and Wiesmann, 1988; Wiesmann and Zeller, 1986; Barclay et al., 1990). As a result of the scheme stochasticity, the branch patterns can be asymmetric (illustrated specifically in Figure 4(b)). Such a configuration results from a positive feedback in the branch selection. The further the branch is from the ignition point, the higher is the branching probability, and so on. In comparing Figures 4(a) and (b), it is shown that the branches are far more numerous closer to the initiation point in the latter. Thus, decreasing λ involves side branching closer to the initiation point. The flash pattern sensitivity to χ is assessed by comparing Figures 4(b) and (c). The decrease of λ allows branches to propagate further (region A2 is reached in Figure 4(c) but not in (b)). We can note that combining the decrease of χ and λ do not influence lightning patterns (compare Figures 4(a) and (c)). In summary, it is shown that tuning λ and χ allows adjustment of the lightning-flash morphology from treeto bush-like patterns. Wiesmann and Zeller (1986) showed that the side branching patterns are controlled by two intrinsic discharge parameters: Ei , the internal electric field and Ec , the critical electric field. Ei is related to the discharge channel resistivity. In dissipating the discharge electrical energy, Ei limits its extension. Increasing Ei involves a branch-number increase by reducing the voltage and thus the electric field at discharge tips. So, relatively more electrical energy is available to break down bonds closer to the initiation point (Barclay et al., 1990). The physical origin of Ec is somewhat difficult to retrieve. In a very crude simplification, Ec is related to the voltage Q. J. R. Meteorol. Soc. 135: 113–124 (2009) DOI: 10.1002/qj

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(a)

(b)

(c)

decreasing χ favours tree-like lightning patterns, as does increasing Ec . Lightning-flash patterns can be quantitatively assessed through their effective fractal dimension, D. Even though the fractal dimension of simulated lightning is bounded through χ, it can differ from the effective fractal dimension D for at least two reasons: the branch bond discretization and the scheme stochasticity. To check D, the correlation method is used (Barclay et al., 1990, compares some methods). Fractal lightning should satisfy M ∼ r D (similar to Equation (2)), where M is the average of the bond numbers included in circular domains of radius r successively centred on each branch node. The fractal dimension D is the slope of a straight line fitted to the (M, r) points plotted on a log–log diagram. The fractal dimension D is estimated separately for the positive and negative branches (Table I), as it could be influenced by the charge density distribution. In Table I, the high correlation coefficient values indicate that the simulated lightning flashes can be considered as fractal objects. Theoretically, D is bounded to 2. When D = 2, the branch pattern completely fills the plane, while for a single-track feature D = 1. Bush-like lightning patterns (dense lightning) in Figures 4(b) and (c) feature D greater than the tree-like patterns (sparse grid nodes close to the branches are free of lightning) in Figure 4(a). The fractal dimensions of positive and negative branches are different. This emphasizes the initial charge structure influence on the discharge geometry. It is also confirms that λ and χ allow gross constraint on the branch morphology (i.e. the fractal dimension). 3.2. 3D simulations

Figure 4. Lightning flashes simulated during 2D one-time-step simulations. The initial charge configuration is identical to that of Figure 3 and is outlined by dash and dash-dotted thin lines. The lightning single track is drawn by a thick dashed line with the initiation point indicated by a diamond. Lightning branches are drawn using solid thick lines. The maximum bond number is prescribed at a distance Lmax (km) as N = Lmax χ λ1−χ . (a) λ = 4.5 and χ = 1.6, (b) λ = 1 and χ = 1.6, and (c) λ = 1 and χ = 1.1.

magnitude applied to the electrode used to generate unidirectional discharges in the DBM. The physical analogy with bi-directional discharges which do not originate from an electrode is not trivial. In DBM, increasing Ec causes the side branch number to decrease and to maintain the leading branch propagation (Barclay et al., 1990). It is worth noting that decreasing λ mimics the Ei increases in favouring bush-like lightning patterns, while c 2009 Royal Meteorological Society Copyright


To explore the simulated lightning-flash patterns along a comprehensive thunderstorm life cycle, a well-known archetype of 3D supercell storms (Klemp and Wilhelmson, 1978) is simulated using M´esoNH. This event constitutes a benchmark of M´esoNH. The storm presents a well-organized dynamic with an intense updraught (> 20 m s−1 ) surrounded by several evaporative downdraughts. The microphysical processes that take place produce a copious number of iced particles with different degrees of riming, but also maintain a sufficient amount of supercooled cloud water to produce rimed particles. These environmental conditions appear to be very favourable to cloud electrification. The simulations were performed on massively parallel machines. The horizontal mesh size is 1 km and the vertical one is 500 m. The horizontal extension of the simulation domain is of 40 km in both the north–south and east–west directions. The domain height is 20 km, and the time step is 1 s. To simulate 1 hour of the storm life, less than 15 minutes are required whatever the input parameters of the lightning-flash scheme. The M´esoNH microphysical scheme includes five cloud particle species: pristine ice crystals, iced aggregates, graupel, cloud liquid water, and rain. A simple non-inductive charging mechanism (Helsdon and Farley, 1987) has been implemented in which bouncing collisions between ice Q. J. R. Meteorol. Soc. 135: 113–124 (2009) DOI: 10.1002/qj

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Table I. Estimated fractal dimensions, D, of positive and negative branch sets from lightning flashes shown in Figure 4.

Figure 4(a) Figure 4(b) Figure 4(c)

λ

χ

4.5 1.0 1.0

1.6 1.6 1.1

Fractal dimension, D Positive Negative 1.66 1.89 1.95

1.82 1.99 1.99

Least-square correlation coefficient Positive Negative 0.99 0.98 0.98

0.98 0.97 0.99

The fractal dimensions are estimated by the slope of a linear fitting. The correlation coefficients indicate how well the fractal model fits the simulated lightning branch patterns.

Figure 5. Perspective view of the simulated supercell thunderstorm. The supercell propagates from northwest (NW) to southeast (SE). Contours in (a) (40 minutes of simulation) and (b) (60 minutes) show the iced (highest) and liquid (lowest) hydrometeor mixing ratios (contour interval = 0.5 g kg−1 ). Superimposed on these diagrams are the isosurfaces of raindrops (2 g kg−1 , lowest), of graupel (2.5 g kg−1 , middle) and of aggregated ice crystals (2.5 g kg−1 ). Contours in (c) (40 minutes of simulation) and (d) (60 minutes) display the total charge density (contour interval = 0.3 nC m−3 ). Dashed curves indicate negative values. Superimposed are the ±1 nC m−3 total charge density isosurfaces (dark (blue) are negative and light (red) positive). This figure is available in colour online at www.interscience.wiley.com/journal/qj

aggregates and graupel lead to electrical charge separation. The larger of two interacting particles gains a positive charge (2 × 10−15 C) if the ambient temperature is lower than −15 ◦ C and a negative one in the opposite situation. For further details on the model set-up and on the simulated electric charge structure, the reader can refer to Molini´e et al. (2002) or Barthe et al. (2005). The hydrometeor and charge configurations are illustrated in Figure 5. Figures 5(a) and (b) illustrate the classic picture of hydrometeor arrangements in thunderclouds. The lighest particles (pristine ice cristals and iced aggregates) are situated in the upper part of the cloud (the upper shaded area). Below is the solid (graupel) and liquid (raindrops) precipitating particles. At 60 min (Figure 5(b)), the shaded areas at upper levels in Figures 5(a) and (b) highlight areas where graupel and iced aggregates interact. Figures 5(c) and (d) show the charge configurations after 40 and 60 min of simulation. They display a dipole with a upper positive charge and a negative one at mid-altitude. Several sparse positive charge c 2009 Royal Meteorological Society Copyright


regions are simulated at the lowest altitudes. Figure 6(b) shows the inter-cloud (IC) and cloud-to-ground (CG) lightning-flash frequency evolution. The first CGs occur 10 minutes after the first ICs were triggered. CGs are less numerous than ICs until the decay stage (around 80 min of simulations). At that time, the lightning frequency reaches its highest levels. Those evolutions are in agreement with the lightning-flash frequency features described by Williams et al. (1989). We do not expect and do not try to achieve better agreement, as the simulated cell has never been observed and the simulated charge configuration serves only to illustrate the lightning-flash scheme capabilities. Nevertheless, the simulation seems realistic. In the following, the lightning-flash parameters, except λ and χ , are set as indicated in Table II. The electric field (Etrig and Eprop ) and charge (Qxc ) thresholds are set in accordance with the charge generation tendency to produce realistic lightning-flash frequencies. This goal of this section is to explore the sensitivity of the thundercloud electrical behaviour to the lightning-flash Q. J. R. Meteorol. Soc. 135: 113–124 (2009) DOI: 10.1002/qj

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Figure 6. Evolution of the lightning-flash frequency during runs (a) λ = 10.0 and χ = 3.0, (b) λ = 3.0 and χ = 3.0, and (c) λ = 10.0 and χ = 0.5. TOTAL are the intra-cloud plus cloud-to-ground (CG) lightning flashes.

Table II. Lightning-flash parameters set for the 3D run. Etrig (kV m−1 )

Eprop (kV m−1 )

Qxc (nC m−3 )

150

10

10−10

patterns. As in section 3.1.2, λ and χ are tuned to grossly constrain lightning-flash patterns. However, it is important to remember that, as the scheme is stochastic, lightning-flash patterns are not entirely constrained. λ and χ constrain only the maximum number of branch bonds, as shown in Figure 7. Comprehensive supercell life-cycle simulations have been performed similar to those illustrated in Figure 5, but with different λ and χ. These two parameters act only in limiting the bond number along branch developments. As shown in Figure 7, the larger the value of χ, the larger is the increase of the bond number with the distance from the lightning initiation point. For a given χ, λ can be used to shift the bond number function. Three configurations of the pair (λ, χ) are presented through three runs a, b and c. The electric field 90th-percentile value, E90 , c 2009 Royal Meteorological Society Copyright


Figure 8. Evolution of the electric field 90th percentile (average over 10 points) for the same combinations of λ and χ as in Figure 6.

computed inside the storm volume (Figure 8) has been chosen as a cloud electrification diagnostic. In run c, E90 displays the highest magnitude over the storm life cycle. Another diagnostic of the cloud electrification level is the lightning-flash frequency (Figure 6). Run c also yields the greatest IC frequencies while the CG frequencies are of the same order of magnitude as those of runs a and b. The D histograms are plotted in Figure 9; those for runs a and b feature two modes around D = 1 and D = 2. None of the lightning flashes in run a and only few in run b feature D greater than 2.5. During run c, D displays several peaks especially for the positive branches and a non-negligible fraction of lightning with D greater than 2.5. The temporal evolutions of D (Figure 10) show that, in run c, the first lightning flashes with D > 2.5 occur between 45 and 50 min simultaneous with the strong E90 increases (Figure 8). The lightning-flash pattern is then suspected to be responsible for the electric field discrepancies between the runs. This is confirmed by the greatest electric charge amounts simulated in run c and depicted in comparing Figures 5(d) (run b) and 11 (run c). The role of the lightning-flash pattern on the thundercloud electrical behaviour has been highlighted. Let us Q. J. R. Meteorol. Soc. 135: 113–124 (2009) DOI: 10.1002/qj

A 3D STOCHASTIC LIGHTNING-FLASH MODEL

121

Fractal dimension

Fractal dimension

Fractal dimension

Figure 9. Lightning-flash fractal dimension histograms for runs with the same combinations of λ and χ as in Figure 6(a, b, c). The fractal dimension bins are 0.1 wide. This figure is available in colour online at www.interscience.wiley.com/journal/qj

now depict the variety of the simulated lightning-flash patterns. The D histograms from runs a and b display two modes. The positive branches yield a first mode around D = 1 (Figure 9). This mode is due to lightning flashes exhibiting only a single track without branching. In the real world, CG lightning striking the ground often possess this feature which can be observed with the naked eye. Such lightning flashes are also drawn in 3D representations of lightning presented in Mazur et al. (1998), Rison et al. (1999), Krehbiel et al. (2000), and Thomas et al. (2000). The second mode of Figure 9(a) or (b) corresponds to D = 2 (available for the two kinds of branches). It is illustrated by negative branches (propagating in a region of positive ambient charge density) in Figure 12(a). If a lightning flash featuring such a D parameter was propagating in a plane, it should completely fill it (i.e. reaching all available grid points) which is compensated in this case by branches propagating in transverse directions (some are horizontals and other verticals). Figure 12(b) shows a lightning flash exhibiting positive branches featuring D = 2.86, among the highest simulated D values. The branches reach almost all the available grid points in the 3D space around the single track. In that case, the simulated lightning flash is close to the Helsdon et al. (1992) model, where the charge density c 2009 Royal Meteorological Society Copyright


is redistributed in an ellipsoidal volume around a bileader.

4.

Discussion

Among runs a, b and c, which one is the most realistic? Certainly, the one which involves the lowest amount of energy. The E90 evolutions indicate that runs a and b are more likely, as the E90 magnitude in run c is much higher than in the other two runs (Figure 8). This is corroborated by the lightning-flash frequency which in run c is twice those of the other two runs (Figure 6). Even though the performed simulations are not intended to reproduce realistic charge configurations, it is obvious that the very large volumes of positive and negative high charge densities in run c (Figure 11), are not realistic. On the contrary, the charge density configurations of run a (Figure 5(c) or (d)) display pancake charge structures in agreement with those deduced from in situ electric field soundings (Marshall et al., 1995; Coleman et al., 2003). Fierro et al. (2006) simulated an idealized supercell thunderstorm which in some aspects is comparable to the one simulated in the current study. Q. J. R. Meteorol. Soc. 135: 113–124 (2009) DOI: 10.1002/qj

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Figure 12. Example of 3D lightning flashes superimposed on contours of the ambient charge density, as in Figures 5(c) and (d). The black line shows the positive and negative single tracks, the upper branches (blue) propagating in the positive ambiant charge density region are negatives and the lower (red) positives neutralize the negative ambiant charge density. (a) at t = 40 min with D = 1.7 (2.0) for positive (negative) branches and (b) at t = 50 min with D = 2.2 (2.8) for positive (negative) branches. This figure is available in colour online at www.interscience.wiley.com/journal/qj Figure 10. Lightning-flash fractal dimension as a function of time for runs with for the same combinations of λ and χ as in Figure 6(a, b, c).

Figure 11. As Figure 5(d), but for run c featuring λ = 10.0 and χ = 0.5. This figure is available in colour online at www.interscience.wiley.com/journal/qj

The MCRM horizontal and vertical resolutions are similar, as well as the maximum vertical updraught velocity (∼ 20 m s−1 ). They tested several non-inductive charging parametrizations. The ones parametrized following Gardiner et al. (1985) and Saunders et al. (1991) produce charge densities (minimum and maximum values around –1.3 nC m−3 and 1.1 nC m−3 ) similar to the current simulations. Before 90 min of simulation, the IC flash rate in Fierro et al. (2006) is between 20 and 40 flashes min−1 , depending on the charging parametrization. This is comparable with the maximum flash rates in runs a and b. Their simulations display far less CGs than the current c 2009 Royal Meteorological Society Copyright


ones. However, in the current simulations, charge density regions extend further towards the ground than those in Fierro et al. (2006). Consequently, among the 3D runs, run c (and so the lightning flashes featuring fractal dimensions D closer to 3 than 2) can be considered the least realistic. Several supports for such a conclusion have been found. Sa˜nudo et al. (1995) had simulated realistic 3D lightning flashes using a DBM. The fractal dimensions of simulated lightning were in the range 1.2 to 2.2. The fractal analysis of lightning flashes in Japan by Hayakawa et al. (2005) reported fractal dimensions between 1.2 and 1.8. Maps of VHF lightning radiation show lightning spreading into horizontal thin layers of cloud (e.g. Coleman et al., 2003; Riousset et al., 2007) and thus have dimensions rather close to 2 or lower. The comprehensive life-cycle simulations of the supercell show that, paradoxically, lightning flashes completely filling the 3D space (D around 3) involve a more chaotic evolution of the cloud electrical behaviour than those with patterns featuring D limited to about 2.5.

5.

Summary

A new lightning-flash scheme is described in this paper. It has been designed to be implemented within mesoscale resolving cloud models. It can be vectorized and parallelized. Q. J. R. Meteorol. Soc. 135: 113–124 (2009) DOI: 10.1002/qj

A 3D STOCHASTIC LIGHTNING-FLASH MODEL

A lightning flash is simulated in two steps. Positive and negative single tracks are first deterministically driven by the electric field vector. Then, a stochastic model is used to compute branch trajectories from the single tracks toward high charge-density regions. Two-dimensional simulations show the model’s ability to produce fractal lightning. The 3D simulations of a comprehensive thunderstorm life cycle are very realistic. They allow assessment of the influence of lightningflash morphologies on the electrical charge distribution in thunderclouds. It is shown that a few bush-like lightning flashes are sufficient to cause chaotic behaviour of the electrical activity in thunderclouds. Thus, treelike lightning flashes are more likely to occur. This is in agreement with lightning-flash observations.

6.

Acknowledgements

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Q. J. R. Meteorol. Soc. 135: 113–124 (2009) DOI: 10.1002/qj