modeling of streamer ignition and propagation in the

Modeling of Streamer Ignition and Propagation in theSystem of Two Approaching Hydrometeors

Jaroslav Jánský1,2 and Victor P. Pasko1

1Communications and Space Sciences Laboratory, Department of Electrical Engineering, School of ElectricalEngineering and Computer Science, Pennsylvania State University, University Park, PA, USA, 2Department ofMathematics and Physics, University of Defence, Brno, Czechia

Abstract First-principles plasma fluid modeling is used for investigation of electrical gas dischargesignited by a configuration of two approaching conducting hydrometeors with typical radii on the order ofseveral millimeters under thunderstorm conditions (i.e., at an elevated location in the Earth's atmospherecorresponding to half of air density at ground level and at applied electric field approximately half ofthat required for avalanche multiplication of electrons in air). It is demonstrated that ultraviolet photonsproduced by the electrical discharges developing due to the electric field enhancement in the gap betweentwo hydrometeors and resultant photoionization in the discharge volume lead to much less stringentconditions for conversion of these discharges to a filamentary streamer form than in the case notaccounting for the effects of photoionization. It is also demonstrated that this photoionization feedback iscritical for understanding and correct description of the subsequent streamer discharges developing on theouter periphery of two hydrometeors whose potential is equalized due to the electrical connectionestablished by the initial streamer discharge between them. The initial streamer ignition between thehydrometeors can be preceded by the corona development, which can have detrimental effect on theignition. However, it is demonstrated that for hydrometeors approaching with a speed of ∼10 m/s the effectof this onset corona is small.

1. IntroductionInitiation of lightning is one of the unsolved problems in lightning discharge physics (Dwyer & Uman, 2014,Section 3, and references therein). At the heart of the problem is the fact that decades of electric field mea-surements made directly inside thunderclouds (Dwyer & Uman, 2014, Section 3.3, and references therein)have failed to find electric field strengths large enough to make the electric discharge we routinely observeinside the thunderclouds in the form of lightning (Dwyer & Uman, 2014, Section 3). One of ideas how thisproblem can be solved is by considering hydrometeors, defined as liquid or ice particles (Dwyer & Uman,2014, Section 3.4). The presence of hydrometeors locally enhances the field near their surfaces, both dueto charges residing on them and due to their polarization (Dwyer & Uman, 2014, Section 3.4). The localincrease of field then can lead to an electrical breakdown. The electrical breakdown (e.g., Dwyer & Uman,2014, Section 2.3) leads to corona discharge (e.g., Dwyer & Uman, 2014, Section 2.4). The corona dischargebehaves differently depending on applied electric fields. The field required for the inception of an electri-cal discharge near the surface of a hydrometeor is less than the field required for streamer formation. It ispossible that the field near the hydrometeor will be reduced by the initial discharge before streamers canform (Dwyer & Uman, 2014, Section 3.4). In present work the term corona discharge at onset stage is usedto describe the form of electrical discharge without streamers.

A variety of models have been developed to study properties of corona initiation from hydrometeors andstreamer propagation (Dwyer & Uman, 2014, Section 3.6, and references therein). In particular, fluid mod-els are useful for description of streamer dynamics as they resolve the electric field and the electron andion densities. In the past, models studied mostly the inception of streamers around a single hydrometeor.Fluid models around a single elongated hydrometeor were used by, for example, Dubinova et al. (2015) andSadighi et al. (2015). Fluid models around a single charged spherical hydrometeor were used by Babich et al.(2016, 2017).

In addition to inception of corona discharge around a single hydrometeor, corona discharge was alsoobserved in experiments with colliding liquid drops (Crabb & Latham, 1974). Blyth et al. (1998) carried out

RESEARCH ARTICLE10.1029/2019JD031337

Key Points:• Ignition criterion for streamers is

quantified• Streamer propagation is simulated

using fluid model• Hydrometeor's speed allows for

streamer ignition

Correspondence to:J. Jánský and V. P. Pasko,[email protected];[email protected]

Citation:Jansky, J., & Pasko, V. P. (2020).Modeling of streamer ignition andpropagation in the system of twoapproaching hydrometeors. Journalof Geophysical Research: Atmospheres,125, e2019JD031337. https://doi.org/10.1029/2019JD031337

Received 11 JUL 2019Accepted 25 FEB 2020Accepted article online 3 MAR 2020

©2020. American Geophysical Union.All Rights Reserved.

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Journal of Geophysical Research: Atmospheres 10.1029/2019JD031337

similar experiments and concluded that collisions of water droplets could indeed generate corona duringthunderstorm conditions. The corona appeared when colliding liquid drops form one long coalesced liq-uid drop. Schroeder et al. (1999) developed a model for corona inception from a long coalesced drop andobtained good agreement with experimental observations. This model was further applied for thunderstormconditions (Solomon et al., 2001), and the authors concluded that such mechanism cannot alone triggerlightning. Cooray et al. (1998) developed a different model for inception of streamers during the interactionof two water drops and suggested another possible scenario when the electric discharge occurs between thetwo water drops. The discharge causes high conducting path between them, which leads to their electri-cal connection. The electrical connection causes the field enhancement on outside of water drops, whichcan lead to corona inception. Cooray et al. (1998) noted that coalescence of hydrometeors would lead to anenhancement of the electric field on their outer periphery similar to the situation when they are connectedby a discharge channel.

Both inception mechanisms using single hydrometeor or two hydrometeors are possible, and there is aclear distinction between them in duration of the enhanced electric field. Single hydrometeors, elongatedor spherical ones carrying net charge, require longer time to increase the field due time required for movinghydrometeor with variable speed of rotation to align with electric field lines or charging the hydrometeor,respectively. On the other hand side the time duration of collision of the hydrometeors is much shorter thanthe duration of ambient electric field enhancement as estimated in section 3.3 We note that field alignmentalso significantly affects the collision of hydrometeors and leads to the reduction of suitable collisions byabout 1 order of magnitude (Cai et al., 2017, section 3.2). For ignition of the discharge an initial electron isrequired. If there are a lot of initial electrons the streamer can ignite between the two hydrometeors, whilethe field around a single hydrometeor will be limited by corona discharge. If there are relatively few initialelectrons, the field around single hydrometeor can increase over the streamer ignition threshold withoutappearance of the corona discharge and allow for streamer ignition once an electron appears in the vicinity.In the present work we focus on cases when colliding hydrometeors connect electrically due to streamersdeveloping in the gap between them.

Cai et al. (2017) performed modeling study of the electrical connection between two hydrometeors withthe same dimension. Cai et al. (2018) extended this study for two hydrometeors with different dimensions.Cai et al. (2018) also evaluated the conditions for inception of streamers on the outside of the electricallyconnected hydrometeors. It is important to note that neither of above mentioned studies for two hydrome-teors employed first-principles plasma fluid models, and all considered some form of approximate criterionfor the corona and streamer ignition.

The criterion for positive corona inception from thundercloud hydrometeors was recently improved by Liuet al. (2012). The authors followed methodology developed in Naidis (2005) for positive corona inception inair with help of photoionization feedback. Photoionization feedback refers to process where the avalancheproduces new generation of seeded electrons for secondary avalanches by photoionization. Naidis (2005)employed the parameter K (ionization integral characterizing the electron avalanche growth) to characterizethe discharge inception. For parameter K higher than the threshold for corona at onset stage, the coronadischarge transitions into the streamer discharge.

The motivation of present work is to improve the understanding of the discharge dynamics in a system oftwo conducting hydrometeors. We complement the robust parametric study of streamer ignition for vari-ous conducting hydrometeor sizes and ambient electric fields reported by Cai et al. (2018). We note thatconducting hydrometeors is a model representation better suitable for liquid particles rather than ice parti-cles. Using plasma fluid modeling, we quantify the ignition criteria used in that work and demonstrate thestreamer development.

2. Model2.1. Scenario Studied in Present WorkWe study a streamer ignition and propagation in a system of two approaching hydrometeors. The scenariostarts as each conducting hydrometeor is polarized due to the ambient electric field inside the thunderstorm.If the two hydrometeors approach each other, the polarization increases even more leading to a significantincrease of the electric field in the gap between them. This field enhancement leads to an increase of electron



Figure 1. Scenario of ignition and propagation of streamers in a system of two approaching hydrometeors. Panel (a)represents ignition of streamer with help of photoionization feedback between two hydrometeors. Panel (b) showsstreamer propagating between two hydrometeors. Panel (c) then shows charge transfer between two hydrometeorsresulting in field increase outside of the system of two electrically connected hydrometeors. Panel (c) also shows theavalanches with photoionization feedback on outside leading to streamer ignition and eventually propagation in openspace as is seen in panel (d).

density due to ionization and can eventually lead to an ignition of the streamer discharge. Figure 1 showsschematically four stages of the discharge ignition and propagation.

For our scenario to start we need just one electron between the hydrometeors. The discussion of free elec-trons is well presented by Dubinova et al. (2015) emphasizing difficulties with generating free electrons inreal thundercloud environment. We note that although the authors rely on cosmic shower event, the mech-anism of detachment at high fields is estimated to be on micro or milliseconds scale. Using time scale ofdetachment 10−6 s and background negative ion density 109 m−3 gives source rate 1015 m−3 · s−1. Resultsfrom our work then give us additional quantities: Separation of two hydrometeors in between hundreds ofmicrometers and millimeters gives volume of interest on the order of 10−10 m3, and relevant time for elec-tron appearance 10−5 s can be estimated from hydrometeor velocity 10 m/s and approach distance of order10−4 m. This estimate gives 1 electron per event. Therefore, one detached electron would serve as an initialelectron for our scenario. Other possibility for creation of free electron is secondary emission of electron byimpact of positive ion to the surface of hydrometeor. However, this process is not sufficiently substantiatedin the existing literature and is not explored in present work. We note that more data on processes withinclouds are necessary to resolve this important issue of availability of electrons within clouds. To minimizethe requirements, we use only one electron for each streamer ignition.



Figure 2. Schematics of the computational domain. Left sphere with radiusR1 is separated by a distance s from sphere with radius R2 in ambientelectric field E⃗.

In the first stage there is an enhanced electric field between hydromete-ors. If the electric field is above the critical field Ek, defined by equality ofionization and two body dissociative attachment coefficient in air (Mor-row & Lowke, 1997), electrons multiply and cause an electron avalanche.The quantity characterizing the electron multiplication is called Meeknumber K and is defined as integral over the line of interest l:

K = ∫l

(𝛼 − 𝜂) dl , (1)

where 𝛼 and 𝜂 are the ionization and two-body dissociative attachmentcoefficients, in our work taken from Morrow and Lowke (1997). There aremultiple criteria that are used in existing literature to determine whether

the electron multiplication in avalanche is sufficient to ignite the streamer discharge. The classical Meekcriterion states that an avalanche to streamer transition happens when K ∼ 18–20 (Raizer, 1991, Section12.3.2). Cai et al. (2017) used a more precise method from Qin et al. (2011) and quantitatively demonstratedthat the avalanche to streamer transition happens at lower electric fields for our case of two closely spacedhydrometeors. Both above mentioned criteria consider evolution of only one avalanche. In air or other gaswith a photoionization effect, the first avalanche also produces electrons due to photoionization and thesesecondary electrons cause additional avalanches, as is schematically shown in Figure 1a. Cumulative effectof all avalanches lowers the Meek number necessary for the ignition. Liu et al. (2012) demonstrated thatK ∼ 7–10 is enough for an ignition of positive corona around single hydrometeor. This effect is called thephotoionization feedback. It is important also to note that Dubinova et al. (2015) demonstrated in theirsimulations the formation of a streamer near an icy hydrometeor for a Meek number of 10, however, withoutexplicitly explaining the effect causing it. In the present work we will determine the Meek number necessaryfor a streamer ignition between two hydrometeors with the photoionization feedback.

After the streamer is ignited at the left hydrometeor as is seen in Figure 1b, the streamer propagates alongthe direction of the electric field to the right hydrometeor. The streamer approaches the right hydrome-teor and causes an electrical connection of two hydrometeors. This electrical connection causes a chargetransfer between them, and both hydrometeors reach the same electric potential. Consequently, the field onthe outside of both hydrometeors is increased. This sudden increase of the electric field causes an increaseof electron multiplication (Meek number), which can lead to a streamer ignition. One of the goals of thepresent work is to determine the threshold Meek number for the streamer ignition accounting for effects ofphotoionization as is shown in Figure 1c.

Once the streamer is ignited on the periphery of two electrically connected hydrometeors, it further prop-agates in an open space. In regions where field will remain higher than E+

cr ∼ 4.4 kV·cm−1·atm−1 (Babichet al., 2016), minimum field required for propagation of positive streamers in air (Bazelyan & Raizer, 2000,pp. 48–50), the streamer will continue its growth, will branch, and eventually the streamer zone will becreated. A large concentration of streamers in streamer zone will cause air heating and streamer to leadertransition. Experiments have shown that leader initiation occurs typically when streamers reach a lengthof about 1 m at near ground air pressures (e.g., Becerra & Cooray, 2006; Raizer, 1991, p. 366). We note thatstreamer to leader transition is mentioned here only to complete description of the possible scenario oflightning leader ignition. The streamer to leader transition is not quantitatively studied in the present work.

2.2. Fluid Model DescriptionFigure 2 shows the schematics of our computational domain. The left hydrometeor is assumed to be a spherewith radius R1. The right hydrometeor is assumed to be a sphere with radius R2. The two hydrometeors areseparated by the gap of length s. The computational domain is cylindrically symmetric with the axis of sym-metry z corresponding to the line connecting the centers of the spheres. The origin of coordinates is placedat the left side of the gap, which equivalently means also the right side of left hydrometeor. The constantambient electric field E⃗ is imposed by introducing flat perfectly conducting electrodes at the edges of thecomputational domain. The electrodes at edges are placed sufficiently far to neglect their direct influenceon the discharge dynamics near the spheres. The left edge is at −1.3 m and the right edge at 1.5 m. The radialsize of the computational domain is 1.9 m.



The model is based on Poisson's equation coupled with continuity equations for charge density inside thehydrometeors 𝜌h, and electrons with density ne, and negative and positive ions with densities nn and np,respectively:

∇⃗ · (𝜀∇⃗V) = −𝜌h −∑

i∈{e,n,p}qi ni , (2)

𝜕𝜌h

𝜕t+ ∇⃗ · (𝜎h E⃗) = −

∑i∈{e,n,p}

qi ∇⃗ · (𝑗i) , (3)

𝜕ne

𝜕t+ ∇⃗ · 𝑗e = Sph + ne 𝛼 |v⃗e| − ne 𝜂 |v⃗e| − ne np 𝛽ep , 𝑗e = −ne 𝜇e E⃗ − De∇⃗ne , (4)

𝜕nn

𝜕t+ ∇⃗ · 𝑗n = ne 𝜂 |v⃗e| − nn np 𝛽np , 𝑗n = −nn 𝜇n E⃗ , (5)

𝜕np

𝜕t+ ∇⃗ · 𝑗p = Sph + ne 𝛼 |v⃗e| − ne np 𝛽ep − nn np 𝛽np , 𝑗p = np 𝜇p E⃗ , (6)

where 𝜀 is permittivity, V is the potential, qi corresponds to the elementary charge of particle of species i,𝜎h is conductivity of hydrometeor, E⃗ = −∇⃗V is the electric field, 𝑗i is the flux of species i, 𝜇i is the abso-lute value of the mobility of species i, Sph is the photoionization term, v⃗e is the drift velocity of electrons,and De is the diffusion coefficient of electrons. The coefficients 𝛽ep and 𝛽np account for the electron-positiveion and negative-positive ion recombination, respectively. Equation (3) is solved at locations of the simu-lation domain where hydrometeor is located, while equations (4)–(6) are solved at locations filled with airoutside of hydrometeors. The interface between air and hydrometeor is included through right-hand side ofequation (3). That means that all impacting fluxes are absorbed into edge cells of hydrometeor. No surfaceemission processes are included in this paper so there is no outflux of charged particles from the hydrom-eteor. Transport and reaction coefficients are assumed to be function of E∕N, where N is the air neutraldensity and are obtained from Morrow and Lowke (1997). The reported simulations are performed at half ofneutral density on the ground N = 0.5 Nground = 1.344× 1019 cm−3 corresponding approximately to altitude6 km above the ground in Earth's atmosphere.

We note that, although the model can handle arbitrary conductivity and permittivity, the present study is afollow-up to work of Cai et al. (2018) and therefore studies the hydrometeors as perfectly conducting spheres.To achieve this goal, the conducting spheres are characterized by conductivity 𝜎h = 1 S/m and relativepermittivity 𝜀r = 1. This leads to a dielectric relaxation time 𝜏drt ∼ 10−11 s. This timescale is at least an orderof magnitude longer than the computational time step, which guarantees numerical stability. On the otherhand, this timescale is shorter by at least an order of magnitude than timescales studied with plasma fluidpart of the model outside of spheres. Each simulation starts by executing the model for time duration of10−9 s without any presence electrons and ions. The steady state electric field corresponding to two perfectlyconducting spheres is reached. Then such electric relaxation guarantees that spheres can be considered asperfect conductor while simulations remain numerically stable.

The computational domain is discretized in the structured rectangular grid. In the radial domain the step is10 μm from axis to r = 1 mm and then the step is expanded following geometric progression with coefficient1.1. In the axial direction the step is 5 μm in the gap and 10 μm on the outside of hydrometeor where thedischarge is ignited. The mesh is then expanded following geometric progression until it reaches the edgeof the domain. The total number of points is nz × nr = 640 × 203 = 129,920.

The finite volume method is used for discretization of equations (2)–(6). The drift-diffusion part of electroncontinuity equation (4) is solved using the improved Scharfetter-Gummel algorithm (Kulikovsky, 1995) withparameter of scheme 𝜖ISG = 0.04, which is recommended by Kulikovsky (1995) as an optimum choice toobtain accurate solutions without numerical oscillations. For source terms in continuity equations, the timeintegration is based on a second-order Runge-Kutta method.

We note that equation (3) for charge density inside hydrometeor has right-hand side that consists of fluxes ofelectrons and positive and negative ions. This term has meaning of boundary flux, and it represents the flow



Figure 3. The Meek number (K) obtained from Poisson's equation andsimple spatial integration of ionization and attachment coefficients(crosses) as a function of separation s of two particles with radii R1 = 3 mmand R2 = 1 mm in background field E = 7.2 kV/cm = Ek∕2 at atmosphericdensity N∕N0 = 0.5. The streamer model is used to determine whether thestreamer is ignited or not ignited. The ignition threshold K = 12 is deductedfrom the three simulations and is shown as a dashed line.

of electrons and positive and negative ions onto the hydrometeor. In simu-lation reported in present work all the particles flowing onto hydrometeorare absorbed and there is no emission (e.g., secondary electron emission)processes present.

In present work, to accommodate execution of the model over long phys-ical time scales a semi-implicit treatment is applied to Poisson's equation(Hagelaar & Kroesen, 2000; Lin et al., 2012; Ventzek et al., 1993) to avoidthe dielectric relaxation time constraint:

∇⃗ ·

[(𝜀 + Δt

∑i∈{e,n,p}

(|qi| 𝜇i ni))

∇⃗V

]= −𝜌h −

∑i∈{e,n,p}

qi ni . (7)

The electric field is evaluated from values obtained after the last time step,taking also into account the correction term with predicted movementof charged species. The semi-implicit version of Poisson's equation (7) isdiscretized using a finite volume approach over the whole computationaldomain. The resulting linear system is solved using an iterative paral-lel semicoarsening multigrid solver SMG included in the HYPRE library(HYPRE, 2016). The time step of simulation is then determined from theminimum between the time step conditions for advection Δtadv <

Δxvi

,

diffusion Δtdif <Δx2

4 Di, and the source term Δtsrc <

1(𝛼−𝜂)|v⃗e| , as Δt =

min(0.5 Δtadv, 0.5 Δtdif, 0.1 Δtsrc), where Δx = min(Δr,Δz).

The photoionization term Sph is modeled using three-exponential Helmholtz model (Bourdon et al., 2007),which efficiently solves photoionization in a form of elliptic partial differential equations requiring sourcesand boundary conditions. For evaluation of photoionization we consider that ultraviolet photons fromgap cannot propagate through the hydrometeor and create new photoionization events on outside of thehydrometeor. A strategy described below is implemented to approximate this effect. We split the whole com-putational domain at the sphere centers, z = −R1 and z = s+R2, into three smaller cylindrical domains. Wecalculate photoionization term Sph in each smaller domain using only the photon sources from that domainbut using the whole computation domain with intention to have remote boundaries. Then the zero bound-ary condition can be used with high precision as all our sources are far from boundaries. The Sph solutionsoutside of each of the smaller domains and inside hydrometeors are simply not used.

3. Results3.1. Discharge Ignition and Propagation Between HydrometeorsIn this section we study the ignition of the streamer discharge between two hydrometeors. In particular, wewant to find the threshold for the ignition. As was mentioned in section 2.1 the threshold can be charac-terized by the electron multiplication, that is, Meek number K. We note that the threshold number K fortwo hydrometeors is expected to be higher than the threshold number for a single hydrometeor 7–9 (Liuet al., 2012, Figure 4) as the surrounding volume is constrained by the second hydrometeor decreasing thecontribution of photoionization. The separation and size of hydrometeors therefore influence the threshold.

Cai et al. (2018) determined that the optimum geometry for a streamer ignition at air neutral density0.5 Nground and ambient electric field 0.5 Ek is R1 = 3 mm and R2 = 1 mm. We focus on this geometry, andwe explore the ignition threshold as a function of decreasing separation. We start at separation s = 0.7 mmand quantify gas discharge dynamics corresponding to this scenario with fluid model. The initial electrondensity corresponding to one electron is placed in one cell next to right hydrometeor for all cases. It appearsthat the streamer is not ignited for this particular separation scenario. Figure 3 shows this simulation as ared cross corresponding to the Meek number K = 10.1 at s = 0.7 mm. It is labeled as “not ignited” as thestreamer discharge did not ignite.

We now decrease the separation to s = 0.6 mm. The Meek number increases to K = 11.4, but the streamer isnot yet ignited. Finally, the separation is decreased to s = 0.5 mm. The Meek number increases to K = 12.7,and the streamer is ignited. Therefore, the threshold is approximated to be 12 for this particular geome-



Figure 4. (a) Electric field and (b–d) electron (full line), positive ion (dashed), and negative ion (dash-dotted) densityprofiles along the z axis between the two particles with radii R1 = 3 mm and R2 = 1 mm separated by s = 0.5 mm inbackground field E = Ek∕2 ≃ 7.2 kV/cm at atmospheric density N∕N0 = 0.5 and at three representative times t = 4, 8,and 10 ns. Cross-sectional view of (e) electric field and (f) logarithm of electron density is shown during streamerpropagation at t = 10 ns.

try. We note that as expected it is much smaller than classical Meek criterion 18–20 but higher than Meeknumber K = 8 for photoionization feedback with single hydrometeor of R = 3 mm at air neutral density0.5 Nground(Naidis, 2005, Figure 1).

We now illustrate the dynamics of streamer discharge for successful ignition at s = 0.5 mm. Figure 4ashows the electric field distribution along the gap between hydrometeors during ignition and propagationof the streamer. Figure 4b shows electron and positive and negative ion density distributions along the gapbetween hydrometeors at t = 4 ns. The electron density is exponentially increasing from the right hydrom-eteor toward the left hydrometeor. The peak value at the left hydrometeor is ∼ 1011 cm−3. Figure 4c showselectron and positive and negative ion density distributions along the gap between hydrometeors at t = 8 ns.



Figure 5. Meek number (K) outside of sphere with radius R2 = 1 mm obtained from Poisson's equation and simplespatial integration of ionization and attachment coefficients (crosses) as function of radius R1 for separation s = 0.4 mmand in background field E = Ek∕2 ≃ 7.2 kV/cm at atmospheric density N∕N0 = 0.5. Streamer model is used to identifywhether the streamer is ignited, and the corresponding ignition estimated threshold K = 8 is shown as dashed line.

The electron density is again exponentially increasing and peaks at ∼ 1013 cm−3. The increase of peak elec-tron density is caused by the photoionization feedback. To summarize, the observed dynamics is due tothe outflow of electrons to left hydrometeor being smaller than number of created electrons in secondaryavalanches initiated from electrons seeded by photoionization. The electron and ion densities are continu-ously increasing until they are able to disturb the external electric field. Figure 4a shows the disturbance att = 8 ns. This moment corresponds to the streamer ignition and is followed by the streamer propagation.The strong electric field enhancement ahead of the streamer, one of the characteristic streamer features, isobserved in Figure 4a at z = 0.25 mm and t = 10 ns. Figure 4d shows electron and positive and negativeion density distributions along the gap between hydrometeors at t = 10 ns. The streamer head is located atz = 0.25 mm and is followed by streamer channel of high electron and positive ion density of ∼ 1013 cm−3.

The cross-sectional view of the electric field and electron density at t = 10 ns is shown in Figures 4e and 4f,respectively. The view shows that radial size of streamer is rather small compared to hydrometeors, about0.1 mm. For t > 10 ns the discharge continues to propagate toward the right hydrometeor and impacts it,which causes electrical connection of both hydrometeors. It is important to note that although resolvingprecisely the streamer impact requires finer mesh and better numerical schemes, for the purposes of thecurrent study only the streamer ignition and the electrical connection of hydrometeors is of primary inter-est and not the precise dynamics of the streamer impact. Therefore, the hydrometeors are now electricallyconnected and we next focus on streamer ignition outside of hydrometeors.

3.2. Discharge Ignition and Propagation Outside of HydrometeorsIn this section we start with study of the ignition of the streamer discharge outside of right hydrometeorwith radius R2. We note that the threshold number K for ignition outside of hydrometeor is expected to becomparable to the threshold for a single hydrometeor 7–9 (Liu et al., 2012, Figure 4).

The electric field outside of right hydrometeor is a function of R1, R2, and s. A variation of each ofthese parameters would contribute to the ignition threshold K. In present work we keep constant radiusR2 = 1 mm of hydrometeor where ignition happens, and separation s = 0.4 mm. We vary the radius of lefthydrometeor R1. We note that a slightly smaller separation 0.4 mm is used instead of 0.5 mm from section3.1. The smaller separation ensured that the discharge is ignited between hydrometeors for all considered R1values, which is a necessary condition for the ignition outside. Figure 5 shows Meek number K as functionof hydrometeor radius R1. Meek number K is integrated over the z axis from right side of hydrometeor untilthe electric field drops below Ek. For bigger radii R1 ≥ 2.8 mm the discharge is ignited and for smaller radiiR1 ≤ 2.7 mm the discharge is not ignited. The corresponding threshold for Meek number lies in betweenthese values and is approximated to be 8 for given hydrometeor geometry.



Figure 6. (a) Electric field and (b) electron (full line), positive ion (dashed), and negative ion (dash-dotted) densityprofiles along the z axis outside of particle with radius R2 = 1 mm connected with particle with radius R1 = 3 mm byconductive streamer channel of length s = 0.5 mm in background field E = Ek∕2 ≃ 7.2 kV/cm at atmospheric densityN∕N0 = 0.5 for three representative times t = 11, 15, and 41.5 ns. The cross-sectional views of the electric field (c, e,and g) and logarithm of electron density (d, f, and h) are shown at three representative times t = 15, 35, and 41.5 ns.



We now illustrate the continuation of dynamics of streamer discharge for hydrometeors with R1 = 3 mm,R2 = 1 mm, and s = 0.5 mm, which is the set of parameters from section 3.1. Figure 6a shows the electricfield distribution outside of the right hydrometeor with R2 = 1 mm. First at t = 11 ns, the time correspond-ing to the electrical connection of hydrometeors, the field is still comparable to the field before connectionat 10 ns; see Figure 4e. Then until t = 15 ns the electric field is increasing. This increase is caused bycharge transfer between the hydrometeors, which lasts for several nanoseconds due to the final conductiv-ity of the discharge channel. The cross-sectional views of the electric field and electron density at t = 15 nsare shown in Figures 6c and 6d, respectively. These figure panels illustrate the discharge channel betweenhydrometeors with low field and high electron density, and the enhanced electric field outside of the righthydrometeor.

We note that as explained in section 2.2 we do not allow photoionization from discharge inside the gapto ionize volume of air through the hydrometeor, outside the hydrometeor system. The initial electron onoutside is input at z = 3.11 mm, where E(z) = Ek after t = 15 ns as an initial condition. The initial dynamicsis not considered to represent reality, and it only provides us roughly with some electron seeds outside of thehydrometeor. We emphasize that these initial conditions do not influence whether the streamer is ignited.Once the field is above the ignition threshold, the electron density is increasing until it disturbs that fieldregardless of its initial value. The only influence of the initial electron density is the delay of the streamerignition. For our particular initial conditions, the electron density increases starting from t = 15 ns and is ofthe order 1010 cm−3 at 35 ns. The cross-sectional view of the electric field and electron density at t = 35 nsis shown in Figures 6e and 6f, respectively. The electron density enhancement at this moment is alreadyvisible, while electric field is still undisturbed. The electron density continues to increase until the electricfield is disturbed, and the discharge starts to propagate away from hydrometeors as shown in Figures 6gand 6h. Figure 6a shows that the electric field along axis for t = 41.5 ns has features of a typical enhancedelectric field ahead of streamer head (z = 2.75 mm) and a decreased field in the discharge channel behind,which is also a representative feature of streamers. Figure 6b shows electron and positive and negative iondensity distributions outside of hydrometeor at t = 41.5 ns. The streamer channel with high electron andpositive ion densities is seen from z = 2.5 to 2.75 mm. The cross-sectional view of the electric field andelectron density at t = 41.5 ns is shown in Figures 6g and 6h, respectively. We can conclude that the streamerhas been formed and is successfully propagating away from the two hydrometeors into the open space.

3.3. Effect of Corona Discharge at Onset StageIn previous sections we have simulated successful streamer ignition and propagation for a given hydrome-teor geometry without considering actual time dynamics of two approaching hydrometeors. As discussed insection 1 and in more detail in Dwyer and Uman (2014), the streamer can be preceded by a corona dischargeat the onset stage that can have detrimental effect on the streamer ignition. For our scenario, described insection 2.1, it means that for hydrometeors at a slightly larger separation than one required for streamer igni-tion only the corona discharge will be produced. That will lead to a reduction of field between hydrometeorsforbidding the streamer ignition. The field reduction is a consequence of charge transfer between particles.It is therefore important to understand conditions when the charge transferred between particles due to thecorona is not high enough to reduce the electric field before the streamer ignition occurs.

Figure 7a shows charge evolution on the particle with radius R1 = 3 mm as a function of time for threedifferent separations. For all cases the charge is increasing first due to initial avalanche to about 10−5 nC.Then the photoionization feedback starts to be seen. For two cases with streamer propagation, namely,s = 500 μm and s = 560 μm, the charge continues to increase representing growing avalanches with help ofphotoionization feedback. Then as streamer ignites the charge steeply increases by 2 orders of magnitudedue to the charge transfer in highly conducting discharge channel. Finally, the two particles reach the sameelectric potential that indicates completion of the charge transfer. The total charge transferred during caseswith streamer propagation is 0.3 nC. The charge Qred ∼ 0.1 nC can therefore be considered as a referencecharge value necessary for a significant reduction of the field between particles.

Figure 7b shows current evolution on the particle with radius R1 = 3 mm as a function of time for threedifferent separations. It is simply a time derivative of the charge from Figure 7a, I = dQ∕dt. Cases withseparation s = 560 μm and s = 570 μm are important as there is streamer ignition for one and not for theother. First of them has parameter K above the threshold for photoionization feedback, and the current isincreasing. Second case has lower K than the threshold and therefore current is decreasing as more electrons



Figure 7. Charge (a) and current (b) on the particle with radius R1 = 3 mm as a function of time for particle R2 = 1 mm, three different separation s = 500 μm(black solid), s = 560 μm (red dashed), and s = 570 μm (blue dash-dotted), and for background field E = 7.2 kV/cm = Ek∕2 at atmospheric density N∕N0 = 0.5.

flow into the hydrometeor than are created by secondary avalanches caused by photoionization. Neither oneof them is in steady state corresponding to corona at onset stage, but a good order of magnitude estimate ofcorona at onset stage current Icor ∼ 10−5 A can be obtained as it should be bounded by these two cases. Anestimate of the corona duration necessary for a significant reduction of field is then

𝜏discharge =Qred

Icor= 10−10 C

10−5 A= 10−5 s . (8)

Having assumed that hydrometeors approach each other with speed v = 10 m/s, we compare above men-tioned time with time of approach of two hydrometeors from separation 570 μm to separation 560 μm(distance Δs = 10 μm):

𝜏approach = Δsv

= 10−5 m10 m/s

= 10−6 s (9)

since 𝜏approach is shorter than 𝜏discharge the streamer will be ignited. However, for hydrometeors with approach-ing speed of 1 m/s the times are comparable and it is possible that the streamer will not be ignited as fieldwill be reduced due to the corona at onset stage.

We note that streamer will not be ignited if we make estimates using the separations studied for ignitionin section 3.1, 500 μm for ignited case and 600 μm for not ignited case. Then the time of approach will besufficient to discharge hydrometeors before the streamer ignition if corona at onset stage would be transfer-ring charge during the whole time of approach. The refinement of separations to 560 and 570 μm serves todecrease the maximum estimate of time duration of the corona at onset stage and provides a quantitativesupport for our hypothesis.

4. ConclusionsThe principal results of this paper are as follows:

1. We have studied ignition of the streamer between two approaching hydrometeors of sizes R1 = 3 mm andR2 = 1 mm for background field E = 7.2 kV/cm = Ek∕2 at atmospheric density N∕N0 = 0.5. We estab-lished that photoionization feedback mechanism allows for streamer ignition at fields corresponding toelectron multiplication K = 12.

2. We have also studied ignition of the streamer outside of the two electrically connected hydromete-ors and established that photoionization feedback mechanism allows for streamer ignition at fieldscorresponding to electron multiplication K = 8.



3. We have successfully simulated streamer ignition and propagation in system of two approaching hydrom-eteors to support the suggested scenario.

4. We have estimated that for hydrometeors approaching with speed of 10 m/s the corona dischargepreceding streamer discharge will not have detrimental effect on streamer ignition.

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AcknowledgmentsThis research was supported by NSFunder Grants AGS-1623780 andAGS-1744099 to Penn State University.Numerical model is fully described inthe text and cited references. All dataare available at Penn State UniversityData Center (https://doi.org/10.26208/3dy3-g353).


https://doi.org/10.1002/2016JD024901

https://doi.org/10.1016/j.jastp.2016.12.010

https://doi.org/10.1088/0022-3727/39/16/028

https://doi.org/10.1029/97JD02554

https://doi.org/10.1002/2017GL073107

https://doi.org/10.1029/2018JD028407

http://www.llnl.gov/CASC/hypre/

https://doi.org/10.1029/2010JA016366

https://doi.org/10.1002/2014JD022724

https://doi.org/10.26208/3dy3-g353

https://doi.org/10.26208/3dy3-g353

modeling of streamer ignition and propagation in the

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