The Electric Fields of Lightning Clouds in Atmospheres of Different Properties

Abstract

In this work, we concentrate on the electric fields produced by any distribution or structure of electric charges in clouds, these quantities are, for the most part, unknowns, and still we extract valuable information from their vectorial properties and working within close distances, especially on the structures that produce pronounced sinks/sources with large divergences of E → and focusing on observation points close to them. We introduce the concept of “at the verge of discharge electric field,” Ę → produced by two specific charge configurations. We detail the role the dielectric character of the cloud plays in allowing/not allowing discharge. We profile the conditions to be met by Ę → to appear and to be sustained for at least a few milliseconds. One objective of this work is to calculate some relevant quantities for a typical thundercloud modeled just at the verge of discharging. A simple electrified cloud model containing many of the features widely reported in the literature is constructed and described with some detail. The charge structures possible are delineated and the electric fields produced described. A few basic calculations are carried out for a thought experiment, moving charges around, we calculate some velocities and accelerations and analyze what we learn from it.

1. Introduction

Clouds as cotton fields taking a diversity of colors, or ragged white milk-like spreads, or tall, very tall cloud compounds attract the sight and inspiration of people everywhere and produce awe in some of us, as shown in Figure 1A . But then, electrified clouds and then their lightnings, thunders, and copious precipitations have scared and fascinated humans since immemorial times [1, 2]. Beyond the gods associated with them in different cultures [3, 4], these thunderclouds sustain a great amount of energy, in the form of hydrodynamic energy and electromagnetic energy. They also carry great amounts of water [5, 6], and they also carry and concentrate on the electric charge, moving it and distributing it in different arrangements until they can carry no more, as show in Figure 1B . The charging and charging processes cannot go forever, since the electric energy, proportional to the total charge accumulated [7, 8, 9] would tend to infinity, then at some point the cloud system becomes electromagnetically unstable, I even say quite unstable. Lightning is fundamentally a process of electrical discharge of an electrified cloud, a thundercloud, but it is not the only one, gigantic jets launched into the sky is another one. Lightning can be very rapid and violent and quite sonorous and luminous [10, 11, 12]. It is now well established [10, 11, 12, 13, 14, 15] that these cloud electric discharges go down to ground, or mountains, or tall buildings, metallic structures, trees, animals, and people [10, 11, 12, 13, 14, 15], and they also go up the sky and can terminate just above the cloud in the sky [16, 17, 18] or go higher and reach the ionosphere (around 90 km above see level) and beyond [18, 19, 20]. These energetic (thousands of joules) discharges frequently are accompained or followed or preceded by light, UV rays, gamma rays, bremsstrahlung [21, 22, 23, 24], and/or X-rays. Figure 1 illustrates some cloud formations and some of these processes.

Figure 1.

Diverse cloud formations and colors and sizes and lightnings: (A) clouds as cotton fields taking a diversity of colors, or ragged white milk-like spreads, or tall, very tall cloud compounds; (B) very extensive cloud formations lightning above cities and the sea; and (C) an assortment of different phenomena directly produced by discharging electrified clouds.

Lightnings are also preceded by torrents of classically accelerated electrons, protons, small ions, even relativistically moving electrons, gamma rays, and UV-vis [20, 21, 22, 23, 24]. These discharges happen frequently from cloud to ground, from cloud to cloud, and from a region, A, to a region, B, within the same electrified cloud. It has been estimated that about 9,000,000 lightning discharges happen worldwide every day [25]. These everyday cloud electromagnetic phenomena bring with them, even more electromagnetic phenomena, including light at different wavelengths, gamma ray and X-ray radiation, elementary particles as electrons, neutrons, and positrons, radiofrequency signals and sound [26, 27], sprites, elves, and glows. All this happens inside the clouds, around the clouds, and kilometers away from the clouds, as shown in Figure 1 . In order to understand the physics of all this, it is necessary to get to know more in detail what a cloud is, its basic charged and neutral components, its turbulent internal motions, and the different charging mechanisms and discharging mechanisms. It is necessary to know the different charge structures and the electric fields, $\overrightarrow{E},$ with their divergences, Div $·\overrightarrow{E}$ = ${\rho }_{\mathit{TOT}}$ , produced by these tremendous moving electrified cloud bodies. Some great amount of electric charge, positive and negative, remains quasistatic for short periods of time (of the order of milliseconds), located in specific regions at different altitudes [25, 26, 27] as represented in Figures 1C and 2 . Other charged elements are much more mobile and eventually can confirm electric currents within conductive patches that form and disappear in micro- or milliseconds [28] as shown in Figures 1B , C and 2 .

Figure 2.

Electrified clouds and some of their constituents: (A) a tall, moving, multicolor, electrified cloud bathed by cosmic radiation (blue tortous rays) and solar rays (yellow ray) with charge distributions located at different altitudes. Water droplets, ice, graupel, cations, and anions are some of the hydrometeors in it; and (B) a conglomerate of charged clouds with intra- ( ) and intercloud ( ) discharging and some other cloud components shown: Water droplet ( ), graupel ( ), ice ( ), hail ( ), molecular ion ( ), snow ( ), and pollutant particle ( ). The density of charge within each cloud is symbolically represented as ρ_i( ${\overrightarrow{\mathrm{r}}}_{\mathrm{i}}^{´}$ ) with $\mathrm{i}=\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d},\dots$ , and conducting patches are shown as ( ).

At the level of individual charged particles, it is necessary to know the pathways, velocities, accelerations, and directions these charges follow during the discharges. We, now, know that all types of charges, prominently electrons, can be involved in the cloud discharges [29, 30, 31, 32]. Charges in clouds include, but are not limited to, electrons, $e^{-}$ , protons, $p^{+}$ , water droplets, $w^{\pm }$ , small and large ice cristals, ${\mathit{ic}}^{\pm }$ , graupel (ice covered by water), $g^{\pm }$ , hail, $h^{\pm }$ , all kinds of typical atmospheric cations, ${\mathit{ci}}^{+},$ and anions, ${\mathit{ai}}^{-}$ , and polarized atoms and molecules, and continuous and permanent elements of cosmic rays [29, 30, 31, 32]. Several of the above-mentioned characteristics are illustrated in Figure 2 . Part of the fundamental electromagnetic problem that an electrified cloud poses (one that can become a thunder cloud) is to know, to determine its charge structure, how it is distributed in cloud space and time, and the electric field(s) it produces. As Griffiths puts it [33] (more in general, not particularized to clouds): The fundamental problem a theory of electromagnetism hopes to solve is this: I hold a bunch of charges here, ${\rho }_1$ or $Q_{\mathit{\text{source}}}$ , (and may be shaken around)—what happens to some other charge, q, over there? The classical solution takes the form of a field theory. We say that the space around an electric charge is permeated by electric and magnetic fields (the electromagnetic “odor,” as it were of the charge). The second charge in the presence of these fields experiences a force (Coulomb force); the fields then transmit the influence from one charge to the other—they mediate the interaction. Refer to the charges in Figure 2 . And we add: The knowledge of such a bunch of charges, ${\rho }_1\left({\overrightarrow{r}}^{´},t\right)$ or $Q_{\mathit{\text{source}}}$ “here” and the fields and forces they produce “there” allow us to determine the trajectories, velocities, and energies, that the other charge(s) q ( $\overrightarrow{r}$ , t) will acquire. From this, we can calculate currents, velocities, accelerations, and radiation fields produced but only when the dielectric and conduction properties of the medium allow [7, 8, 9, 33]. We want to apply this to electrified clouds. In this chapter, we want to pose this fundamental problem for electrified clouds and contribute with some basic calculations toward its solution. In particular, we want to be as explicit as possible with the source charges (distributions), their accumulation “here,” the electric forces, $\overrightarrow{F}$ , and electric fields, $\overrightarrow{E}$ , they produce, and how these forces and electric fields affect (many) other charges, q, “over there” and their discharge and accompanying glows. A cloud that has accumulated a great amount of charge can discharge it toward other clouds, ground, mountains, metallic objects, and ionosphere. When the source charges in electrified clouds are just too much to be held, they discharge toward other charged conducting objects. This can be accompanied by light and sound, and then we have lightning. Lightning continues to be a mystery to science [1, 2, 20, 25]. It is not clear at all how it is initiated, nor how are reached the values of electric field required for lightning to initiate and the many effects lightning produce above and below the cloud. The charging processes, the discharging processes, and the electric fields involved are at the center of the unsolved questions [10, 11, 12, 25]. We should stress that for more than a century now, a great number of researchers have worked on these, and their insights, measurements, contributions, and acquired knowledge are vast [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. We, certainly, drink from this vast reservoir of knowledge. Yet, it is recognized that these fundamental problems continue to be open, and they figure at the top of a list of 10 basic unanswered questions [25] in the physics of electrified clouds research.

Here, we construct a simple electrified cloud model containing many of the features widely reported in the literature, and we describe it in some detail. Our description is as complete and detailed as possible, then we perform a few basic calculations on it and perform a thought experiment, moving charges around inside this model cloud, and we calculate some quantities and analyze what we learn from it. But first we want to mention, in some detail, that electric charges are really very common in this world and how charges are produced in different materials and, of course, in clouds that become heavily electrified.

2. Charges are everywhere in the world and the universe

Electric charge is a fundamental feature of a great number of material structures, small, large, mesoscopic, microscopic, of this world, and indeed of the universe [34, 35, 36, 37, 38]. Some examples are water droplets, little and large ice crystals, cell membranes, cut fingernails, ashes from volcanoes, pollutant particles, the earth, the ionosphere, clouds, balloons, atoms, electrons, protons, amber, fur, glass, straw, and small pieces of wood. Charged, electrified, and objects have been known for millenia. The Greeks were the first to describe some of them, like amber and fur, and they also discovered how to electrically charge them, by simple friction! By simply friction one against the other, they produced the effect of electrostatics. They also discovered that there are two types of “electricity” that later were named “positive” and “negative” and that positive-positive, +q $\leftrightarrow$ +q, (negative-negative, −q $\leftrightarrow$ −q) charged bodies repeal and that positive-negative, +q ⇋ −q, and negative-positive, −q ⇋ +q bodies attract [34, 35, 36, 37]. Natural, strongly electrical phenomena, out of man’s direct manipulation, as clouds and thunderclouds and lightning were also known from antiquity. In several cultures, there was the god of thunder [39, 40]. It was also known that electrified bodies can be discharged and that charge can “move” and migrate can be transfered from one place to another, constituting the electric current, I = $∆$ q/ $∆$ t. Latter on, glass, water flowing, pieces of cloths, rubber, rugs, metal knobs, comb(s), metal cages of washing machines and of refrigerators, sea water, wet floor, our hands, our body, pollutant particles, etc. [41, 42] were, then, by experimenting, included in that list of chargeable systems. When a lot of charge is accumulated by any of these ways, irrespective of the particular mechanism, we take the total charge, Q = ${\sum }_i^n\mathit{qi}$ when charge is an accumulation of discrete elements and Q = ${\int }_q\mathit{dQ}$ when modeled as a continuous distribution of charge. It is also very convenient to use its volumetric density, $\rho$ = Q/V = (1/V) $\sum _i^n\mathit{qi}$ or $\rho$ = (1/V) ${\int }_q\mathit{dQ}$ and their surface density, $\sigma$ = Q/S = (1/S) ${\sum }_i^n\mathit{qi}$ or $\sigma$ = (1/S) ${\int }_q\mathit{dQ}$ and their linear density, $\lambda$ = Q/ $\ell$ = (1/ $\ell$ ) ${\sum }_i^n\mathit{qi}$ or $\lambda$ = (1/ $\ell$ ) ${\int }_q\mathit{dQ}$ .

With the execution of specific experiments and development of technology, more charged objects were known like the popular electrostatic bottles, as shown in Figure 3 [43, 44]. It is very impressive how they can be charged and how they can be discharged in just a small fraction of a second, electric current, I, traveling through the air and producing a spark, light, and sound, and then it came the “famous” Benjamin Franklin’s kite, as shown in Figure 3E , [44, 45], and then the whole of the early history of the research on charged clouds and their ways of discharging [46, 47, 48] mainly into ground or to a tree, or to a church bell (metallic), or on another nearby cloud. During discharge, electric current will travel from the cloud to ground and from ground to cloud, or from cloud to cloud producing, not always, light and sound. Several other thundercloud and lightning events were observed, reported, and “described” many years ago, and some of them are bolts, sprites, glows, blue jets, and gigantic jets [1, 2, 10, 11, 12, 13, 14, 15, 16]. In more recent times, novel discharge mechanisms and electromagnetic emissions have been observed and measured from ground, planes, aerostatic globes, and satellites [10, 11, 12, 13, 14, 15, 16]. From the more physical sciences view, the very same Coulomb Balance of 1785 (modified from the Cavendish balance used to measure, quantitate, the force between charged bodies) allowed Charles Coulomb to find $\overrightarrow{F}$ = K $q_1{q}_2\widehat{r}$ / $r^2$ . This universal law indicates that the force between two charged bodies is inversely proportional to the square of their distance and to the product of their charges and is directed along the line that joints them. It also includes the experimental fact that charges of the same sign repeal and charges of opposite sign attract. To do the experiments, Coulomb had to charge, electrify, the different bodies he used, operation that he did, mainly, by friction. The electroscope was invented, and it made possible some more quantifications [49]. Several of these developments are pictured in Figure 3 .

Figure 3.

Charging and discharging processes in a variety of systems presented from present-day machines to the oldest ones: (A) a laboratory-controlled discharge device, 1990s; (B) a Van de Graaff electrostatic generator, 1910; (C) Volta’s battery presentation to Napoleon, 1810; (D) a basic electroscope, 1777; (E) a painting showing Benjamin Franklin’s kite experiment, $\sim$ 175; and (F) Amber friction with fur, a pictorial representation of Tales de Mileto discovery $\sim$ 600 BC and a plastic comb attracting small pieces of paper, (generalizes the amber-fur electrostatics).

Later, the Van de Graaff electrostatic machine [50] was invented to specifically charge conducting bodies like spherical shells, and so on. More developments in the knowledge of electrical charge, its motion, and some applications constitute the invention of the battery [51] and its whole development. Faraday introduced the concept of electric field that is defined as $\overrightarrow{E}$ = lim(q- > 0) $\overrightarrow{F}$ /q, taking the Coulomb force just mentioned, and we have $\overrightarrow{E}$ = lim(q- > 0) [KQ $\widehat{r}$ / $r^2]$ . Conductors and nonconductors were then defined [7, 8, 9, 52], and microscopic mobile units of charge, electrons, were suspected to form a part of the intimate nature of matter [53]. Now, we know that the atom is full of charge specifically distributed, the nucleus is positive charged, +q, and the “revolving” electrons are negatively charged, −q. An atom is electrically neutral, but charged species, ions, are very common, and some examples are $O_2^{-}$ , $H^{+}$ , $K^{+}$ , ${\mathit{Cl}}^{-}$ , and so on. The whole material universe is constructed from electrons, protons, nuclei with protons and neutrons inside, atoms, then molecules, the chemical bond, minerals, rocks, hydrogen species, oxygen species, organic, molecules, polymers, and so on, and they are in all places, corners of the universe in the form of gasses, mesoscopic and galactic plasmas, rocky planets, moons, and atmospheres. Physical processes as friction, collisions, scatterings, breakage, fractionation, and intense electric fields are universal phenomena and separate charges from their parent matter; hence, free charge can appear potentially in every corner of the universe, generating diverging and/or rotational electric fields. Stellar rocky bodies with some kind of atmosphere can concentrate on the electric charge in somewhat stable arrangements, and free electrons generated by cosmic radiation can be accelerated in those atmospheres. Coming back to earth, we see below that such acceleration of electrons already takes place in electrified earth-bound clouds.

3. Charges can be produced in many different ways

To understand better the charging processes in clouds, we mention, first, how charges can be produced in many different ways: the most common ones are by friction, fractionation, breakage (breaking apart polymers and small pieces of paper), and others include thermoionization, just heating of a conducting cable with the passage of current, heating a gas, air, application of an intense electric field, UV-vis excitation/photoionization (this is one of the reasons we have life in this planet in the first place), photoionization in the atmosphere [54], electrolysis [55], battery and fuel cell mechanisms [56], xerox copying, ionizing radiation (NO, H, $O_2$ , $H_2$ O, organic molecules, and nucleic basis are very sensitive to ionizing radiation), cosmic radiation, collisions, and near collisions of many kinds: electron-atom, electron-molecule, atom-atom, atom-molecule, elementary particle-atom/molecule, scattering of gamma and/or X-rays, and UV rays scattered by atoms or molecules [53], and so forth.

In neutral gasses charging mechanisms, charge separation include elevated temperatures promoting more atom/molecule frenetic motion, more and more energetic collisions of atoms/molecules, molecular friction (then the asymmetric breaking of them), very strong turbulent fluxes (turbulent winds in diverse directions) promoting, again, energetic collisions where linear and angular momentum, energy, and charge are exchanged or transferred, and exposure to energetic gamma rays, X-rays, and/or cosmic radiation, application of a high-voltage difference, $∆V$ , or equivalently a strong electric field, $\overrightarrow{E}$ = − $\nabla V$ (very strong, of the order of hundreds of thousand Volts/mt), could be enough to break molecules and to break polarized molecules, or at least rip them from one electron, leaving behind a cationic, heavy, species. Discharged gasses of detectors and lamps are good examples of these systems [57, 58]. If the gas is basically dielectric, a moderate to high electric field will only polarize the molecules of the system producing a total macroscopic polarization vector $\overrightarrow{P}$ that points in opposite direction from $\overrightarrow{E}$ . More charging mechanisms are man-made accelerators, or nuclear plants with heavy radiation, and cosmic rays, and naturally radioactive elements present in the medium can produce very energetic electrons or positrons or neutros from any of these bombardment machines. These accelerated species, mainly, but not exclusively, electrons, will collide with and/or scatter from bigger molecules of the medium and produce more ionized species and more fast-moving electrons, and they have the possibility of producing avalanches of electrons and charged species [59, 60]. When the gas is mostly isolated and calm, the less the above charge-producing processes take place and the gas is mostly neutral.

In plasmas-charging mechanisms, charge separation include the same ones present in a gaseous system, but now there is an important manner conductivity, $\sigma$ . This conductivity is due to permanently present, replenishing, anions (−, include electrons, $O_2^{-}$ , other oxygen and hydrogen species) and cations (+, include positively charged species, molecules, small ice crystals, positrons, protons, etc.). They do not need to be permanent species, and they could be transient with an active production mechanism, for example, cosmic radiation as in the ionosphere. This conduction property provides the plasma with very particular electromagnetic properties that nonconducting gases do not have. In fact, the Maxwell equations valid for a plasma are as follows:

$$\mathrm{Div}.\overrightarrow{E}={\rho }_{\mathit{tot}}/{ϵ}_o$$ E1

$$\mathrm{Rot}\times \overrightarrow{E}=-\frac{\partial \overrightarrow{B}}{\partial t}$$ E2

$$\mathrm{Div}.\overrightarrow{B}=0$$ E3

$$\mathrm{Rot}\times \overrightarrow{H}={\mu }_o\left(\overrightarrow{J}f+\frac{\partial \overrightarrow{P}}{\partial t}+\mathrm{Rot}\times \overrightarrow{M}+\frac{ϵ\partial \overrightarrow{E}}{\partial t}\right)$$ E4

where ${ϵ}_o$ is the free space electrical permitivity, $ϵ$ is the medium permitivity that could easily be larger than ${ϵ}_o$ by a factor of 10, 20, or even more depending on the content of polarizable molecules in the medium. And the conductivity is given by $\sigma$ [35, 58].

4. In electrified clouds, charging mechanisms are complex

A cloud is a very complex hydrodynamic and electrodynamic system, and it is heterogeneously conformed. A wealth of phenomena actually occurs inside and on the surroundings of a simple or complex cloud. Just to mention some: they may aggregate looking as calmly cotton fields, but then again they can become thunderous and vertiginously electric active releasing thousands of joules of energy in their lightnings, they can show an halo-glow above them, they can produce stripes and Saint Elmo’s fire, their internal energy can accelerate electrons to relativistic velocities and runaway electrons, and then cascades of even more electrons in relativistic regimes, they can produce electromagnetic radiation fields, like bremsstrahlung, and so forth.

A cloud contains water, water vapor, a great amount of water microdroplets, little ice crystals, graupels, hail that normally are not present in neutral gases nor plasmas, atmospheric air with its regular composition, anionic species, free electrons, free protons, and the charged constituents of cosmic rays that traverse it. These elements carry a lot of the charge. Yet, for the most part, a cloud is nonconducting and its dielectric components make up the great proportion (> 93%) of its composition. We consider that our cloud under consideration contains a great number of minuscule pollutant particles. Pollutants in clouds are important because they contribute to attachment of free charge particles and to its dielectric character [6, 7].

A cloud is always moving and is in vertiginous motion in all directions, and many ice crystals, anions, cations, graupels, and water droplets are in the upward, downward, and turbulent air currents. Sometimes, it contains strong air currents forming divergences, curls and, of course, turbulence, and very importantly, also, vertical upward and downward hydrodynamic currents, and they can be, frequently, kilometers tall, as is the case of cumulonimbus. These constituents move, friction, collide energetically, and a multitude of charge separation and charge transfer processes take place and have been identified, quantified, and reported for decades now [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. A cloud is always immersed on the solar ultraviolet bath, the cosmic rays, natural radioactive decay earth’s products and the natural electric field present in the atmosphere at all altitudes.

5. Charging mechanisms: charge separation in electrified clouds

Clouds, being mainly nonconductive, can become electrified, and they sustain frequently very large quantities of electric charge, both positive, ${\rho }^{+}$ , and negative, ${\rho }^{-}$ . They become electrified-charged by motion, collisions, fracture and friction of ice little crystals, water droplets, graupel snow, air molecules, and so on with tiny ice crystals (they end up with charge in their surface). Another electrification mechanism is by experiencing atmospheric electric fields, just polarizing atoms and molecules, and even breaking down some of these polarized molecules/atoms. Radioactive products can produce charged species, X-rays, and gamma rays and can dislodge electrons from their atoms/molecules. Then, we have free electrons on the one hand and cations on the other hand. Ultraviolet-visible (UV-vis) radiation from the sun and cosmic gamma rays can contribute, daily, continuously, to the production of free charge in clouds.

Very recently [61], it has been found that water microdroplets show large electric fields just outside their surface and uncompensated charge resides on their surface [62], no collisions, nor radiation needed. It is a natural phenomenon that charge concentrates, uncompensated, on the surface of these water microdroplets. It is also known that small ice particles carry charge (or equivalently, radical species) on their surfaces [10, 11, 12]. The exact production processes of this charge on these water particles continue to be investigated at the present time. Pollutant molecules also show the ability to catch, trap, on their surfaces free charge, anions and/or cations, as already mentioned.

Now, in clouds, there might appear, for short periods of time, zones showing some conductivity, $\sigma$ , and this means charge mobility at least for short intervals of time appear concentrated stochastically here and there [63].

In summary, the charged elements in an electrified, nonconductive, cloud include graupel, large and small, water droplets, ice crystals, some in needle-like form, cloud drops, and ionic species normally present in the atmosphere, charged pollutant species, free electrons, free protons, and the charged constituents of cosmic rays and the charged products they produce within the domain of a cloud. And these charges are produced by several mechanisms that include, but are not limited to, noninductive processes such as collisions, breakage, fragmentation, and inductive processes such as polarization of molecules, redistribution of charge on the surface of water microdroplets, electric fields attracting and concentrating heavy charged-particles, and so on.

Hence, electrified clouds display definite regions that concentrate electric charge, positive in some regions, ${\rho }^{+}$ , and negative in others, ${\rho }^{-}$ . Then, most clouds posses a rather complicated charge structure, electric field lines emanate from ${\rho }^{+}$ regions (sources) and sink into ${\rho }^{-}$ regions (sinks). Hence, the charge structure, the electric fields, $\overrightarrow{E}$ , they produce with complex electric field line maps, and their dielectric character along with their conductivity (spots-like) property provide an electrified cloud with very particular electromagnetic properties that nonconducting gases do not have. Below, we go into some detail into these electromagnetic properties.

These charges can form long and wide charge distributions, some quasi-planar, others more volumetric, which locations are labeled as ${\overrightarrow{r}}^{´}$ , and these charges tend to accumulate in more or less well-defined regions inside the cloud; see Figure 4 for details of a typical electrified cloud structure.

Figure 4.

Essential electromagnetic features of a moving electrified cloud include quantitative features: (A) vector positions of its charge distributions; (B) distances between the “centers” of charged regions; (C) specific regions of accumulation of charge; and (D) position of observation measurement points, p close to the cloud and Q far away from the cloud. It shows some of its basic elements such as water microdroplets, small pieces of ice, dipolar molecules, and small conducting inclusions. The electromagnetic heterogeneity of the cloud and the accumulation of charge, ${\rho }^{+},{\rho }^{-}$ are enhanced. The location of an observation point (or measurement location) is ${\overrightarrow{r}}_p$ (green). The location of ${\rho }_1^{+}$ is given by the vector position ${\overrightarrow{r}}_1^{´}$ , the location of ${\rho }_2^{+}$ is given by the vector position ${\overrightarrow{r}}_2^{´}$ , and the location of ${\rho }_3^{-}$ is given by the vector position ${\overrightarrow{r}}_3^{´}$ , and the prime on $\overrightarrow{r}$ always indicates that we are dealing with the position of source charge, which in turn is the source of the electric field and the electric potential this cloud carries. Any conceivable charge structure in a cloud is denoted in this work by { ${\rho }^{+}$ , ${\rho }^{-}$ }^TOT , and it is an unknown of the system.

6. Distribution of free electric charge in clouds

The distribution of free electric charge in clouds is positive in one or more regions and negative in other regions. Figure 4 shows such charge and spatial configuration and vector distances. Our cloud, any cloud that can sustain free charge in it, in any possible charge structure configuration, can have distributions of charge, let say:

$$\rho \left(\overrightarrow{r}\right)=\left\{\begin{array}{c}{\rho }_1^{+}\left({\overrightarrow{r}}_1^{´}\right),{\overrightarrow{r}}_1^{´}\mathrm{ꞓ}\text{source charge Region}1\\ {\rho }_2^{+}\left({\overrightarrow{r}}_2^{´}\right),{\overrightarrow{r}}_2^{´}\mathrm{ꞓ}\text{source charge Region}2\\ {\rho }_3^{-}\left({\overrightarrow{r}}_3^{´}\right),{\overrightarrow{r}}_3^{´}\mathrm{ꞓ}\text{source charge Region}3\end{array}\right.$$ E5

similarly for ${\rho }_4^{-}$ ( ${\overrightarrow{r}}_4^{´}$ ), ${\rho }_5^{-}$ ( ${\overrightarrow{r}}_5^{´}$ ), and so on. The electric fields produced, per unit volumen of charge, are ${\overrightarrow{E}}_i$ = K ${\rho }_i$ ( $\overrightarrow{r}$ ) $\widehat{r}$ / $r^2$ , where k = 1/4 $\pi {ϵ}_0$ and $i$ = 1, 2, 3, 4, 5, …. If the distribution is discrete, ${\mathrm{\rho }}^{\pm }\left({\overrightarrow{r}}^{´}\right)$ = (1/V) ${\sum }_i^N{q}_i$ , or the corresponding integral if the density of charge is modeled continuosly. We locate the charges in the cloud and indicate a reference system as shown in Figure 4 . Then, the electric field per unit volume of charge is at the observation point:

$${\overrightarrow{r}}_2,\left\{\begin{array}{c}{\overrightarrow{E}}^{+}\left({\overrightarrow{r}}_p,t\right)=\mathrm{k}\left({\mathrm{\rho }}_1^{+}\left({{\overrightarrow{r}}^{´}}_1\right)/{R_{1p}}^2\right){\widehat{r}}_{12}\text{for}{{\mathrm{\rho }}_1}^{+}\left({{\overrightarrow{r}}_1}^{´}\right)\\ {\overrightarrow{E}}^{-}\left({\overrightarrow{r}}_p,t\right)=\mathrm{k}\left({{\mathrm{\rho }}_2}^{-}\left({{\overrightarrow{r}}_2}^{´}\right)/{R_{2p}}^2\right){\widehat{r}}_{2p}\text{for}{{\mathrm{\rho }}^{-}}_2\left({\overrightarrow{r}}^{´}\right)\\ {\overrightarrow{E}}^{+}\left({\overrightarrow{r}}_p,t\right)=\mathrm{k}\left({\mathrm{\rho }}_3^{+}\left({{\overrightarrow{r}}^{´}}_3\right)/{R_{3p}}^2\right){\widehat{r}}_{3p},\text{for}{{\mathrm{\rho }}_3}^{+}\left({{\overrightarrow{r}}_3}^{´}\right)\end{array}\right.$$ E6

| ${\overrightarrow{R}}_{\mathit{ip}}$ | = $R_{\mathit{ip}}$ = ${\overrightarrow{r}}_p$ − ${\overrightarrow{r}}_i$ , and ${\overrightarrow{r}}_1$ = ${\overrightarrow{r}}^{´}$ is the source/sink region where ${\rho }_{\mathit{\text{free}}}^{\pm }$ is located and ${\overrightarrow{r}}_p$ are positions of the observer (where the field has been evaluated). The total field at the observation point “p” is the sum of the fields in (6).

Partial information on ρ ( ${\overrightarrow{r}}^{´}$ ) would produce only partial information on $\overrightarrow{E}$ , its divergence, and its electric energy, W $\propto {\left|\overrightarrow{E}\right|}^2,$ and an electromagnetic analysis is incomplete. The measurements of electric fields have given valuable information on the average values and their quite large variations-fluctuations. Conventional breakdown field values (3000 kv/m) are known as the breakeven field (202 kv/m) and the runaway field (280 kv/m) values [10, 12, 25].

We must note that ρ( ${\overrightarrow{\mathrm{r}}}^{´}$ ) in this work always represent an already electrified, or charged region in a cloud irrespective of its extent, its particular mathematical form, being it a discrete conglomerate of charge, or a “modeled” continuous distribution of charge, ${\mathrm{\rho }}^{+}$ indicates a distribution, or extension, or structure of positive charge from which divergent electric field lines emanate, and ${\rho }^{-}$ indicates a distribution of negative charge in which the electric field lines converge, sink. Any conceivable charge structure in a cloud is denoted in this work by { ${\rho }^{+}$ , ${\rho }^{-}$ }^TOT, and it is an unknown of the cloud system. It is extremely hard to determine ρ ( ${\overrightarrow{r}}^{´}$ ). Observations for decades have given only partial knowledge of it. Nowadays, a fuller, more detailed determination of ρ ( ${\overrightarrow{r}}^{´},t$ ) and of the electric field, $\overrightarrow{E}({\overrightarrow{r}}_p$ , t), it produces at the observation position ${\overrightarrow{r}}_p$ continue to be at the top of the electrified cloud-lightning research. Another important electromagnetic property of clouds to be considered is its dielectric character, and we go into it, next.

Clouds contain dielectrics. Some of the molecules and atoms that compose a cloud are dielectric (some very strong dielectric and some weak dielectric), some other components are already electrified, and others can be electrified by noninductive mechanisms. Let us look at these dielectric molecules (which produce bound charges) with a bit more detail. Let us think just on water molecules, water microdroplets, pollutant particles, and little aggregates of $H_{2}O$ ). ρ( ${\overrightarrow{r}}^{´}$ ) will apply an electric field $\overrightarrow{E}({\overrightarrow{r}}_d,t)$ to these molecules at their position, ${\overrightarrow{r}}_d$ , at time t and produce a separation of the centers of positive charge from the center of negative charge, producing an atomic/molecular dipole moment given by $\overrightarrow{p}$ = q $\overrightarrow{\ell }$ . For thousands and thousands of these dielectric molecules lumped, packed, together and experiencing the same electric field $\overrightarrow{E}$ , a macroscopic, total, electric moment is formed $\overrightarrow{\mathcal{P}}$ = ${\sum }_i^n{\overrightarrow{p}}_i,\mathit{\text{with}}n\ge {10}^4$ or much more greater, reaching nano-, micro-, millimetric, and/or larger dimensions. Then, a macroscopic density of electric moment is constructed, $\overrightarrow{P}$ = $\overrightarrow{\mathcal{P}}$ /vol [7, 8, 9]. When $\overrightarrow{P}$ is not distributed uniformly in the volume of the aggregate cloud, which is a likely scenario, then $\overrightarrow{P}$ becomes divergent and Div $·\overrightarrow{P}\ne$ 0; hence, a volumetric bound charge, ${\rho }_b$ = Div $·\overrightarrow{P}$ , appears. In addition, a surface-bound charge, $\overrightarrow{P}·\widehat{n}$ |_surface= ${\sigma }_b$ , appear on all the boundaries of the dielectric parts of the molecule, or aggregate, and this can happen also to water microdroplets [62]. ${\rho }_b$ and ${\sigma }_b$ can only be determined, theoretically, when the polarization vector $\overrightarrow{P}$ is known. Experimentally, the measurement of the electric field is just outside the dielectric molecule, and aggregate or microwater droplet will allow us to determine $\overrightarrow{P}$ and the volumetric and surface distributions of the bound charges. This knowledge will help us to understand their electric behavior in front of other charges [7, 8] and/or immersed in electric fields.

In real systems such as clouds, thunderclouds, plasmas, dielectrics in clouds, or for electronics, to know the distribution of free charge is challenging by itself, to know the distribution of bound charges is even more challenging, then we would need to calculate or evaluate both, the electric field vector, $\overrightarrow{E}$ , and the macroscopic polarization vector $\overrightarrow{P}$ . However, using the first Maxwell equation in the form, Div $·\overrightarrow{E}$ = ${\rho }_{\mathit{\text{free}}}$ / $ϵ$ , we need to know in advance only ${\rho }_f$ (and not ${\rho }_b$ , this is a relief), the total electric permitivity, $ϵ$ = ${ϵ}_o{ϵ}_r$ = ${ϵ}_o(1+{\chi }_E$ ) of the medium, then the vector field $\overrightarrow{D}$ = $ϵ\overrightarrow{E}$ is evaluated and from it, $\overrightarrow{E}$ = $\overrightarrow{D}$ / $ϵ$ . The algebraic process is simple, and it applies equally well to a molecule, an aggregate of dielectrics, a cloud region, or a complete cloud. When the dielectric medium is sufficiently diluted and it extends for some micrometers or for kilometers, then $ϵ$ could approach ${ϵ}_0$ , and we can consider to be working with a standard clear air (in the lab, or outdoors with unpolluted air, etc.) or as when we have free charges in vacuum. Otherwise, some consideration should be paid to the polarizability of the medium since $\overrightarrow{P}$ = ${ϵ}_o{\chi }_E\overrightarrow{E}$ . For linear and homogeneous dielectric materials, the three vectors go along the same directions, but when the dielectrics are not linear, $\overrightarrow{P}$ and $\overrightarrow{E}$ are no longer colinear and the response becomes more elaborate [7, 8, 9]. The existence of the vector $\overrightarrow{P}$ brings about an electric field produced by $\overrightarrow{p}$ = q $\overrightarrow{\ell }$ , that is, $\overrightarrow{E}$ p that opposes in direction to the original $\overrightarrow{E}$ (produced by free charges, ${\rho }_f$ ); hence, the total electric field in the midst of the dielectric portions of the system is reduced from ${\overrightarrow{E}}_i$ to, ${\overrightarrow{E}}_{\mathit{TOT}}$ = ${\overrightarrow{E}}_i$ – ${\overrightarrow{E}}_p$ and breakdown of the dielectric constant would be more difficult to achieve, and the transport of some free charge and free currents, ${\overrightarrow{J}}_f$ = $\frac{1}{A}\frac{\mathit{dq}}{\mathit{dt}}\widehat{n}$ , through the dielectric regions would be more difficult to establish and of less magnitude. Hence, let us suppose we have a cloud region with free charge density ${\rho }^{+}$ facing another free charge density ${\rho }^{-}$ with dielectric molecules, aggregates between them, but not totally filling the space in between; hence, there will be small regions, filaments, and islands, of nondielectric matter. The zones more clearly dielectrically occupying most of the cloud volume (let us say <90%), even covering space in between the charged regions, would leave only small regions, islands, and filamentary spaces with $\overrightarrow{p}\to$ 0 approaching the free space condition in which $\overrightarrow{E}$ p $\to$ 0, and the value of electric field is ${\overrightarrow{E}}_i$ not reduced by $\overrightarrow{E}p$ . So, these nondielectric spaces would be more favorable for electric currents, ${\overrightarrow{J}}_f$ , to form and flow. It is, already, very well known that dielectric matter is not a favorable medium to sustain electric currents [7, 8, 9]. If polarizable molecules in the cloud are moving, as they are, the possible trajectories of electric currents also change.

So, in a cloud the total charge is, ${\mathrm{\rho }}_{\mathrm{tot}}$ , and it is composed of free charge produced by ionization, application of an electric field “ ${\overrightarrow{\mathrm{E}}}_{\mathrm{i}}$ ”, vertiginous vertical upward/downward hydrodynamic currents, friction, collisions, cosmic rays, solar radiation, etc., and bound charge produced by polarization, $\overrightarrow{P}$ = ${ϵ}_o{\chi }_E\overrightarrow{E}$ , of molecules in the medium, provoked by the applied ${\overrightarrow{E}}_i$ field.

So, for the above, we want to consider free charge only: if we want to consider the structure, distribution, of free charge only, that is { ${\rho }^{+}$ , ${\rho }^{-}$ }^free $\equiv$ ${\rho }_f$ , this quantity is more amenable to work with Maxwell equations. For example, Gauss Law $\nabla ·\overrightarrow{E}$ = ${\rho }_f$ / $ϵ$ can be put in terms of ${\rho }_f$ , and then we will be dealing with sources or sinks of electric field lines, but only of free charge. And if not known, it can be modeled and/or inferred (partially) from measurements.

If, in addition, the cloud contains no magnetizable elements, then the magnetization vector, $\overrightarrow{M}$ , is zero, and if the Maxwell displacement current is very small compared to the free currents in our turbulent thundercloud, then we neglect the term $\frac{ϵ\partial \overrightarrow{E}}{\partial t}$ in (4) and we are left with the reduced Maxwell equations, (7)–(10) for this kind of electrified cloud with dielectric material in it and some free charge able to move, and considering that a cloud is a very heterogeneous structure changing very rapidly in time:

$$\mathrm{Div}·\overrightarrow{E}={\rho }_f/ϵ$$ E7

$$\mathrm{Rot}\times \overrightarrow{E}=-\frac{\partial \overrightarrow{B}}{\partial t}$$ E8

$$\mathrm{Div}·\overrightarrow{B}=0$$ E9

$$\mathrm{Rot}\times \overrightarrow{B}=\left(\overrightarrow{J}f+\frac{\partial \overrightarrow{P}}{\partial t}\right)$$ E10

The constitutive equations are $\overrightarrow{P}$ = ( $ϵ-{ϵ}_o$ ) $\overrightarrow{E}$ , $\overrightarrow{D}$ = $ϵ\overrightarrow{E}$ , and $\overrightarrow{J}$ = $\sigma \overrightarrow{E}$ , and conductivity in the regions that sustain plasma like matter could be modeled as before [33]:

$$\sigma =e^2\left(\raisebox{1ex}{$N^{+}{t}^{+}$}\!\left/ \!\raisebox{-1ex}{$m^{+}$}\right.+\raisebox{1ex}{$N^{-}{t}^{-}$}\!\left/ \!\raisebox{-1ex}{$m^{-}$}\right.\right)$$ E11

where all quantities, for our very dynamic cloud, are very strong functions of space and time. The particular values of the parameters and charge distributions and electric fields at time $t_1$ in a locale A are not necessarily the same at the same locale at time $t_2$ > $t_1$ , with $t_2$ − $t_1\ll$ 1 millisecond, and this leaves the theoretical treatment with Maxwell equations as “snapshots” of the resulting fields, currents, etc., as quasistatic frames as time evolves.

7. Clouds interacting with more clouds and its environment

A cloud with a typical charge structure, electric field configuration, with its dielectric and conducting character and in the presence of other charged clouds and electromagnetic environment was schematically shown in Figure 2 . The distribution of free charge with positive charge on the upper part of the cloud ${\rho }_1^{+}$ is the source of divergent electric field lines. Negative, extended, distribution of charge in the middle section of the cloud, ${\rho }_2^{-}$ , is a large, convergent sink of electric field lines, and another smaller distribution of positive charge in the lower part of the cloud and somewhat toward one side, ${\rho }_3^{+}$ , left, or right, were located in a reference frame, and the electric fields they produced were given also in Figure 4 . Metallic towers, an airplane flying nearby, a mountain, trees, ground, and lakes are also considered. The central cloud could be very tall and active with wind divergences and curls, upward convection currents, then downward and turbulence, and cumulonimbus clouds are just one example.

This cloud has also other neighboring clouds, having at least another electrified cloud, ${\rho }_{\mathit{\text{near}}}^{\pm }$ nearby that can be electrified (carrying even more charge and producing more electric fields that move some other free charge) and energetic exchange of their charge can occur, sometimes accompanied with lightnings. Below the continental clouds, there is always ground, which is for the most part charged positively. Above the cloud there is another charged volumetric region and with conductivity, the ionosphere, $\sigma$ , just as mentioned before.

More about clouds interacting with clouds: A cloud is not only a reservoir of great amounts of water, ice, water vapor, atmospheric gases, and electrical charge-producing electric fields. It is a tremendously dynamic structure that contains megas and megas of Joules of hydrodynamic energy and gigas and gigas of Joules of electromagnetic energy. The energy, W, of a configuration of charges in an electrified cloud with n charges located at, mutual, distances $r_{\mathit{ij}}\approx$ $r_{\mathit{jk}}$ in a quasistatic ensemble is W = $k{\sum }_{i<j}{q}_i{q}_j/r_{\mathit{ij}}$ = $q_N{V}_{N-1}$ with its electric potencial $V_{N-1}$ = K $(q_1/r_{1N}+q_2/r_{2N}$ + $\frac{q_3}{r_{3N}}+\frac{q_4}{r_{4N}}+\dots q_{N-1}/r_{\left(N-1\right)N}).$ N could reach trillions and trillions of charged particles. Such energy is released and channeled in accord with the electromagnetic environment, EME. EME is composed of ground, mountains, trees, other electrified clouds, dielectric media, sea surface, ionosphere, temperatures, pressures, pollutants (varying with height), planes flying near or inside clouds, and metallic structures (be them tall, small, tubular, flat, and so forth). For an electromagnetic study of an electrified cloud, it is indispensable to know some physical parameters of its EME. For example, just to mention a couple of examples, the ground is a huge reservoir of positive and negative charge probably, mainly, due to the great amount of minerals, nutrients, dirt, and water in it. The ionosphere is a fully conducting medium with a lot of free charge (mainly electrons) in it. An adequate electromagnetic treatment of an electrified cloud should include its EME. The standard atmosphere has also plenty of electric charge and not only in the ionosphere that is up high several kilometers (around 90 km) above ground, and this air region is continuously exposed to cosmic radiation, the sun’s ultraviolet radiation, and the electric fields of electrified clouds. Hence, molecules and atoms get energetically excited, “broken down”, and ionized, captions and anions are formed, and free charged particles, mainly electrons and protons, are produced and constitute the charge in that gaseous medium.

8. Two cloud charge structure models that can produce high electric fields in their vicinity just momentarily and locally

In this section, we construct two simple cloud charge structure models that can produce high electric fields in their vicinity just momentarily and locally, schematically presented in Figures 5 and 6 . Within these, a discharge process is being cooked (build) up. Many charge densities-structures in clouds ${\rho }_1^{+}$ , ${\rho }_2^{-}$ , ${\rho }_3^{++}$ , ${\rho }_4^{-}$ , ${\rho }_5^{--}$ , etc., are possible. Moreover, clouds are, by no means static structures, they move, change form in their insides and outsides, in question of microseconds, by internal currents (up and down hot-cold streams) and because of the wind. This makes the clouds and their charge densities in very dynamic structures.

Figure 5.

Pointy charge structures, ${\check{\rho }}^{+}$ , and ${\check{\rho }}^{-}$ developed in a moving cloud. Some important redistribution of free and bound charge starts to take place. Free charge, $e^{-}$ , $p^{+}$ , anions, and/or cations already in the body of the cloud will get strongly attracted to these force field lines and will move accordingly, as $\overrightarrow{F}$ = q $\overrightarrow{E}$ = m $\overrightarrow{a}$ , and then they will accelerate as, $\overrightarrow{a}$ = q $\overrightarrow{E}$ /m. they continue to accelerate as long as collisions do not stop them. This regime is called enlarged-effective mean free path, $\ell$ .

Figure 6.

A momentarily intensified electric field. A scenario in which a kind of capacitor is formed for which the ${\stackrel{̿}{\rho }}^{+}$ distribution gets much more closer to ${\stackrel{̿}{\rho }}^{-}$ due to wind forces. $\overrightarrow{E}$ increases between them as [1/r]² , the electric field in between is the sum of the field produced by ${\stackrel{̿}{\rho }}^{+}$ and the field produced by ${\stackrel{̿}{\rho }}^{-}$ , and charged species are attracted to this $\overrightarrow{\stackrel{̿}{E}}$ field and then accelerated.

Within this dynamics, let us suppose that the charge densities move such that “pointed” distributions of charge, ${\check{\rho }}^{+}$ , ${\check{\rho }}^{-}$ appear (and dissappear) in a region $a$ of a cloud, then hot spots of large electric fields, $\overrightarrow{\check{\mathrm{E}}}$ , and large divergences, DIV $·\overrightarrow{\check{\mathrm{E}}}$ , emerge for just a fraction of a second. Also, in another close by cloud or another region b in the same cloud, other charge distributions, ${\stackrel{=}{\rho }}^{+}$ , and ${\stackrel{=}{\rho }}^{-}$ , can get closer, kind of capacitor type of structure, due, again, to strong winds and rotationals and upliftings (down-liftings), creating, again, a zone with very high electric field, $\overrightarrow{\stackrel{=}{E}}$ . Both scenarios are shown in Figures 5 and 6 , respectively. Other charge structures are possible that elicit, locally and momentarily, high electric fields. The creation of strong electric fields within electrified clouds is a pre-requisite for discharging and formation of leaders, streamers, and lightning, the appearance of nonrelativistic and relativistic moving electrons, hence, the rapid increase of conductivity, emission of gamma and X-rays, and so on. We see in more detail these two possibilities now.

We envision here, case A) the emergence of electric field hot spots as a result of, by chance, the formation of pointy charge “processes”, ${\check{\rho }}^{+}$ , and ${\check{\rho }}^{-}$ , coming out from some charged regions, as shown in Figure 5 . This makes $\overrightarrow{\check{\mathrm{E}}}$ larger close to them and DIV $·\overrightarrow{\check{\mathrm{E}}}$ also larger; hence, more field lines get out of the positive “pointy” charge density and close to it more field lines cross a unit area.

More field lines enter into ${\check{\rho }}^{-}$ , sink, and close to them more field lines converge, attracting more positive mobile charges. This, in turn, increases the potential difference between these regions and regions of less ${\rho }^{\pm }$ . The electric force that a single charge q of mass m can experience in those electric field hot spot regions gets, consequently, larger, and if $\overrightarrow{F}$ = q $\overrightarrow{\check{\mathrm{E}}}$ , its acceleration is $\overrightarrow{a}$ = q $\overrightarrow{\check{\mathrm{E}}}$ /m. If we disregard collisions for the moment and consider an enlarged-effective mean free path, $\ell ,$ for these moving charges, then there are not, by definition, important damping processes that stop this increasingly accelerated motion. We take an enlarged effective mean free path as a composed trajectory in which collisions and gazing scatterings do not, appreciably, affect greatly the acceleration, nor the kinetic energy of the particle and the continuous work of the electric field, W = q $\overrightarrow{E}·$ d $\overrightarrow{\ell }$ , keeps moving the charge.

In scenario B, ${\stackrel{=}{\mathrm{\rho }}}^{+}$ gets much closer to ${\stackrel{=}{\mathrm{\rho }}}^{-}$ as pictured in Figure 6 . A kind of capacitor is formed for which the ${\stackrel{=}{\rho }}^{+}$ distribution gets much closer to ${\stackrel{=}{\rho }}^{-}$ due to wind forces that push one to the other. Short duration fluctuations of R $({\rho }^{+},{\rho }^{-}$ ) in Figure 6 increases $\overrightarrow{E}$ between them as [1/R( ${\rho }^{+},{\rho }^{-}$ )]², the electric field in between is the sum of the field produced by ${\stackrel{=}{\rho }}^{+}$ and the field produced by ${\stackrel{=}{\rho }}^{-}$ , it becomes very high and then again, it attracts free charge that happens to be in that space, or close by at that moment, t*. The Coulomb force they feel is, again, $\overrightarrow{F}$ = q $\overrightarrow{\stackrel{=}{E}},$ and its acceleration is $\overrightarrow{a}$ = q $\overrightarrow{\stackrel{=}{E}}$ /m. They accelerate with no damping inside the elongated effective mean free path, $\ell$ . This increasingly accelerated motion can continue without being slowed down in this regime. We propose that the electric work made by the $\overrightarrow{\stackrel{=}{E}}$ field, W = q $\overrightarrow{\stackrel{=}{E}}·$ d $\overrightarrow{\ell }$ is large enough to keep some, but not all, electrons accelerated, these electrons will become runaway electrons in the next stage, few microseconds ahead.

But, it just so happens that this motion can also be maintained and even increased by positive feedback in which the loss, by collisions, of moving energy is compensated by gain of energy in other elastic collisions in which more massive particles transfer linear momentum and energy to these lighter charged particles. Both scenarios can contribute greatly the cosmic rays, and apparently, more specifically their secondary products at lower altitudes, well below the lower layer of the ionosphere, well into the stratosphere.

If this intense electric field window, in both scenarios, last for some microseconds, a lot of charged particles can be moved on along these force field lines and can reach distances of a fraction of a kilometer or more. Then, we can talk about electric current flowing down to lower electric potential or of positive charge traveling, current, J+, toward the negative pole or “electrode,” and/or of electrons traveling toward the positive pole or “electrode,” conforming a negative current, J-. Of course, both kinds of currents can happen simultaneously, or sequentially. The end point could simply be the air, or the ground, or a metallic structure, a tree, even the ionosphere. We see that the liberation of electric energy W is quite large, since part of the charge structure is being disassembled, let’s say, just as an order of magnitude illustration, ${10}^{12}$ charges (electron equivalents) are being “send” (ideally) to infinity (toward electric potential zero), then W will be ${10}^{12}$ times the work done to remove one single charge from the cloud charge structure to infinity [7, 8, 9]. Let us suppose this unit work is about 0.23 nano Joule, then in order to dismantle ${10}^{12}$ particles from the ensemble, the energy spend would be (0.23× ${10}^{-9}$ )× ${10}^{12}$ Joule = 2.3x ${10}^3$ Joule = 2.3 thousand Joules, that is quite a bit of energy. Where all this energy goes to?

The intense field values reached, in both scenarios, just before the beginning of these J+ and J- currents, and the expenditure of the above-mentioned electric energy W are defined as the “at the verge of discharge electric fields,” and we label them as $\overrightarrow{\mathit{Ę}}$ . Then, we are here with the maximum source and sink of electric field lines in both cases. We want to concentrate more on scenario B. Notice that here $\overrightarrow{\underset{̨}{E}}$ = ${\overrightarrow{E}}^{+}$ + ${\overrightarrow{E}}^{-}$ in the in-between zone. Since dielectrics reduce the total electric field in spaces where $\overrightarrow{P}\ne$ 0, as: ${\overrightarrow{E}}_{\mathit{TOT}}$ = $\overrightarrow{\mathit{Ę}}$ − $\overrightarrow{E}p$ , then within the dielectric matter $\overrightarrow{E}$ is not increasing toward $\overrightarrow{\underset{̨}{E}}$ in there. However, if the dielectric matter in the cloud is inhomogeneously distributed and $\overrightarrow{P}$ varies greatly in the interior of the cloud, v. gr., covering the space with big and small dielectric patches, then some regions in between patches are almost $\overrightarrow{P}$ -free slender-tortuous elongated regions in which $ϵ\to$ ${ϵ}_0$ and the electric field, still, is $\overrightarrow{\underset{̨}{E}}$ = ${\overrightarrow{E}}^{+}$ + ${\overrightarrow{E}}^{-}$ with no ${\overrightarrow{E}}_p$ reduction. These two scenarios pinpoint very specific local-cloud “spaces” that fulfill particular conditions in which $\overrightarrow{E}$ can reach the “on the verge of discharge field,” $\overrightarrow{\underset{̨}{E}}$ . Two more fundamental ingredients for discharge to occur are as follows: (A) the presence of a lot of free light charged particles, mainly electrons, protons, and small ions, to be transported, and (B) an increased conductivity in these locations at time t*, both go hand on hand.

Once we are right at the verge of discharge, in these locations, with an increased conductivity, and a corresponding resistivity able to produce a lot of heat from Joule effect, what happens next?? well, if in the following few microseconds charge ${\rho }_b^{+}\left({\overrightarrow{r}}_b^{´}\right)$ , and ${\rho }_d^{-}\left({\overrightarrow{r}}_d^{´}\right)$ drops and/or gets redistributed again, or gets separated, larger R values, and then $\overrightarrow{E}$ diminishes quickly and potential discharge is inhibited, as when a huge sneeze gets frustrated! and the cloud continues in the charged state. Otherwise, the discharge process, really, takes off. The above makes us think of the many inhibited discharges that are hold-on in the last microseconds, because one of the above condition ceased to exist.

9. Then the discharge “Proceeds”

Discharge occurs triggered by not-so-well-understood mechanisms, but some likely mechanisms are as follows: (A) simply large amplitude variations of electric fields; (B) cosmic ray energetic events triggering the production of bare, accelerated electrons; (C) energetic ionizations by atomic/molecular collisions; (D) breakdown of the dielectric constant of cloud-air at some locations nearby, or between ${\rho }^{+}$ and ${\rho }^{-}$ cloud charged regions; and (E) increase of $\overrightarrow{E}$ intensity due to a sudden increase, surge, of charge in ${\rho }^{+}$ and ${\rho }^{-}$ regions, since $\left|\overrightarrow{E}\right|\propto {\mathrm{\rho }}^{\pm }$ . But, it is also posible to increase $\overrightarrow{E}$ by diminishing the distances, | $\overrightarrow{r}$ | = $r$ = ${\overrightarrow{r}}_2$ − ${\overrightarrow{r}}_1$ , among the charged regions. We go into more detail here. (F) E increase due to charge distance decrease. Movement of ${\rho }^{+}$ ( ${\rho }^{-}$ ) toward ${\rho }^{-}$ ( ${\rho }^{+-}$ ) as in scenario B above due to a strong air dynamics fluctuation, turbulence, rotations, convection, so that, short duration fluctuations of R $({\rho }^{+},{\rho }^{-}$ ) increases $\overrightarrow{E}$ in-between as 1/[R( ${\rho }^{+},{\rho }^{-}$ )]², so, if R goes from $R_0$ to (1/4) $R_0$ then E goes from $E_0$ to 16 $E_0$ and if $E_0$ = 120 kvolts/m, then the increased field would be now $\underset{̨}{E}$ = 1920 kvolts/m, and this momentary field is quite enough to produce the breaking of some polarizable molecules. Some ions, cations, and free-unbound electrons are produced and will immediately get accelerated toward/away from the closest charged region ${\rho }^{\pm }$ of opposite sign. Even in the case of just reducing the distance by half, if R goes from $R_0$ to (1/2) $R_0$ , then E goes from $E_0$ to 4 $E_0$ and if $E_0$ = 120 kvolts/m, then the increased field would be now $\underset{̨}{E}$ = 480 kvolts/m. This field is larger than the 320 kvolts/m of the reported breakdown field [25]. One of the objectives of this work has been to construct plausible scenarios, electromagnetic conditions, on which large intensity electric fields $\overrightarrow{\underset{̨}{\mathrm{E}}}$ , with durations of at least some micro-, or milliseconds, can develop in the in-between and/or around the charged cloud regions, be them on the upper/center/lower part of the cloud, or toward the sideways, as shown in Figures 5 and 6 .

In Figure 7A , we have imagined five charged particles at five different locations in this electrified environment (well within the cloud), a free electron, a positron, a positive ion, a proton, and a polarized molecule, experiencing the presence of the electric field $\overrightarrow{E}$ , and the vector distances should properly be measured from the coordinate system arbitrarily situated at $ℴ$ . All of these charges experience Coulomb force at their location. It should be pointed out that not always it is trivial (or easy) to calculate such a field $\overrightarrow{E}$ . In fact, it continues to be an open problem in thundercloud systems.

Figure 7.

We have imagined five electrical objects located at seven different positions in this electrified environment. They all experience the presence of the electric field $\overrightarrow{E}$ ; hence, a coulomb force depends on their location P1, P2, …, P5, well inside the cloud: (A) original charge configuration with typical distributions of charge and distances; (B) the charge distributions have been moved to get some closer and others farther away, the electric field and coulomb forces change accordingly. A zone of higher ${\overrightarrow{E}}^{+}$ field is denoted $B^{+}$ and another region with weakened ${\overrightarrow{E}}^{-}$ field is denoted $B^{-}$ .

Now, we will do a “Thought” experiment: If the relative distance of the field sources and the charged particles is changed, several of the forces experienced vary even drastically, the force field lines get distorted (they follow different paths now), and the electric energies will change correspondingly. The electric potential lines are also modified, but it is always obeying $\overrightarrow{E}·$ d ${\overrightarrow{\ell }}_{\mathit{pot}}$ = 0, as shown in Figure 7B . Above shows distances modified, and electric field and electric potentials have changed with respect to panel A. Divergences and rotationals in panel A have changed, and now stronger divergences appear and other rotationals appear in panel B. For example, see the regions marked $B^{+}$ and $B^{-}$ with corresponding electric fields ${\overrightarrow{E}}^{+}$ and ${\overrightarrow{E}}^{-}$ and electric potentials $V^{+}$ and $V^{-}$ . These spatial changes bring about several electromagnetic consequences, some are as follows:

Region $B^{+}$ / $B^{-}$ with more/less force field lines concentrate on more/less electric energy (measured in Jules) per unit volume since W/vol = ( $\raisebox{1ex}{$ϵ$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ ) $E^2$ and since $E_{B+}$ >> $E_{B-}$ then $W_{B+}$ >> $W_{B-}$ . In fact, since E $\alpha$ (1/ $r^2$ ), then W $\alpha$ (1/ $r^4$ ), and the electric energy is a very strong function of the distance between charged bodies. If distance R changes just by half, R $\to$ R/2, then energy density will increase 16 times in region $B^{+}$ .

1. Consider two identical charges $\stackrel{\sim }{q}$ and $\overline{q}$ (two electrons, two positrons, two protons, two anions, and two cations) of equal mass, $\stackrel{\sim }{m}$ . We put $\stackrel{\sim }{q}$ in region $B^{+}$ at a point p at half the distance $\stackrel{\sim }{R}$ from the sources, and the identical charge $\overline{q}$ is placed in region $B^{-}$ , at a similar distance from its sources. $\stackrel{\sim }{q}$ experiences a much stronger Coulomb force, ${\overrightarrow{F}}^{+}$ = $\stackrel{\sim }{q}{\overrightarrow{E}}^{+}$ than the identical charge $\overline{q}$ placed in region $B^{-}$ , where ${\overrightarrow{F}}^{-}$ = $\overline{q}{\overrightarrow{E}}^{-}$ . The histories of subsequent motion of $\stackrel{\sim }{q}$ and of $\overline{q}$ will be very different. $\overline{q}$ could eventually recombine, and $\stackrel{\sim }{q}$ could become relativistic and could produce an avalanche of more charged particles.

2. If allowed to move, both charges will follow the corresponding field line directions, and if allowed to move for a time interval $\mathrm{\Delta }t$ without collisions or damping, then they will be accelerated by ${\overrightarrow{F}}^{+}$ = $\stackrel{\sim }{m}{\overrightarrow{a}}^{+}$ = $\stackrel{\sim }{q}{\overrightarrow{E}}^{+}$ in region $B^{+},$ and by ${\overrightarrow{F}}^{-}=\stackrel{\sim }{m}{\overrightarrow{\mathrm{a}}}^{-}$ in $B^{-}$ region. Then if $E^{+}$ = $\eta E^{-}$ with, for example, $\eta$ = 2, 10, 100, 1000 and so on (we are modeling, arbitrarily, intense electric fields, $E^{+}$ , here twice, five times, one order of magnitude, two orders of magnitude, three orders of magnitude larger than the weaker field, $E^{-}$ ), then ${\mathit{ma}}^{+}$ = $\stackrel{\sim }{q}{E}^{+}$ = $\stackrel{\sim }{q}\eta E^{-}$ = $\eta$ ( $\stackrel{\sim }{q}{E}^{-}$ ) = $\eta$ m $a^{-}$ ; hence $a^{+}$ = $\eta a^{-}$ , so the same charged particle placed in region $B^{+}$ in a stronger electric field will accelerate $\eta$ times more than if put in region $B^{-}$ with the weaker electric field $E^{-}$ , $E^{+}$ / $E^{-}$ = $\eta$ = $a^{+}$ / $a^{-}$ . Moreover, | ${\overrightarrow{F}}^{+}\mid$ / $\mid {\overrightarrow{F}}^{-}$ |= $\eta$ . This helps to understand the existence of slow-moving charges and fast-moving charges in the same electrified cloud. This applies equally well to light and heavy charged particles.

3. When known the intensity of the electric fields $E^{\pm }$ , then $a^{\pm }$ = ( $\stackrel{\sim }{q}/\stackrel{\sim }{m}$ ) $E^{\pm }$ can also be determined if the mass of the particle is known.

4. If we consider the enlarged-effective mean free path, $\ell ,$ (as defined above) with no loss of energy, no friction, as if flying in free space, and if $\ell$ is about one meter, tens, or hundreds of meters, then charges will reach classical velocities of $v^2$ = $v_0^2$ + 2 $a\ell$ , or $v$ = $v_0$ + $a$ t. As a manner of numerical example, let us suppose an electron starts this accelerated motion inside an electric field E of just 1 kvolts/m with an initial velocity, $v_0$ = 1000 m/sec or less, and $\ell$ =(0.1 Km)L, where L can be less or greater than 1 is a convenient adjusting distance parameter of the mean free path, then the acceleration $a$ = qE/m is going to be (1.6 × ${10}^{-19}$ Coulomb) (1 kvolts/m)/(9.11 × ${10}^{-31}$ Kg), then $a$ = 1.756 × ${10}^{14}$ [m/ ${\mathit{sec}}^2$ ].

   Then $v^2$ = (1000 m/sec)² + 2x1.756 × ${10}^{14}$ [m/ ${\mathit{sec}}^2$ ]. (100 L) [m], taking square root we obtain, $v\approx$ 1.87 $\sqrt{L}$ × ${10}^8$ m/sec, and this velocity is a fraction of the velocity of light, (1.87 $\sqrt{L}$ × ${10}^8$ )/(3x ${10}^8$ m/sec) = 0.624 $\sqrt{L}$ , impresive! It should be noted that in this calculation the adjusting distance parameter L is just equal to 1, and the initial velocity with which the electron enters the region of $\overrightarrow{E}$ is a very small contribution and can be safely neglected. This classical calculation of acceleration and velocity is not correct for relativistic velocities but helps to give an idea of the tremendous accelerations and velocities that electrons can reach under these circumstances.

5. The above means also that a great number of electrons ( $\ge {10}^{10}$ ) with small, medium, and high velocities just drifting (going by in any direction) in the vicinity of $\overrightarrow{E}$ is going to be accelerated to reach velocities at around 0.624 c, and many slow-moving electrons just drifting by easily fullfil this initial condition. Since $v$ = $v_i$ + $a$ t, then the time to reach the final velocity (in this elongated effective mean free path length) is t = ( $v$ − $v_i$ )/ $a$ = 1.87 $\sqrt{L}$ × ${10}^8$ − 1000) (m/sec)/(1.756 × ${10}^{14})$ [m/sec2] $\cong$ 1.065 $\sqrt{L}\mathit{\text{μsec}}$ , a very short time!. We can envision many electrons drifting by in the region of $\overrightarrow{E}$ of just 1000 volts/m and get this acceleration and reach a velocity 0.6234c in just a little more than a microsecond. Here, L continues to be equal to 1.

6. The gain in kinetic energy in this accelerated motion along $\ell$ is $K$ – $K_0$ = ½ ${\mathit{mV}}^2$ − ½ $m{V_0}^2\cong$ ½ ${\mathit{mV}}^2$ since ${V_0}^2$ << $V$ ², then $∆$ k $\cong$ ½ m $V$ ² = 16Lx ${10}^{-15}$ Joule = 99.84 L Mev. Let us suppose the mean free path is not as large as 0.5 Km, because of small energy collisions, scatterings, frictions and so on, and let us model a more realistic mean free path of a fifth of the more idealized one, then $L$ = (1/5); hence, the velocity reduction would be $V\approx$ 1.87 $\sqrt{1/5}$ × ${10}^8$ m/sec = 0.28c, and its kinetic energy go down from 99.84 Mev to 19.97 Mev. So in a more frictional environment which reduces $\ell$ from 100 meters to just 20 meters, the velocity of the electron goes down to a weak relativistic regime with about 20 Mev of kinetic energy. If collisions and energy losses become even more important and the L value becomes L = 1/20, so $\ell$ = 100 m/20 = 5 m, then we have $V\approx$ = 1.87 $\sqrt{L}$ × ${10}^8$ m/sec = 0.0935 ${10}^8$ m/sec and the corresponding kinetic energy is now 99.84Mev/20 = 4.99 Mev. Velocity is now not relativistic, yet it carries an important energy with it.

7. Again, the initial velocity can be, and has been, safely neglected. The acceleration would be exactly the same, and the time to travel $\ell$ would be the same microseconds since it does not depend on $\ell$ either. So, the quotient of kinetic energy gains for the case with L = 1 with respect to L = 1/5, or 1/20 is ½ $mV$ ² /½ $mV$ ²L = 1/L = 5, or 20, and the shorter mean free path $\ell$ modeling collisions, friction, etc., impacts directly in the velocity not reaching relativistic regimes and reaching lower kinetic energies by a factor (1/L).

8. The heavier the charged particle, the less accelerated it becomes since $\mathit{a\alpha }$ $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$m$}\right.$ , and the larger the charge, the bigger the acceleration. Electrons accelerate and reach even relativistic velocities in time intervals of a few microseconds in fields of about 1000 V/m in diluted (almost free) space, no collisions, no frictions, and no damping. Very importantly, if the particles possess already a significant initial velocity, let’s say $V_0$ = 0.1c, then again $V_f^2$ >> $V_i^2$ and our calculation still holds.

9. For nonrelativistic motions let us take $V_f$ = ${10}^{-2}$ (3 × ${10}^8\frac{m}{\mathit{sec}})$ = 0.01c, and $V_i$ = 0.001c, then [ $V_f$ ² − $V_i$ ²]^1/2 = [(9x ${10}^{10}$ )]^1/2(99)^1/2 $\frac{m}{\mathit{sec}}$ . Hence $a$ = ( $V_f^2$ − $V_i^2$ )/2S $\approx$ (3/L) × ${10}^3\frac{\mathit{km}}{{\mathit{sec}}^2}$ , again the acceleration is mainly determined by the final velocity reached and not by its initial velocity, and this permits all kinds of wandering and drifting particles entering the zone of acceleration to reach at the end of this interval similar accelerations in similar intervals of time. This works as a velocity-acceleration funnel that also directs the “charged beam” into a particular, well-defined direction, the direction of $\overrightarrow{E}$ . If the beam does not encounter many “obstacles” (collisions and scatterings), then we have an electrical current traveling some tens or hundreds of meters down that trajectory.

10. Also, $\raisebox{1ex}{$d{\overrightarrow{P}}^{+}$}\!\left/ \!\raisebox{-1ex}{$\mathit{dt}$}\right.$ = ${\overrightarrow{F}}^{+}$ = q ${\overrightarrow{E}}^{+}$ , and $\raisebox{1ex}{$d{\overrightarrow{P}}^{-}$}\!\left/ \!\raisebox{-1ex}{$\mathit{dt}$}\right.$ = ${\overrightarrow{F}}^{-}$ = q ${\overrightarrow{E}}^{-}$ , and ( $\raisebox{1ex}{$\mathit{dP}+$}\!\left/ \!\raisebox{-1ex}{$\mathit{dt}$}\right.$ )/( $\raisebox{1ex}{$\mathit{dP}-$}\!\left/ \!\raisebox{-1ex}{$\mathit{dt}$}\right.$ ) = $E^{+}$ / $E^{-}$ = $\eta$ = 2, 10, 100, 1000. Hence, the charge q moving in region $B^{+}$ increases its linear momentum per unit time much more than the charge moving in region $B^{-}$ . Under collisions and scatterings, the charges moving in $B^{+}$ region can transfer much more linear momentum and energy to other charges and “spread” motion and energy to other charged particles, moving classically or relativistically.

11. All these points from (A) to (K) indicate in one way or another the powerful effect that a larger electric field and its direction has on N charges moving in it as compared with the same charges moving in weaker electric fields.

12. From (A) to (K), seems to us, that are conditions for a lightning starting in the cloud and directed to any number of possible endings and directions.

More dynamics and electromagnetic features can come out from the above basic calculations that can be analyzed and contrasted to field measurements.

10. Conclusions

Different transient charge distributions have been considered and their electric fields discussed, and these quantities are, for the most part unknowns, still we extracted valuable information from their vectorial properties, working within close distances, on the structures that produce pronounced sinks/sources with large divergences of $\overrightarrow{E}$ and focusing on observation points close to them. We introduced the concept of “at the verge of discharge electric field, $\overrightarrow{\mathit{Ę}}$ produced by two specific charge configurations”. The role the dielectric character of the cloud plays in allowing/not allowing discharge was analyzed. We profiled the conditions to be met by $\overrightarrow{\mathit{Ę}}$ to appear and to be sustained for at least a few milliseconds. A simple, typical, electrified cloud model was constructed and described in detail. Then a few basic calculations are carried out for a thought experiment, moving charges around inside this model cloud, and we calculated velocities, accelerations, and kinetic energies for long- and short-elongated effective mean free paths, fast and slow electrons resulted in this dynamics. The acceleration of charges works irrespective of initial velocities, what is important is a light mass and an intense electric field. We analyzed what we learn from it.