Studies of the Formation of Summer Precipitation and Thunderstorm Electricity

N. S. Shishkin

Introduction

The formation of precipitation and thunderstorm electricity is a very complex physical process, which has attracted the attention of many investigators from the earliest steps of science. This problem was studied by the great Russian scientist M. V. Lomonosov, who in the middle of the 18th century put forward the first theory of thunderstorm phenomena, correctly connecting these phenomena with the charging of cloud droplets and with the development of ascending motions in clouds^1,3. The greatest meteorologist A. I. Voeikov^5 devoted much attention to the question of precipitation formation. At the end of the 19th century, anticipating modern views, he pointed to the action of gravity as the principal cause of the coalescence of cloud droplets, ensuring their growth to the size of raindrops.

However, these ideas did not receive proper development for many years.

In the science of precipitation, in the 1930s of our century the ideas of Bergeron and Findeisen became established; at their basis lies the assumption that, for precipitation to fall, the appearance of ice crystals in the cloud is necessary*).

Bergeron and Findeisen substantiated this assumption by two experimental facts.

First, numerous investigations of clouds carried out in the temperate climatic zone showed that precipitation for the most part falls from clouds whose upper part is located above the level of the zero isotherm, and the detection in them

) In recent years the priority in this (erroneous) assertion has been disputed by the Americans. Byers (BAMS 28, No. 3 (1947)) writes that already in 1927, in the book Theory of the Thunderstorm*, Hall stated: “A thunderstorm is impossible until the top of the cloud freezes, or, considering the problem more broadly, actual rain or snow cannot fall unless the top of the cloud freezes.”

the solid phase of water almost always serves as a guarantee of precipitation.

Secondly, laboratory investigations have shown that the elasticity of saturated vapor over ice is lower than over water at negative air temperature. Therefore, in a region where the solid and liquid phases of water coexist, there must occur a diffusion distillation of water vapor from water particles to ice particles, ensuring the rapid growth of the latter at the expense of the former.

Ice crystals (“ice nuclei”), in Bergeron’s opinion,^3 are always contained in the atmosphere either as remnants of former ice clouds, or are formed “by sublimation as a result of cooling caused by uplift, radiation, or expansion of a layer whose dew point already at the very beginning lay considerably below \(0^\circ\text{C}\)”*).

Bergeron denied the possibility of the appearance of ice particles by sublimation within water clouds, believing that “all nuclei have already been completely used for the formation of droplets.”

According to Bergeron, precipitation is formed in the case when a cloud, in its vertical development, reaches the “level of ice nuclei.” Crystals that have entered the environment of cloud droplets grow at the expense of “vigorous condensation” and, upon melting, form raindrops.

Thus, in Bergeron’s opinion, the release of precipitation is connected with an external cause and, consequently, could not be explained by the internal laws governing the development of the cloud.

At first he made an exception for drizzling rains, which could form by means of condensation alone. In 1933,^3 he also made an exception for tropical downpours, since it was known to him that in India there had been cases of downpours falling from purely water clouds. However, he did not attempt to give any serious explanation of this phenomenon, confining himself only to a reference to the possibility of the action of certain electrical processes or of the action, in the upper part of the cloud, of the Reynolds effect (cooling of droplets by radiation). Truly, as M. V. Lomonosov said: “They write obscurely about what they obscurely imagine.” Bergeron found no new explanations in 1949 either.^23

In attempting to remove the above-mentioned fundamental shortcoming of Bergeron’s theory, Findeisen^24, ^25 introduced the concept of a “sublimation level,” which was to replace the “level of ice nuclei.” He believed that in the air, along with condensation nuclei, there exist special sublimation nuclei capable of leading, under

*) Bergeron also mentions the possibility of the appearance of ice particles as a result of the freezing of supercooled cloud droplets.

the development of a water cloud above the level of the zero isotherm to the formation of ice crystals in it*).

In addition, in 1939 he supplemented Bergeron’s scheme by calculating the growth of large drops (formed during the melting of ice crystals) through coagulation with small cloud droplets. He showed that the growth of these particles, for a given structure of the small-droplet cloud, is practically proportional to the length of the path traversed while falling through the cloud.

Taking into account the additions to the theory made by Findeisen, the scheme of precipitation formation assumed the following form. When moist air is lifted to the condensation level, water droplets are formed in it; these subsequently grow by condensation, but cannot grow to large sizes because the condensation process slows as the droplets become larger. After the cloud has reached, in its development, the sublimation level (approximately the isotherm of \(-10^\circ\)C), ice crystals appear in it, formed by sublimation on special nuclei. Through the distillation of water vapor from the drops, these crystals rapidly grow to a size significantly exceeding that of the cloud droplets, after which they begin to grow by coagulation with the cloud droplets. Falling into a region of positive temperatures, they melt and turn into raindrops.

It seemed that the problem of precipitation had found its fundamental solution. The scheme described came to be called the Bergeron–Findeisen theory of precipitation; it received almost universal recognition. True, in individual articles doubts were expressed about the reality of sublimation nuclei, but these were not given due significance. Even numerous studies showing that, in the tropics, heavy showers often fall from manifestly water clouds did not prompt supporters of the Bergeron–Findeisen theory to reconsider the fundamental bases of this theory.

We have already mentioned that Bergeron and Findeisen denied the possibility of cloud droplets growing to the size of large raindrops. From their point of view, ice particles are a bridge connecting cloud and raindrops. Without such a bridge, according to Bergeron and Findeisen, the resolution of precipitation is impossible. The exceptions mentioned do not follow in any way from the theory itself. But in purely water clouds there is no such bridge; consequently, it is not necessary. If Bergeron and Findeisen had said that the appearance of ice particles in a water cloud favors the falling of precipitation, as a particular regularity, they would have been right. The elevation of a particular

*) Unlike Bergeron, he sought the cause of the release of precipitation in the very process of cloud development, but the process of growth of cloud droplets, according to Findeisen as well, is not capable of ensuring the falling of rain.

to elevate a regularity to the rank of a universal law was their profound error, one of fundamental significance.

The same error is also contained in Langmuir’s work^38 on the theory of precipitation (1948). Langmuir made a step forward in comparison with Findeisen in the study of coagulation, taking into account the aerodynamic conditions of droplet collision.

In all the works mentioned, there is no consistent solution of the question of the role of updrafts in clouds. The conditions for the melting of ice particles are not investigated.

The adherents of the Bergeron–Findeisen theory were also unable to solve the problem of thunderstorm electricity, which is closely connected with the problem of the formation of summer precipitation. To solve these problems it is necessary first of all to turn to the study of the processes of growth of cloud droplets. The physical foundations for the development of the theory of summer precipitation and thunderstorm electricity have been created by studies of clouds carried out by Soviet scientists^4, 8, 15.

From average characteristics of the sizes of cloud droplets, V. A. Zaitsev, I. I. Chestnaya, A. M. Borovikov, and others proceeded to the study of changes in the spectrum of cloud droplets (i.e., in their size distribution) with height during the development of a cloud.

This proved to be a leap forward in the study of cloudiness. It turned out that the spectrum of cloud droplets changes in such a way that, as the cloud grows upward, differences in droplet sizes increase, providing the onset of coagulation growth of large cloud droplets through coalescence with smaller droplets. As size increases, the rate of coagulation growth increases, and with prolonged vertical development of cloudiness there are no obstacles to the formation of raindrops even without the participation of the ice phase. If crystals appear in a water cloud, this promotes the formation of precipitation, but is not a necessary condition for its release.

In the present article we devote special attention to the physical processes occurring in water clouds.

I. EXPERIMENTAL DATA ON CLOUDS AND THEIR ANALYSIS

1. Data on the structure and development of clouds

Work on the investigation of the microstructure of clouds began in the USSR in 1935.

In 1935–1939 the investigations were carried out under mountain conditions, in the Caucasus (in the region of Gagra and on the slopes of Elbrus), by G. I. Tarayan, S. M. Katchenkov, E. S. Selezneva, and others.

From 1946 onward, aircraft studies of the structure of clouds of various forms began. They were carried out by E. S. Selezneva, V. A. Zaitsev, I. I. Chestnaya, A. M. Borovikov, and others.

Microphotographing of drops has shown that at the base of developing water clouds of all forms there occur only small drops, whose radius in the lower layer of the cloud (50–100 m thick) does not exceed 10–13 μ. The size distribution curve of the drops has a sharp peak at \(r = 2\text{–}3\ \mu\).

As one ascends upward, the sizes of the drops increase, and the peak becomes more and more blurred. In the region of large drop sizes a long “tail” appears in the spectrum. At a height of 400–500 m from the base of the cloud, drops of radius 25–30 μ are encountered, and at a height of 1000 m—drops of radius 100 μ and more.

The scheme of the distribution of drops with height for a cumulus cloud (according to V. A. Zaitsev) is shown in Fig. 1. A more or less analogous picture, according to the data of A. M. Borovikov,^4 occurs also in developing stratiform clouds.

Fig. 1.

The reasons for the increase in the difference in drop sizes with height are the following:

1) In the lower part of the cloud, along with the condensational growth of already formed drops as the cloud mass rises, new small drops arise on those condensation nuclei which had not previously participated in the process.

2) Large drops grow at the expense of the diffusional transfer of water vapor from small drops, over whose surface the elasticity of the saturating vapor is greater than over large ones.

3) After the radius of large drops reaches the value \(r \approx 15\ \mu\) (see Section II, 2), their coagulation with small drops begins, further increasing the diversity of drops by size. Near the boundaries of the cloud the drops are usually small because of evaporation.

The number of drops per unit volume in the lower part of a developing cloud (from several tens of meters to several hundreds of meters from the cloud base) increases with height owing to the appearance of new drops through condensation. The maximum number of drops varies within wide limits: from several

tens to a thousand and more per \(1\ \mathrm{cm}^3\). Subsequently, as ascent proceeds, the number of droplets per unit volume gradually decreases, both owing to the evaporation of small droplets during the diffusional transfer of water vapor from small droplets to large ones, and owing to coagulation. At a height of \(1\)—\(2\ \mathrm{km}\) above the base of a cloud, the concentration of droplets usually does not exceed several tens of droplets per cubic centimeter.

Measurements of the water content of a cloud, i.e. the amount of droplet-liquid water per unit volume, carried out by V. A. Zaitsev\(^8\), showed that it first increases as ascent proceeds upward, and then decreases in that part of the cloud where evaporation makes itself felt in the course of its development.

Fig. 2. Water content in a cumulus cloud, measured during flight on July 24, 1948 (10 h 28 m—11 h 45 m).

A characteristic example of the distribution of water content with height for cumulus clouds is given in Fig. 2. An analogous picture also occurs in stratus clouds. In the layer \(50\)—\(100\ \mathrm{m}\) from the base of the cloud the water content does not exceed \(0.05\)—\(0.10\ \mathrm{g}/\mathrm{m}^3\). In the central part of clouds the water content may reach values of \(2\)—\(3\ \mathrm{g}/\mathrm{m}^3\) and more. Toward the boundaries of the cloud the water content decreases.

The increase of the water content of a cloud with height is connected with the condensation of new portions of water vapor as the cloud mass rises. The mass of condensing moisture is easy to calculate, assuming that the temperature in the cloud decreases with height according to the moist-adiabatic law and that the humidity of the air in the rising cloud mass is close to saturation.

The dependence of the vapor pressure of saturated water vapor on temperature is determined, as is known, by Magnus’ empirical formula:

\[ E = 6.10 \cdot 10^{\frac{7.45t}{235+t}}\ \mathrm{mb}. \tag{1} \]

With a decrease in temperature \(t\), the vapor pressure \(E\) decreases.

The dependence of the amount of saturated water vapor per \(1\ \mathrm{kg}\) of moist air on the pressure \(p\) is calculated by the formula

\[ S=\frac{623E}{p-0.377E}\ \mathrm{g}/\mathrm{kg}. \tag{2} \]

With a decrease in pressure \(S\) increases.

The results of our calculations for the case when the cloud base is at an altitude of 1000 m above sea level and the temperature at this level is \(+6^\circ\text{C}\) are given in Table I and in Fig. 3.

Table I

Water content of convective clouds according to theoretical calculations

A. With the cloud-base height 1000 m above sea level and the temperature at this level \(+6^\circ\text{C}\)
Height above sea level, m	1000	1150	1350	1550	1750	2000	3000	4000	5000	6000	7000
Water content, g/m³	0	0.29	0.60	1.08	1.40	1.78	2.80	3.69	3.93	4.17	4.07

B. With the cloud-base height 760 m above sea level and the temperature at this level \(+21^\circ\text{C}\)
Height above sea level, m	760	910	1220	1520	1830	2130	2440	2650
Water content, g/m³	0	0.43	1.07	1.68	2.24	2.78	3.29	3.59

There also are given the results of an analogous calculation by Langmuir \(^{28}\) for tropical clouds with a cloud-base height of 760 m and a temperature at this level of \(+21^\circ\text{C}\).

The values of water content calculated by Langmuir exceed our values for the corresponding heights by approximately a factor of 1.5, which is connected above all with the higher temperature of the cloud.

Beginning at an altitude of 6–6.5 km above sea level, the water content of the cloud, calculated in g/m³, decreases with height owing to the considerable decrease in air density.

Fig. 3. Theoretically calculated values of cloud water content.

As a characteristic drop size, which will be used below, it is convenient to take the radius \(r_m\) of the drops that make the greatest contribution to the water content.

In accordance with the type of process that mainly determines the change in drop size, a water cloud may be divided into three zones:

1) The condensation zone—the lower part of the cloud, several hundred meters thick (excluding parts of the cloud close to the lateral boundary), where condensation plays the principal role in the growth of drops.

2) The coagulation zone—the upper part of the cloud (excluding parts of the cloud close to the lateral and upper boundaries), where drops whose radius exceeds \(r_m\) grow mainly through coagulation, while condensation plays a substantial role only for the fine-droplet part of the cloud during its ascent.

3) The evaporation zone—the cloud layer close to the lateral and upper boundaries, where the drop size decreases owing to evaporation; the cloud mass in this zone has a fine-droplet structure (the radius of the drops, as a rule, does not exceed \(20\text{–}25\,\mu\)), and the water content is usually \(0.1\text{–}0.3\ \mathrm{g/m^3}\).

If ice crystals appear in the upper part of an initially water cloud, then up to a radius of \(50\text{–}60\,\mu\) (see Fig. 9) the distillation of water vapor from drops to crystals (at an absolute supersaturation \(\varepsilon \approx 10^{-7}\)) is of substantial importance (see Sections II, 2). Larger particles, and particles in clouds of mixed structure, grow more rapidly owing to coagulation.

Finally, in a purely ice cloud the crystals grow through sublimation of water vapor during the ascent of the cloud mass and through the coagulation of crystals into flakes.

There is no detailed information on the vertical motions of air inside clouds or on the laws governing their development.

Inside cumulus clouds, as can readily be verified by visual observations, vertical motions have the character of irregularly developing jets\(^ {17}\), arising now in one part, now in another part of the cloud and, under favorable conditions, penetrating its entire thickness. On reaching the upper boundary of the cloud, the jets lead to the formation of rapidly growing protuberances, which usually soon cease their development.

If jets arise in large numbers, they cause rapid growth of the entire cloud. Records of the vertical surges of gliders and aircraft in cumulus clouds show that vertical-motion velocities of up to \(15\text{–}17\ \mathrm{m/sec}\) occur in them. The average rate of vertical development of cumulus clouds is usually small (less than \(1\ \mathrm{m/sec}\)) and only in the case of thunderstorm and hail clouds can it attain large values.

In stratiform clouds the velocities of vertical motion, as a rule, do not exceed a few centimeters per second and

only in individual cases can reach tens of centimeters per second, but detailed information about them is likewise lacking. The insufficiently studied rate of the vertical development of cloudiness is at present the chief obstacle to constructing a theory of precipitation formation.

2. Radar Data on Clouds

A powerful means of investigating the structure of clouds is the method of radar, which first arose in the Soviet Union through the works of L. I. Mandelstam and N. D. Papaleksi. Existing types of radar make it possible to detect accumulations of raindrops and crystals in the air. With the aid of radars, while remaining on the ground, one can investigate the formation of precipitation centers inside clouds and their development in time and in space up to the fall of precipitation to the earth.

Radio echoes from clouds and precipitation are studied with the aid of horizontal- and vertical-scan tubes. A horizontal-scan tube makes it possible to obtain a simultaneous image, in horizontal projection, of all regions containing sufficiently large drops and “illuminated” by the radar beam. Usually, to obtain a picture of precipitation, the radar antenna is rotated about a vertical axis. On the fluorescent surface of the horizontal-scan tube, precipitation regions are represented by bright spots.

By changing the angle of inclination of the antenna to the vertical axis, one can obtain a complete picture of the distribution of masses of large drops in the surrounding space within the range of the radar. Distances from the center of the tube give, on the corresponding scale, the true distances of the precipitation zones from the radar. For convenience, scale concentric circles are drawn on the tube every 5–10 km. Figures 4, 5, and 6 give photographs of spots on the horizontal-scan tube from precipitation regions of various origin, borrowed from Maynard’s article [31].

Thermal thunderstorms give bright spots arranged in a disorderly manner (Fig. 4). Precipitation of a cold front gives systems of bright spots in the form of long bands (Fig. 5), from whose displacement one can judge the motion of the front. Sometimes cold-front systems consist of several precipitation bands, each up to 100–150 km long and 15–30 km wide, following one another at a distance of 15–45 km. Precipitation of a warm front gives a diffuse spot, which may reach very large dimensions. The brightness of the spot is usually small. The picture from typhoons (Fig. 6) is characterized by the dark “eye” of the storm and by bright curved bands with feather-like edges.

Observing the motion of precipitation zones, one can predict the approach of storms, which is of great importance for the national economy.

Very interesting information about the process of precipitation formation can be obtained with the aid of radar having a vertical-scan tube. The radio echo on the vertical-scan tube

Fig. 4. Radio echo from a heat thunderstorm.

at its inception either has the form of separate spots of small horizontal extent, or is depicted as a bright horizontally oriented band of considerable length. As the cloud develops, the sizes of the spots increase; from the bright band, vertical bands of lesser intensity extend upward and downward.

In convective clouds the horizontal extent of the radio-echo spot is usually small, whereas the vertical dimensions reach several kilometers and may exceed the horizontal dimensions severalfold. The characteristic development of the radio echo for the case of a heavy shower on July 27, 1946, in Florida (USA) is shown in Fig. 7, borrowed from the article by Byers and Coons1.

Fig. 5. Radio echo from a cold front.

Fig. 6. Radio echo from a typhoon.

Fig. 7. Development of radio echo from the thunderstorm of July 27, 1946, on the vertical-scan tube.

Similar photographs of a thunderstorm on the vertical-scan tube are given in the article by G. V. Rozenberg.^10

Such a form of radio echo is not difficult to explain from the physical point of view. Let us consider the case of a water cloud. As was already indicated above, in a developing cloud the drops grow larger with height. The growth of drops becomes especially rapid after they have grown sufficiently by condensation for coagulation of the drops with one another to begin. Depending on the velocity of the ascending currents, at the corresponding height there appears a sufficiently large number of large drops to give rise to a radio echo. If the conditions for the development of ascending currents are statistically the same for a large cloud system, then the masses of large drops arise practically simultaneously over large areas, and the radio echo on the vertical-scan tube assumes the appearance of a bright horizontal band.

Separate powerful ascending convective currents lead to the upward transport of cloud masses containing large drops, owing to which vertical bands develop upward from the bright horizontal band. The fall of large drops relative to the level where the bright horizontal band has arisen, in regions of weaker ascending motions or in regions of descending motions, leads to the appearance of vertical bands developing downward. Naturally, when these bands reach the ground level this corresponds to precipitation on the ground.

Analogous arguments may also be applied to a cloud containing ice particles. The fact of the frequent coincidence in height of the bright band with the level of the zero isotherm gave supporters of the Bergeron–Findeisen theory^21,23 grounds for considering the radiolocation data as confirming this theory. However, the assertion that radio echoes can be given only by clouds in which an ice phase is present cannot be regarded as correct.

Interesting data on the development of radio echoes on the vertical-scan tube for thunderstorms in the USA are given by Workman and Reynolds.^34 In the observations described, radio echoes usually appeared near the isotherm \(-10^\circ\mathrm{C}\). As the cloud developed, the upper boundary of the radio echo moved upward at a speed of from 2 to 8 m/sec for different thunderstorms (the mean value proved to be 4 m/sec). When the band reached a vertical thickness of the order of 4 km (near the isotherm \(-30^\circ\mathrm{C}\)), its upward development ceased. The average time from the moment of appearance of the radio echo until it reached the upper point was 12 min. After this the top of the radio echo began to descend. The mean speed was likewise 4 m/sec, with greater ascent speeds in the early stage of radio-echo development corresponding to greater descent speeds. This is quite natural if one takes into account that, at high velocities of ascending motions, precipitation particles may arise

larger size, having also a greater rate of fall (see Section II, 2). Thunderstorm phenomena usually began after the radio echo had reached its maximum height, which also agrees well with the theory (see Section III).

3. Electrical processes in clouds

According to Ya. I. Frenkel, cloud particles acquire an electric charge by capturing ions present in the air. As studies by Simpson and his collaborators have shown \(^{32,33}\), the typical distribution of charges in a cloud is as follows: the upper part of the cloud has a positive space charge, the lower part a negative space charge; sometimes in this part of the cloud there are relatively small regions in which positive charges predominate.

In accordance with this distribution of charges, the electric field inside the cloud is for the most part positive, while the field beneath the cloud is negative. The field was recorded by means of a special instrument—an altielectrograph—carried up into the cloud by a pilot balloon. The magnitude of the electric-field strength inside the cloud usually does not exceed, in order of magnitude, \(100\ \text{V}/\text{cm}\). But when the altielectrograph passed through thunderstorm clouds, regions with considerably greater field strengths were noted, producing sparks in the instrument. Sparks occurred mainly at levels from 3 to 8 km above the earth’s surface, with a positive sign of the field.

The electric field at the earth’s surface changes sharply when thunderstorm clouds pass. Before the passage of a cloud the field is usually positive. When the cloud is over the given point, the field changes sign. Beneath the central part of the cloud positive values of the field are sometimes encountered, if in the lower part of the cloud there are regions of positively charged drops. After the cloud has passed, the field again becomes positive. Under a cloudless sky the field strength near the ground is small and is, in order of magnitude, \(1\ \text{V}/\text{cm}\). Under clouds, field strengths of several hundred \(\text{V}/\text{cm}\) are encountered.

Changes in the field during the passage of a thunderstorm cloud occur in jumps. Israel’s observations \(^{37}\), carried out with a field variograph, showed that sharp fluctuations of the field strength as a thunderstorm approaches may begin 2 hours before the first lightning discharge and continue after the end of the thunderstorm for another 3–4 hours. The average time interval between the beginning of the potential jumps and the beginning of the thunderstorm is, according to a number of observations, 55 min, and between the end of the thunderstorm and the end of the potential jumps, 61 min. The strongest field fluctuations are recorded during spark discharges (lightning).

The number of pulses per unit time is, before the onset of a thunderstorm, 0.5–1.0 pulses per minute (with a receiver sensitivity of 4.5 V/m); it then increases to 2–3 pulses per minute beneath the center of the storm and, after the center of the storm has passed, gradually decreases. The numbers of pulses of the two signs are almost the same. Thus, during the thunderstorm of August 12, 1941, in Potsdam, over the course of 3 hours, 155 negative pulses and 120 positive ones were recorded. Jumps of the electric field near the ground were also observed for non-thunderstorm clouds in the absence of lightning.

Consequently, it may be considered that weak forms of thunderstorm activity are characteristic of any clouds. Lightning discharges are only the culmination point in the development of the process, reached under especially favorable conditions.

For explaining thunderstorm phenomena it is especially important to know the regularities of the charging of cloud particles and precipitation particles. The charges of raindrops and snowflakes began to be studied long ago. The total charge brought by precipitation is measured by collecting the precipitation in an electrically insulated vessel connected to an electrometer. If the opening of the vessel is made small and the vessel is connected to a very sensitive electrometer, then the charge of individual raindrops or snowflakes can be measured.

Fig. 8. Frequency of occurrence of the charge of large cloud drops (after Gunn).

Studies show that precipitation can have a charge of either positive or negative sign. The average charge brought by precipitation is, in order of magnitude, 1 CGSE per 1 cm³ of water. According to ground-based data, the greatest total amount of electricity (up to 20 CGSE/cm³) is brought by thundershowers. The maximum values of the specific charge of rainwater or snow (the charge per unit mass of precipitation), calculated from the charges of individual particles¹¹, reach 150–200 CGSE/cm³.

Recently Gunn²⁶ developed a method that makes it possible to measure the charge of individual raindrops from an aircraft. The instrument is based on the induction principle: a drop passing through a well-insulated ring induces in it a charge, recorded by an oscillograph. The sensitivity of the instrument made it possible to register drops with a charge on the order of 0.01 CGSE and greater. According to Gunn’s data, individual large drops inside a rain-producing cloud,

had a charge of up to \(0.25\) CGSE; the mean charge was \(0.03\text{--}0.04\) CGSE. A graph of the recurrence frequency of droplet charges is given in Fig. 8.

The free charge on a considerable number of droplets was so large that the field at their surface reached breakdown values. Indeed, if a droplet of radius \(500\,\mu\) has a charge of \(0.1\) CGSE, then the field at its surface is equal to \(12\,000\) V/cm, i.e., close to the breakdown field.

If the concentration of anomalously charged particles is large, then the field of their charges can quite well explain the occurrence of thunderstorm discharges. This question is considered theoretically in Section III.

II. QUANTITATIVE THEORY OF PRECIPITATION FOR A CLOUD WITH A UNIFORM UPDRAFT

1. Physical premises of the theory

It is generally known that, under real atmospheric conditions, the formation of cloud droplets requires the presence in the air of so-called condensation nuclei of size \(10^{-7}\text{--}10^{-5}\) cm. Their number in the near-ground layer of air may reach several thousand and even several tens of thousands per \(1\ \text{cm}^3\). Like all Brownian particles, they can coagulate with one another, which, along with the difference in the mass of the primary “dry” nuclei, leads to the appearance of a certain spectrum of their sizes.

The transformation of nuclei into cloud droplets occurs when air is lifted to a level where the humidity of the air reaches a critical value sufficient for the onset of condensation and dependent on the size and hygroscopicity of the nuclei. Condensation begins with the most active nuclei. If, despite condensation, the relative humidity increases as the air rises, then less active nuclei are also drawn into the condensation process. At the level where the number of droplets per unit volume is maximal, the supersaturation has its maximum value; it may reach several percent. With further ascent the supersaturation decreases; however, the cloud droplets continue to grow in the region of upward currents even at very small values of supersaturation. The experimental fact established by V. A. Zaitsev for cumulus clouds—that in the lower part of a cloud, where condensation plays the main role, the distribution of droplets by size depends practically only on the height above the cloud base—makes it possible to calculate supersaturations. We shall assume that the supersaturation in the cloud (with the exception of that part where the number of droplets per unit volume increases with height) depends only on the velocity of the upward current and, for a given velocity, is constant.

In the initial stage of the existence of a cloud droplet, its growth takes place mainly through diffusion processes—direct condensation in a supersaturated atmosphere and diffusive transfer of water vapor from small drops to large ones, or from warm drops to cold ones, owing to the difference in the elasticities of the saturating vapor over the surfaces of the particles. The diversity of droplet sizes, as was already indicated in Section I, increases with height. After droplets of radius \(r \approx 15\,\mu\) appear in the cloud, their coagulation growth begins through the coalescence of droplets of different sizes falling at different velocities.

We neglect the processes of Brownian coagulation of cloud droplets, their coalescence under the action of hydrodynamic and electrical forces, and air turbulence. Gravitational coagulation ensures an ever-accelerating increase in the radius of droplets up to the size of raindrops, whereas condensation growth slows as the droplet size increases.

However, in order for this coagulation growth to lead to a substantial increase in droplet sizes, it is necessary, just as for diffusive growth, that sufficiently prolonged upward motions develop in the clouds. The greater the velocity of the ascending current, the more the droplets in the cloud can grow through coagulation.

We carried out calculations under the assumption that in the cloud there is a continuous and uniform ascending current and that the droplets are distributed by size in accordance with Smoluchowski’s asymptotic formula:

\[ n(r)=\frac{25 q_w}{4\pi r_m^6}\,r^2 e^{-\frac{5}{3}\frac{r^3}{r_m^3}}, \tag{3} \]

where \(q_w\) is the water content of the cloud, and \(r_m\) is the radius of the droplets that make the greatest contribution to the water content. Generally speaking, this formula is applicable to condensation nuclei, and even then not entirely rigorously, since they grow not only through Brownian coagulation but also through condensation. However, the character of the initial growth of cloud droplets described above contributes to the fact that their size distribution retains, in general outline, the same form; and for calculations not claiming great accuracy, formula (3) may be used. Accounting for the joint action of condensation and coagulation processes in the presence of an ascending current makes it possible to calculate the growth of droplets with height and with time\(^{18,19}\) (see Section II, 2).

Proceeding from the assumption that droplets which have reached the upper point of their trajectory do not subsequently collide with one another while falling, one can theoretically find the number of rain...

drops (see Section II, 3). Thus, the assumptions indicated above make it possible, in the main, to solve the problem of precipitation from purely water clouds. The most essential point for the further development of the theory is the investigation of the laws governing upward motions in clouds.

As for the role of the ice phase, as was already said in the introduction, its appearance accelerates the diffusion stage of growth, since the growth of ice particles as a result of the distillation of water vapor from drops proceeds considerably faster than the growth of water drops due to condensation, when the velocities of the ascending current are not too large. A quantitative calculation is easily carried out if one assumes that the supersaturation with respect to ice particles is increased, in comparison with the case of drops, by the difference in the elasticities of saturated vapor over water and over ice (see Section II, 4).

For \(r \approx 50\text{--}60\,\mu\) and water content \(q_w = 1\ \mathrm{g}/\mathrm{m}^3\), the particles grow as a result of coagulation faster than by condensation, and the rate of growth practically does not depend on whether the particle is liquid or solid, up to those sizes at which the shape of falling liquid particles begins to be noticeably deformed.

2. Growth of Drops Due to Condensation and Coagulation in a Cloud with an Ascending Current

Let us calculate the growth of a drop in a cloud with a uniform and continuous ascending current under the simultaneous action of condensation and coagulation.

A drop in such a cloud, in the initial period of its existence, rises upward, being carried along by the ascending current, until it reaches such a size that its falling velocity becomes equal to the velocity of the ascending current. With further enlargement the drop begins to descend.

The change in the height of the drop above the base of the cloud is described by the differential equation (if its falling velocity is regarded as Stokesian)

\[ \frac{dz}{dt}=u-\frac{2}{9}\frac{\rho g}{\eta}r^2; \tag{4} \]

where \(\rho\) is the density of water, \(g\) is the acceleration of gravity, \(\eta\) is the viscosity of air, \(r\) is the radius of the drop, and \(u\) is the velocity of the ascending current.

At the upper point of the trajectory

\[ r=\sqrt{\frac{9\eta u}{2\rho g}}. \tag{5} \]

For large drops it is necessary to take into account the deviation from Stokes’ law.

To solve equation (4), let us proceed in the left-hand side to differentiation with respect to \(r\), and let the rate of increase of the radius under the simultaneous action of condensation and coagulation be defined as the sum of the rates of growth due to each of these processes:

\[ \frac{dr}{dt}=\left(\frac{dr}{dt}\right)_{\mathrm{cond}}+\left(\frac{dr}{dt}\right)_{\mathrm{coag}}. \tag{6} \]

Finding the condensation rate of growth in a dense cloud, when the distances between droplets are small, is a difficult problem that has not yet been solved. We shall confine ourselves to considering the condensation growth of an isolated droplet at a specified supersaturation.

As is known from theory and from experiments in the Wilson chamber, the square of the droplet radius changes, at constant supersaturation, proportionally to time \(t\)*:

\[ r^{2}=2\varepsilon Dt, \tag{7} \]

where \(\varepsilon=\dfrac{q_{\mathrm{v}}-q_{0}}{\rho}\) is the absolute supersaturation in dimensionless units, \(q_{\mathrm{v}}\) is the density of water vapor in the surrounding space, \(q_{0}\) is the density of saturated water vapor over the surface of the droplet, and \(D\) is the coefficient of diffusion of water vapor in air.

The quantity \(\varepsilon\) is found from the experimental values for the condensation growth of \(r_{m}\) with height. From the experimental fact that, for different clouds, \(r_{m}\) at a given height has practically identical values, there follows the above-indicated assumption of constant supersaturation at a constant velocity of the ascending current,

\[ \frac{\varepsilon}{u}=\mathrm{const}. \tag{8} \]

*) With the exception of the initial period of droplet growth, when it is necessary to use the formula

\[ t=\frac{1}{2\varepsilon D}\left[ r^{2}+2\frac{\alpha}{\xi}r+\frac{\alpha^{2}}{\xi}\ln\frac{r}{\alpha} -\left(\frac{1}{6}-\frac{\beta^{3}}{\alpha^{3}}\right)\frac{2\alpha^{3}}{\xi r}+\ldots \right], \tag{7'} \]

which was obtained by us by the method of M. E. Shvets \(^{16}\), taking into account the dependence of the density of saturated water vapor over the surface of a droplet on its radius and on the concentration of the impurity. In formula (7′), \(\alpha\) and \(\beta\) are quantities entering into the Thomson and Raoult formulas and having the dimension of length:

\[ \alpha=\frac{2\sigma}{\rho R_{w}T},\qquad \beta^{3}=\frac{3m}{4\pi\rho}\frac{\mu_{\mathrm{v}}}{\mu_{\mathrm{p}}} \]

(\(\sigma\) is the surface tension, \(R_{w}\) is the gas constant for water vapor, \(T\) is the absolute temperature, \(m\) is the mass of the impurity in the droplet, and \(\mu_{\mathrm{v}}\) and \(\mu_{\mathrm{p}}\) are the molecular weights of water and of the impurity),

\[ \xi=\frac{q_{\mathrm{v}}-q_{0}}{q_{0}} \]

is the relative supersaturation.

Comparison with V. A. Zaitsev’s data showed that the constant is approximately equal to \(3\cdot 10^{-11}\). Consequently, at \(u=10\ \text{cm/sec}\), \(\varepsilon=3\cdot 10^{-10}\); at \(u=100\ \text{cm/sec}\), \(\varepsilon=3\cdot 10^{-9}\), etc.

The values of the rate of growth of drops due to condensation at different \(\varepsilon\) are given in Fig. 9. The curves were calculated by formula \((7')\). An explanation of the coagulation curves shown in Fig. 9 will be given below.

Fig. 9. Rate of growth of drops due to coagulation at a water content of \(1\ \text{g}/\text{m}^3\) and due to condensation at different supersaturations.

If a drop grows only by condensation, then integration of equation (4) gives\(^7\)

\[ z=\frac{r^2}{2\varepsilon D}\left(u-\frac{\rho g}{9\eta}r^2\right). \tag{9} \]

The highest point of the trajectory is found from the formula

\[ H=\frac{9\eta u^2}{8\rho g\varepsilon D}, \tag{10} \]

and the radius of the drop at the moment it falls out of the cloud is

\[ r=\sqrt{\frac{9\eta u}{\rho g}}. \tag{11} \]

At \(u=10\ \text{cm/sec}\) a drop can grow in the cloud by condensation only up to \(r=40\ \mu\), and the highest point of its trajectory is located at an altitude of \(3\ \text{km}\). At \(u=70\ \text{cm/sec}\), a drop growing by condensation would have, on falling out of the cloud, a radius of \(170\ \mu\), and it would have to rise inside the cloud to an altitude of \(36\ \text{km}\) over the course of almost 9 hours.

As already indicated above, calculation by the condensation formulas can be carried out only up to \(r=15\ \mu\). Thereafter it is necessary to take coagulation growth into account.

The rate of growth of a drop due to gravitational coagulation in a cloud of polydisperse structure is determined by the expression

\[ \left(\frac{dr}{dt}\right)_{\text{coag}}=\int ESn_1\,\Delta v\,\Delta r\,dr_1, \tag{12} \]

where \(E\) is the coagulation coefficient, \(S=\pi(r+r_1)^2\) is the effective collision cross-section, \(n_1\) is the number of cloud drops of radius \(r_1\) per unit volume, \(\Delta v\) is the difference in the fall velocities of drops with radii \(r\) and \(r_1\), and \(\Delta r\) is the increase in the radius of the large drop upon coalescence with one drop of radius \(r_1\).

The integration is carried out over all sizes of cloud drops \((r_1<r)\) with which a drop of radius \(r\) can coagulate.

According to Langmuir\(^ {28}\), the collision of two spherical drops moving in a viscous medium is possible if the coefficient of inertia satisfies the condition

\[ k=\frac{\lambda_1}{r}\geqslant 1.214, \tag{13} \]

where \(\lambda_1\) is the inertial path length of the small drop; \(r\) is the radius of the large drop. The quantity \(\lambda_1\) is determined as follows. Let a drop of radius \(r_1\) fall in a viscous medium with velocity \(v_0\). If the resistance of the medium is assumed to be Stokesian, then the velocity of the drop’s motion will decrease with time according to the exponential law

\[ v=v_0e^{-\frac{t}{\tau_1}}, \]

where \(\tau_1=\dfrac{2}{9}\dfrac{\rho}{g}r_1^2\) is the relaxation time of inertial motion, i.e., the time during which the velocity of the drop’s motion decreases by a factor of \(e\).

The path traversed by the drop by inertia is equal to

\[ \lambda_1=\int_0^\infty v_0e^{-\frac{t}{\tau_1}}\,dt=v_0\tau_1. \]

When two drops with radii \(r\) and \(r_1\) move under the action of gravity in a viscous medium, their relative velocity at the initial moment of convergence, when the trajectory may still be regarded as undistorted, is equal to

\[ v_0=g(\tau-\tau_1). \]

Consequently, the inertial path of motion of a drop of radius \(r_1\) relative to a drop of radius \(r\) will be

\[ \lambda_1=g\tau_1(\tau-\tau_1). \]

After substituting the value of \(\lambda_1\) into (13), we find the condition for coagulation of drops

\[ r\sqrt{\frac{1}{2}-\sqrt{\frac{1}{4}-\frac{R^3}{r^3}}}\leq r_1\leq r\sqrt{\frac{1}{2}+\sqrt{\frac{1}{4}-\frac{R^3}{r^3}}}, \tag{14} \]

where

\[ R=\sqrt[3]{\frac{1.214}{g}\left(\frac{9\eta}{2\rho}\right)^2}\simeq 9\,\mu \]

is the characteristic radius for gravitational coagulation. The region of coagulation is shown in Fig. 10.

Fig. 10. Region of coagulation of cloud drops (not hatched).

We took the coagulation coefficient \(E\) to be equal to[^19]

\[ E=\left(\frac{k-1.214}{k}\right)^2, \tag{15} \]

which gives values close to the Langmuir collision coefficient obtained experimentally. The coefficient of effectiveness of collisions within a cloud we assumed to be equal to unity (in accordance with the experiments of B. V. Deryagin and P. S. Prokhorov[^6][^9], which confirmed this for a considerable interval of fall velocities of drops in saturated air), i.e., we assumed that every collision of drops leads to their coalescence.

For small drops, to which Stokes’ law is applicable, integration of (12) at constant \(q_w\) and \(r_m\) gives

\[ \left(\frac{dr}{dt}\right)_{\mathrm{coag}} = \frac{\rho g}{18\eta}\, \frac{q_w r_m^4}{r^3}\, P\left(\frac{r}{r_m}\right), \tag{16} \]

where

\[ \begin{aligned} P\left(\frac{r}{r_m}\right)= \Biggl\{& \left[1-(1+x)e^{-x}\right]\frac{r^4}{r_m^4} +2\left(\frac{3}{5}\right)^{\frac13}\gamma\left(\frac73,x\right)\frac{r^3}{r_m^3} -\frac65\left[2!-(x^2+2x+2!)e^{-x}\right]\frac{r}{r_m} \\ &-\left(\frac35\right)^{\frac43}\gamma\left(\frac{10}{3},x\right) -\Biggl[ 2\left(\frac53\right)^{\frac23}\gamma\left(\frac43,x\right)\frac{r^3}{r_m^3} \\ &\qquad +4\left(\frac53\right)^{\frac13}\gamma\left(\frac53,x\right)\frac{r^2}{r_m^2} +2\frac{r}{r_m} -2(1+x)e^{-x}\frac{r}{r_m} \Biggr]\frac{R^3}{r_m^3} \\ &+\Biggl[ \left(\frac53\right)^{\frac43}\gamma\left(\frac23,x\right)\frac{r^2}{r_m^2} \\ &\qquad -2\left(\frac53\right)^2 e^{-\frac53\frac{r^3}{r_m^3}} Ei\left(\frac53\frac{r^3}{r_m^3}-x\right)\frac{r^4}{r_m^4} \\ &\qquad +2\left(\frac53\right)^{\frac23} \sum_{k=0}^{\infty} \left(\frac35\frac{r_m^3}{r^3}\right)^k \gamma\left(\frac{3k+4}{3},x\right) \\ &\qquad +2\left(\frac53\right)^{\frac13}\frac{r_m}{r} \sum_{k=0}^{\infty} \left(\frac35\frac{r_m^3}{r^3}\right)^k \gamma\left(\frac{3k+5}{3},x\right) \Biggr]\frac{R^6}{r_m^6} \Biggr\}_{x_0}^{x_1}; \end{aligned} \]

\[ x=\frac53\frac{r_1^3}{r_m^3}, \]
and the limits for \(r_1\) are given by inequality (14).

The result of the calculation for two values, \(r_m=10\mu\) and \(15\mu\), with cloud liquid-water content \(q_w=1\ \mathrm{g}/\mathrm{m}^3\), is presented graphically in Fig. 9*), where the dotted coagulation curves are calculated by formula (16), and the solid curves under the assumption that a large drop of radius \(r\) coagulates only with drops \(r_1\leqslant r_m\). In the latter case the solution of the problem of coagulation growth is carried through analytically to the end.

If in (16) the upper limit is taken to be \(r_m\), then the function \(P\left(\frac{r}{r_m}\right)\) becomes a polynomial of the 4th degree. Having found the roots of this

*) The first comparison of the curves of the rate of condensation growth with our curves for the rate of coagulation growth was made by B. V. Kirokhnin.

of the polynomial and decomposing the fraction \(\dfrac{r^2}{P\left(\dfrac{r}{r_m}\right)}\) into simple terms, we obtain an easily integrable differential equation. The result for the growth rate is in this case, as is evident from the graph, underestimated. For large drops the solution may be obtained by numerical integration, taking into account the deviation of the fall velocity from the Stokes value.

Using (7) and (16), we obtain for the growth rate due to condensation and coagulation

\[ \frac{dr}{dt} = \frac{\varepsilon D}{r} + \frac{\rho g}{18\eta} - \frac{q_w r_m^4}{r^3}\, P\!\left(\frac{r}{r_m}\right). \tag{17} \]

This equation is not entirely rigorous, since the first term on the right was derived under the assumption that the drop is isolated, whereas the second term assumes collisions of drops. However, for calculations not claiming high accuracy it may be used, especially since both terms must be taken into account only for a limited interval of drop sizes. In the initial period of drop growth (up to \(r=15\mu\)) the second term on the right is equal to zero; for large drops the first term may be neglected.

Passing in (4) to differentiation with respect to \(r\) and using (17), we obtain

\[ \frac{dz}{dr} = -\frac{4}{q_w}\, \frac{ \dfrac{r^4}{r_m^4} - \dfrac{9\eta u}{2\rho g r_m^2}\, \dfrac{r^2}{r_m^2} }{ P'\!\left(\dfrac{r}{r_m}\right) }, \tag{18} \]

where the function \(P'\!\left(\dfrac{r}{r_m}\right)\) also includes the condensation term.

Solving this equation, we find the “trajectories” of a drop in the cloud, \(z(r)\), and solving (17), the dependence \(r(t)\). When collisions of a drop of radius \(r\) only with small cloud droplets \(r_1 \leqslant r_m\) are taken into account, the solution of equation (18) has the form

\[ z=-4\,\frac{r_m}{q_w}\, \Phi\!\left(\frac{r}{r_m}\right). \tag{19} \]

The form of the function \(\Phi\!\left(\dfrac{r}{r_m}\right)\) for the case of a cloud with an ascending current \(u=10\ \text{cm/sec}\) is given in Table II for several values of \(r_m\).

Calculations by these formulas may be carried out in such intervals of sizes that the expressions under the logarithm sign are positive. This restriction is connected with the choice of the form of the coagulation coefficient.

Table II

Form of the function \(\Phi\left(\dfrac{r}{r_m}\right)=\Phi(y)\), entering into formula (19), for a cloud with an upward-current velocity \(u=10\ \mathrm{cm/sec}\).

\(r_m\), in \(\mu\)	\(\left(1-\dfrac{8}{3}e^{-5/3}\right)\cdot \Phi(y)\)
5	\(y+74{,}5\lg(y-19{,}0)-10{,}0\lg(y-9{,}10)-0{,}72\lg\left(y^2+0{,}63y+1{,}10\right)+0{,}42\,\operatorname{arc\,tg}\dfrac{2y+0{,}63}{2{,}00}\)
10	\(y-4{,}71\lg(y-2{,}16)+6{,}25\lg(y-1{,}26)+1{,}58\lg\left(y^2+1{,}39y+0{,}84\right)-0{,}89\,\operatorname{arc\,tg}\dfrac{2y+1{,}39}{1{,}40}\)
15	\(y-1{,}17\lg(y-1{,}21)+0{,}64\lg(y+0{,}25)-0{,}34\lg\left(y^2+1{,}48y+0{,}82\right)-3{,}67\,\operatorname{arc\,tg}\dfrac{2y+1{,}48}{1{,}04}\)
20	\(y-0{,}39\lg(y-1{,}01)+2{,}96\lg(y+0{,}55)-2{,}60\lg\left(y^2+1{,}60y+0{,}82\right)-2{,}72\,\operatorname{arc\,tg}\dfrac{2y+1{,}60}{0{,}85}\)

The result of the successive calculation (the course of \(r_m\) with height was calculated by numerical integration according to analogous formulas for \(r=r_m\)) is shown in Fig. 11 for two values of the initial radius of a drop at the level of 100 m from the base of the cloud: \(r_0=5{,}5\mu\) and \(r_0=10\mu\).

Fig. 11. Growth of drops with height in a cloud with an upward current
\(u=10\ \mathrm{cm/sec}\).

N. S. SHISHKIN

For comparison, in the left part of the graph curves are drawn for the purely condensation growth of drops. In solving the problem with allowance for collisions of drops of all sizes, the radius of the initial raindrops increases from \(150\mu\) to \(330\mu\).

A calculation of the time by formula (17), carried out by numerical integration, shows that rain may begin in the case of a cloud with an updraft velocity \(u = 10\ \text{cm/sec}\) 6.3 hours after the formation of the cloud.

The result of the calculations for the case \(u = 70\ \text{cm/sec}\), where we were able to compare our theoretical data with observational data, is given in Figs. 12 and 13. The graphs were calculated for the initial raindrops under the following initial conditions: height of the cloud base 1000 m above sea level, temperature at this level \(+6^\circ\text{C}\), radius of drops at a level 100 m from the cloud base \(r_0 = 10\mu\).

From the figures it is seen that rain may begin with the fall through the cloud base of drops of radius \(r \approx 1200\mu\) 1 h 20 min after the formation of the cloud. The minimum height of the cloudiness producing precipitation must be 2100 m, and the total height of the cloudiness at the moment of rainfall is equal to 3400 m.

Fig. 12. Growth of drops with height in a cloud with an updraft \(u = 70\ \text{cm/sec}\).

On May 26, 1950, we were able to observe from the Koltushi district, near Leningrad, the development of a ridge of cumulonimbus clouds of a slowly moving cold front from the moment of its formation until the fall of rain.

On that day in the morning there was fog in Leningrad, which dissipated after sunrise. At about 8 a.m. solar time, in the southwest direction from Koltushi, a ridge of rapidly growing cumulus clouds appeared. The clouds and the horizon were clearly visible the whole time. The vertical extent of the clouds could be judged from angular distances and from the known level of the cloud base, established by aircraft sounding (\(1.2\)—\(1.3\) km above sea level; the temperature at this level was \(+10.5^\circ\text{C}\)).

The vertical development of the clouds proceeded unevenly, through the appearance of rapidly growing bulges on their upper …

surface. After the bulge was detected, the duration of its growth was usually 3–4 min, while the growth rate reached 4 m/sec.

By 9 h 35 min a number of peaks had spread out and an almost flat upper surface had formed. The height of the upper boundary of the cloudiness at 9 h 41 min was about 5 km (the temperature at this level was −10°C).

The mean rate of vertical development was 70–80 cm/sec.

At 9 h 58 min, beneath the cloud, the formation of fall streaks was observed, which developed down to the ground. Consequently, less than 2 hours after the appearance of the cloudiness were required for rain to form.

Fig. 13. Growth of drops with time in a cloud with an ascending current
\(u = 70\ \text{cm/sec}\).

This figure agrees satisfactorily with our result if one takes into account that, from the precipitation of the first raindrops through the cloud base, as calculated by us, to the moment when a noticeable fall streak appears, at least 10–15 min must elapse. Some discrepancy is quite explainable by the fact that we did not take into account the evaporation of drops, which may play an important role, especially in the initial period of the cloud’s existence, and the nonuniformity of the cloud’s development.

Our results may also be compared with Lengmuir’s data\(^{29}\) on the development process near Albuquerque (USA, New Mexico) on 21 July 1949 of a powerful Cb, for which simultaneous visual observation and radar observation were carried out.

The cumulus cloud appeared at 8 h 30 min near the Manzano mountain range. Until 9 h 57 min the cloud slowly grew in height at an average rate of about 80 cm/sec up to the level of 7.9 km (temperature −23°C). From this moment the vertical velocity increased to 3.7 m/sec. At 10 h 06 min, when the cloud top had a height of 11 km, a radio echo was detected by radar at a height of 6 km (i.e., judging from Lengmuir’s data, approximately at a height of 2.3 km above the cloud base, at a temperature of −9°C). During the first 6 min the top of the echo rose to a level of about 10 km with

with an average velocity \(u=11\ \mathrm{m/sec}\), which indicates an updraft velocity of up to \(15—17\ \mathrm{m/sec}\).

At 10 h 10 min, i.e., 4 min after the detection of the radio echo, the first lightning flash was visible. Soon a heavy rain was observed under the cloud.

Let us assume that drops of radius \(100—200\ \mu\) can produce a radio echo when their concentration is sufficiently large. According to our graph, the first drops of this size could appear \(1\) h \(10\) min—\(1\) h \(15\) min after the formation of the cloud, and some additional time was required for their concentration to increase to a value capable of producing a radio echo. The actual time was \(1\) h \(36\) min. The agreement may again be regarded as satisfactory.

Thunderstorm discharges, as we shall see in Section III, can form in the region where the radius of the drops reaches, approximately, values of the order of \(1000\ \mu\). According to Fig. 13, the formation of such drops requires \(3—5\) min after the drops have reached the size indicated for the onset of a radio echo (this time depends little on the velocity of the updraft), which agrees excellently with the actual result.

Langmuir believes that in the case he described the release from the ground of AgI smoke, whose particles at negative temperatures can serve as nuclei for the formation of ice particles, had a substantial effect on the occurrence of rain. Our data show that rain could have fallen without any participation of the ice phase (in the calculations its influence was not taken into account), and consequently without the effect of AgI.

3. Calculation of the Number of Raindrops and the Intensity of Precipitation

As we have already indicated above, the number of raindrops for a cloud with a uniform updraft can be calculated, without claiming accuracy, with the aid of the assumption that drops falling relative to the ground do not collide with one another.

The basis for such an assumption is the small number of raindrops per unit volume and the difference in their fall velocities. Collisions of falling drops with one another would accelerate their fallout and would increase the intensity of precipitation in the initial period of rain.

Thus, in order to determine the number of raindrops, it is sufficient to count the number of cloud drops that reach, per unit time, the maximum height at which their radius is equal to \(r_{\mathrm{cr}}=\sqrt{\dfrac{9\eta u}{2\rho g}}\). To represent visually the process of rain development in space and in time, we carried out a calculation

growth of drops which, at the level of 100 m from the base of the cloud, have different radii, at the velocity of the ascending current \(u=10\ \mathrm{cm/sec}\). The results of the calculation are presented in Figs. 14 and 15.

Fig. 14. Growth with height of drops of different initial sizes in a cloud with an ascending current \(u=10\ \mathrm{cm/sec}\).

Fig. 15. Growth with time of drops of different initial sizes in a cloud with an ascending current \(u=10\ \mathrm{cm/sec}\).

For the moment when drops that had \(r_0=10\ \mu\) reach their upper point, the number of raindrops may be considered equal to zero.

For the \(k\)-th stage of the calculation, the number of drops with \(r \geqslant r_{\mathrm{cr}}\) formed in a column of unit cross-sectional area per unit time will be equal to

\[ N_k = u\left( n_k e^{-\frac{5}{3}\frac{r_{\mathrm{cr}}^3}{r_m^3}} - n_{k-1} e^{-\frac{5}{3}\frac{R_{k-1}^3}{r_m^3}} \right), \tag{20} \]

where

\[ n_k=\frac{5q_w}{4\pi r_m^3} \]

is the total number of drops per unit volume at the level where \(r_k=r_{\mathrm{cr}}\), and \(r_m\) is the radius of the drops making the greatest contribution to the water content at this level; \(R_{k-1}\) is the radius of the drops which reached the size \(r=r_{\mathrm{cr}}\) at the end of the preceding stage of the calculation and began to fall to lower levels.

Fig. 16. Growth of rain intensity with time for a cloud with an ascending current \(u=10\ \mathrm{cm/sec}\).

The value \(R_{k-1}\) can be obtained directly from the graph \(r(t)\); for this it is sufficient, from the value \(r_{k-1}=r_{\mathrm{cr}}\) on the corresponding curve, to draw a line parallel to the ordinate axis. The point of intersection with the curve for the preceding stage will give the value \(R_{k-1}\). For the calculation stages adopted by us, the second term in equation (20) may be neglected.

Multiplying the number of drops \(N_k\) by the volume of a drop of the corresponding size \(r_k\) when it falls through the base of the cloud, and summing over all drop sizes that occur in the rain at the given instant of time, we obtain the rain intensity

\[ I=\sum_k \frac{4}{3}\pi r_k^3 N_k. \tag{21} \]

The graph of the growth of rain intensity with time for a cloud in the case \(u=10\ \mathrm{cm/sec}\) is given in Fig. 16.

During the first 40 min of rain its intensity slowly increases to \(1\ \mathrm{mm/hour}\), and during the following 20 min it reaches a value of \(8.5\ \mathrm{mm/hour}\). If the ascending current continues, the rain intensity may attain very large values. Under real conditions, however, after the onset of rain the ascending currents usually cease, or even descending currents develop*). Therefore anomalously large rain intensities are not reached at small velocities of the ascending current.

At large \(u\), the increase of rain intensity with time proceeds considerably faster, and the sizes of the raindrops are large. This is characteristic of summer shower precipitation.

4. The role of the ice phase

As was already indicated in the introduction, ice particles in a cloud of mixed structure, in the initial period of their existence, grow much more rapidly than droplets.

The calculation of the growth of spherical ice particles can be carried out analogously to the calculation of droplet growth described in Section II, 2, with the sole difference that the supersaturation \(s\) must be increased by the amount of the ratio of the difference between the densities of the saturated vapor over water and over ice to the density of ice.

Ice particles may appear in a cloud either in the form of frozen droplets or in the form of certain ice embryos. We have not studied the regularities of the occurrence of the ice phase, but our calculations make it possible to estimate the influence of its appearance on the formation of precipitation.

Let us suppose first that in the region of the upper part of the trajectory of the first precipitation particles (at negative temperatures) the droplets freeze. Experiments show that in clouds the freezing of droplets begins at temperatures considerably below \(0^\circ\mathrm{C}\), and, other conditions being equal, the largest droplets freeze first. Therefore, for the calculation one may assume that all droplets except the largest remain in the liquid supercooled state.

An analysis of the “trajectories” \(z(r)\) of the first precipitation particles shows that at small velocities of the ascending current only comparatively large droplets can enter the region of sufficiently low temperatures. But it was already indicated above that, for \(q_w = 1\ \mathrm{g/m^3}\), ice particles with \(r > 50\text{–}60\,\mu\) grow practically at the same rate as droplets, since the growth takes place mainly at the expense of coagulation. Therefore the freezing of large droplets under the indicated conditions will have practically no effect on the growth of the particles.

*) In addition, in calculating the washing out of the cloud during the rain process, we did not take it into account.

At large velocities of the ascending current, the supersaturation for drops is comparable in order of magnitude with the supersaturation over ice particles in a cloud of mixed structure. Therefore the freezing of drops likewise will have practically no effect on their growth up to those sizes at which the drops will be shattered, whereas ice spherical particles can grow to very large sizes without breaking up.

Consequently, in a continuous ascending current, the calculation of the growth of solid spherical particles formed by the freezing of drops in the upper part of the trajectory may be carried out (if temperature effects are not taken into account and great accuracy is not pursued) as though they were liquid.

Fig. 17. Growth with height of the first precipitation particles for a cloud with an ascending current \(u = 5\ \text{m/sec}\).

An example of the “trajectory” \(z(r)\) of the first precipitation particles for a cloud with an ascending current \(u = 5\ \text{m/sec}\), with cloud-base height \(1\ \text{km}\) and temperature at this level \(+6^\circ\text{C}\), is shown in Fig. 17.

The appearance of ice particles at levels considerably lower than the summit of the trajectory of the first precipitation particles can lead to a noticeable acceleration of precipitation fallout in the case of a cloud with a uniform ascending current. Thus, at \(u = 70\ \text{cm/sec}\), the formation of ice particles at the level of the zero isotherm can lead to precipitation fallout 20–25 min earlier than in the case of a purely water cloud. However, the role of ice particles may be most significant in the case of the cessation of the vertical development of the cloud, when the natural process in a water cloud does not ensure the formation of rain, whereas with the appearance of ice particles the conditions for precipitation may prove sufficient. In temperate latitudes this, apparently, occurs very often.

Let us turn to the question of the melting of ice particles. Using the graphs \(z(r)\) and assuming that the melting of ice particles is connected mainly with coagulation in the cloud below the level of the \(0^\circ\text{C}\) isotherm, it is easy to calculate in which case the particles will fall out in the form of hail and in which case—in the form of rain.

Suppose that at the level of the zero isotherm the mass of a spherical ice particle is equal to \(\mu_0\) and the temperature is equal to \(-\Delta t_i\).

Then, for melting the particle, the required amount of heat is

\[ Q_{\text{melt}}=\mu_0(q_{\text{melt}}+c_{\ell}\Delta t_{\ell}), \tag{22} \]

where \(q_{\text{melt}}\) is the latent heat of fusion of ice, and \(c_{\ell}\) is the heat capacity of ice.

The heat obtained through the coagulation of the ice particle with droplets having temperature \(\Delta t_{\mathrm{w}}\) (we assume that all the heat goes into melting the ice particle) is equal to

\[ Q_{\mathrm{w}}=(\mu-\mu_0)c_{\mathrm{w}}\Delta t_{\mathrm{w}}, \tag{23} \]

where \(\mu\) is the mass of the particle at the moment it falls out of the cloud, and \(c_{\mathrm{w}}\) is the heat capacity of water.

Equating the two expressions, i.e., assuming that by the moment it falls out of the cloud the ice particle has completely melted, we obtain

\[ \frac{r^3}{r_0^3} = \frac{\rho_{\ell}}{\rho_{\mathrm{w}}} \left( 1+ \frac{q_{\text{melt}}+c_{\ell}\Delta t_{\ell}}{c_{\mathrm{w}}\Delta t_{\mathrm{w}}} \right), \tag{24} \]

where \(\rho_{\ell}\) and \(\rho_{\mathrm{w}}\) are the densities of ice and water.

Assuming that inside the cloud there is a moist-adiabatic temperature gradient, and substituting, for the purpose of estimation, mean values of the temperatures of the ice particles and droplets in place of \(\Delta t\), we can calculate, for the given conditions, the values of \(r_0\). If the corresponding value on the curve \(z(r)\) lies above the actual zero isotherm, then the ice particle certainly cannot melt, even if all the heat of the droplets goes into melting.

For example, for \(u=70\ \mathrm{cm/sec}\) (see Fig. 12), melting an ice particle requires \(0.3\ \mathrm{cal}\). From the influx of heat from the droplets with which the particle coagulates in the layer below the \(0^\circ\mathrm{C}\) isotherm level, it will receive only \(0.08\ \mathrm{cal}\). Consequently, at a cloud-base temperature of \(+6^\circ\mathrm{C}\), the ice particle will not be able to melt, and the precipitation will fall as hail. A rough estimate shows that precipitation in the form of rain (if freezing of droplets occurs in the upper part of their trajectory) can fall only in the case when the temperature at cloud-base level is \(+9^\circ\mathrm{C}\). For \(u=5\ \mathrm{m/sec}\), the corresponding critical temperature at cloud base is on the order of \(+17^\circ\mathrm{C}\). The appearance of ice particles in the form of embryos would somewhat change these values.

Finally, Fig. 17 shows that the formation of comparatively large hail does not at all require anomalously high velocities of the updraft. Hail with particle diameters on the order of \(1\ \mathrm{cm}\) can form already at an updraft velocity of \(5\ \mathrm{m/sec}\), if it is sufficiently long-lasting.

III. THUNDERSTORM ELECTRICITY

1. Charging of Cloud Droplets

The process of charging cloud droplets, like the process of growth of their size, may be divided into two stages: diffusional and coagulation. The diffusional stage, according to Ya. I. Frenkelʹ2, consists predominantly in the capture by cloud droplets of negative air ions.

Water is a dipolar liquid; in its surface layer the dipolar molecules are oriented with their negative ends outward. Passage through the double electric layer is facilitated for negative ions and hindered for positive ions.

The potential jump in the double electric layer is equal to

\[ \xi = 4\pi n\delta \bar p, \tag{25} \]

where \(n\) is the number of water molecules per unit volume, \(\delta\) is the thickness of the surface layer, and \(\bar p\) is the mean value of the projection of the molecular dipole moment on the inward normal to the surface.

Taking \(n = 3\cdot 10^{23}\ \mathrm{cm}^{-3}\), \(p = 10^{-19}\), we obtain

\[ \xi \approx 0.3\ \mathrm{V}. \]

The preferential capture of negative ions continues until the potential jump is compensated by the Coulomb field.

In the equilibrium state

\[ \xi = -\frac{q}{r}, \]

whence we obtain for the equilibrium charge of a droplet

\[ q = -\xi r. \tag{26} \]

The potential \(\xi\) is called the electrokinetic potential. According to formula (26), one elementary charge \(e = 4.8\cdot 10^{-10}\) CGSE is the equilibrium charge of a droplet (condensation nucleus) with radius \(r = 5\cdot 10^{-7}\ \mathrm{cm}\). A droplet with radius \(r = 10^{-3}\ \mathrm{cm}\) has an equilibrium charge of \(2000e\). Charges of this order of magnitude are indeed possessed by fog droplets[^11].

After a cloud droplet begins to coagulate with other cloud droplets, the growth of its charge is accelerated, since the charge increases at the expense of coagulation, roughly speaking, pro-

proportional to the increase in mass, i.e., proportional to the third power of the radius, whereas the equilibrium charge is proportional to the radius[^14]. The coagulation growth of the charge in the dense part of a cloud cannot be compensated by the capture of ions of positive sign along the path of fall of a large droplet.

Under favorable conditions, above all when sufficiently powerful vertical motions are present in clouds, the charge of droplets may attain anomalously large values, ensuring, at a sufficient concentration of large droplets, the formation in clouds of discharge values of the electric-field intensity of the order of 30,000 V/cm.

The calculation of the coagulation growth of the charge is carried out as follows. Suppose that small cloud droplets (up to \(r_1 = r_m\)) have the equilibrium charge \(q_1\), while large droplets coagulate only with droplets whose radius satisfies the condition:

\[ r_1 \leq r_m. \]

The rate of growth of the charge of a large drop falling through a polydisperse cloud will be equal to[^20]

\[ \frac{dq}{dt}=\int_{r_0}^{r_m} E S n_1 q_1 \Delta v\,dr_1 \tag{27} \]

(the notation is the same as in Section II). After substituting all the expressions entering into (27) and integrating, we obtain

\[ -\frac{dq}{dt} = \frac{2\pi \delta g}{9\eta}\, Q q_w r_m^4 Q_4\!\left(\frac{r}{r_m}\right), \tag{28} \]

where

\[ Q= \frac{\displaystyle\int_{0}^{r_m}\xi r_1 n_1\,dr_1} {\displaystyle\int_{0}^{r_m}\frac{4}{3}\pi r_1^3 n_1\,dr_1} = \frac{3}{4\pi} \left(\frac{5}{3}\right)^{2/3} \frac{\gamma\!\left(\frac{4}{3},\frac{5}{3}\right)} {1-\frac{8}{3}e^{-5/3}}\, \frac{\xi}{r_m^2}, \]

is the specific charge of small equilibrium-charged droplets, which is a function of \(r_m\).

For \(r_m=10\,\mu\), \(Q=565\) CGSE; for \(r_m=15\,\mu\), \(Q=250\) CGSE.

\[ Q_4\!\left(\frac{r}{r_m}\right) \]

is a polynomial of the fourth degree with respect to \(\frac{r}{r_m}\), [[unclear: the remainder of the line is obscured/illegible]].

determined from the relation

\[ \frac{\gamma\!\left(\frac{4}{3},\,\frac{5}{3}\right)} {1-\frac{8}{3}e^{-\frac{5}{3}}}\,Q_4 = \gamma\!\left(\frac{4}{3},\,\frac{5}{3}\right)\frac{r^4}{r_m^4} +2\left(\frac{3}{5}\right)^{\frac{1}{3}} \left[ \gamma\!\left(\frac{5}{3},\,\frac{5}{3}\right) -\frac{5}{3}\gamma\!\left(\frac{2}{3},\,\frac{5}{3}\right)\frac{R^3}{r_m^3} \right]\frac{r^3}{r_m^3} -\left(\frac{5}{3}\right)^{\frac{4}{3}}\frac{R^3}{r_m^3} \left[ \frac{12}{5}\left(1-e^{-\frac{5}{3}}\right) -\frac{R^3}{r_m^3}E_i\!\left(-\frac{5}{3}\right) +\frac{R^3}{r_m^3}E_i\!\left(-\frac{5}{3}\frac{r_0^3}{r_m^3}\right) \right]\frac{r^2}{r_m^2} \]
\[ -2\left[ \frac{3}{5}\gamma\!\left(\frac{7}{3},\,\frac{5}{3}\right) +\gamma\!\left(\frac{4}{3},\,\frac{5}{3}\right)\frac{R^3}{r_m^3} -5\left(\frac{5}{3}\right)^{\frac{1}{3}}e^{-\frac{5}{3}}\frac{R^6}{r_m^6} -5\gamma\!\left(\frac{4}{3},\,\frac{5}{3}\right)\frac{R^6}{r_m^6} \right]\frac{r}{r_m} \]
\[ -\left[ \left(\frac{3}{5}\right)^{\frac{4}{3}}\gamma\!\left(\frac{8}{3},\,\frac{5}{3}\right) -2\left(\frac{5}{3}\right)^{\frac{2}{3}} \gamma\!\left(\frac{2}{3},\,\frac{5}{3}\right)\frac{R^6}{r_m^6} \right]. \]

As is easy to see, \(r_0\) need be taken into account only in the term with the exponential integral function \(E_i\). For further calculations it is convenient, in the left-hand side of equation (28), to pass to differentiation with respect to \(r\); the rate of increase of the radius has already been calculated in Section II (one need only set the upper limit equal to \(r_m\)).

With the aid of (16) and (28) we obtain

\[ \frac{dq}{dr}=4\pi r^3 Q\,\frac{Q_4}{P_4}. \tag{29} \]

If \(\frac{r}{r_m}\) is large, then one may restrict oneself to the first terms in the expressions for \(Q_4\) and \(P_4\), and we arrive at the simple equation

\[ \frac{dq}{dr}=4\pi Qr^3. \tag{29′} \]

Integrating, we obtain the trivial formula

\[ q=\frac{4}{3}\pi r^3 Q \tag{30} \]

—the charge of the drop is equal to its volume multiplied by the specific charge.

The solution of equation (29) in the general case is written as follows:

\[ q=\frac{4}{3}\pi r_m^3 Q\cdot \varphi\left(\frac{r}{r_m}\right). \tag{31} \]

The function \(\varphi\left(\frac{r}{r_m}\right)\), for water content \(q_w=1\ \mathrm{g/m^3}\) and for \(r_m=10\mu\) and \(15\mu\), has the form

\[ \begin{aligned} \varphi_{10}(y)= {}& y^3+0.06y^3+18.5y+105\lg(y-3.09)-{}\\ &{}-0.12\lg(y-0.29)+6.14\lg(y^3+1.36y+2.56)-{}\\ &{}-9.27\operatorname{arc\,tg}\frac{y+0.68}{1.45}; \end{aligned} \]

\[ \begin{aligned} \varphi_{15}(y)= {}& y^3-0.52y^3+1.77y+1.80\lg(y-1.05)-{}\\ &{}-0.02\lg(y+0.23)-0.14\lg(y^2+1.14y+0.62)-{}\\ &{}-1.00\operatorname{arc\,tg}\frac{y+0.57}{0.55}. \end{aligned} \]

The increase in the charge of drops for different \(r_m\) is given in Fig. 18. Comparing the data on the dependence of the charge of a drop on radius with the data on the dependence of the radius of a drop on its height above the base of the cloud, we obtain a relation between the charge of the drop and its position in the cloud.

In the case of a cloud with an ascending current \(u=10\ \mathrm{cm/sec}\), the result of the calculation is shown graphically in Fig. 19. Curve 1 gives the charge of drops with initial radius at the level of 100 m from the base of the cloud \(r_0=10\mu\), curve 2—for an initial radius of \(5.5\mu\).

Fig. 18. Increase in the charge of large drops due to coagulation with small, equally charged cloud drops at constant values of \(r_m\).

As a drop rises from the base of the cloud to the upper point of its trajectory, its charge gradually increases to \(3\cdot10^{-6}\) CGSE. The maximum charge, which is probably overestimated, since we do not take into account the capture of positive ions along the path of fall, nor

exceeds \(0.05\,CGSE\). Such a charge of the drops can hardly ensure the occurrence of thunderstorm phenomena.

For a cloud with an updraft \(u=1\ \text{m/sec}\), the result of calculating the charge of the drops is given in Fig. 20.

Fig. 19. Change of the charge of a drop with height in a cloud with an updraft
\(u=10\ \text{cm/sec}\).

Charges of drops of the order of \(0.1\,CGSE\) (the field at the surface of the drops is close to the breakdown value) are attained, as is seen from the graph, at a height of about \(3\ \text{km}\) above the base of the cloud. This is, in the present case, the most thunderstorm-dangerous region.

Our calculations were made, as already indicated, without taking into account the capture of ions along the path of fall, which could lead to an overestimate of the magnitude of the charge. The slowing of coagulation growth of the charge, noticeable in Figs. 19 and 20, with the continuing growth of the charge in the lower part of the cloud upon capture of ions, may lead here to the formation of positively charged regions, often observed in thunderstorm clouds.

In the case of a cloud with a uniform updraft, the region of anomalously charged drops has the form of a horizontal layer. The thickness of the layer plays the role of a kind of “path length” of the drops, over which the charge of the drops increases from its equilibrium value to values ensuring the onset of thunderstorm phenomena.

Fig. 20. Change of the charge of a drop with height in a cloud with an updraft \(u=1\ \text{m/sec}\).

Within this layer, negatively charged drops experience a force directed upward and hindering the fall of the drops. On a large drop, whose charge is approximately determined by formula (30), acts ...

there acts an electric force

\[ F = Eq = EQm, \tag{32} \]

where \(m\) is the mass of the drop.

In the case when the condition

\[ Eq + F_{\mathrm{tr}} = mg \tag{33} \]

is satisfied (\(F_{\mathrm{tr}}\) is the force of air resistance for the moving drop), the resultant force is equal to zero, and the drop will move with constant velocity relative to the air.

At \(E=\dfrac{mg}{q}=\dfrac{g}{Q}\), an anomalously charged drop will move together with the air. The intensity of such a field is, for

\[ Q \simeq 10^{3}\,CGSE \]

\[ E_{\mathrm{cr}} \simeq 3000\ \mathrm{V/cm}. \]

But for small, equilibrium-charged cloud drops, whose charge is proportional to the first power of the radius, this field will not be critical.

For them the critical field is determined by the equation

\[ E'_{\mathrm{cr}}=\frac{mg}{q}=\frac{4\pi g r_1^{2}}{3\xi}. \tag{34} \]

For \(r_1=10\,\mu\), \(E'_{\mathrm{cr}}=1200\ \mathrm{V/cm}\); for \(r_1=20\,\mu\), \(E'_{\mathrm{cr}}=5000\ \mathrm{V/cm}\);

for \(r_1=30\,\mu\), \(E'_{\mathrm{cr}}=10\,000\ \mathrm{V/cm}\).

Thus, despite the fact that under the action of an electric field the motion of the drops will also change, between large and small drops there will always exist a difference in velocities of motion, and their coalescence may continue, although, apparently, at \(E \simeq E_{\mathrm{cr}}\) the coagulation process slows down.

We do not take into account here the electric forces of interaction of charged drops. This should be additionally investigated, especially for the case of anomalously charged drops.

2. Stationary electric field of a cloud

The falling of drops under the action of gravity leads to a macroscopic separation of charges. In the upper part of the cloud a layer is formed that contains predominantly positive volume charges; in the lower part of the cloud a predominance of negative charges is created.

If the current caused by the falling of drops is compensated by the conduction current in the electric field that arises when the charges are separated,

in an electric field, a stationary electrical state of the cloud is attained. Its theory was given by Ya. I. Frenkel’2.

The equation describing the stationary state of a homogeneous cloud can be written in the form

\[ qnv=\lambda E, \tag{35} \]

where \(n\) is the number of drops in a unit volume, \(v\) is the falling velocity of the drops, and \(\lambda\) is the conductivity of the air in the cloud.

For a fine-droplet cloud the falling velocity of the drops is determined by the formula

\[ v=\frac{mg-qE}{6\pi\eta r}. \tag{36} \]

For the intensity of the stationary electric field we finally obtain

\[ E=\frac{\frac{4}{3}\pi r^{3} n\rho g\,\xi}{6\pi\eta\lambda+\xi^{2}r}, \tag{37} \]

where \(\rho\) is the density of water.

The second term in the denominator is much smaller than the first, and instead of (37) one may write the approximate formula

\[ E=\frac{q_w g\xi}{6\pi\eta\lambda}, \tag{38} \]

where \(q_w\) is the water content of the cloud.

In the case of a polydisperse cloud the total water content is equal to the sum of the “partial” water contents. If all the drops are charged in equilibrium, then the electric field of the cloud does not depend on the size of the drops and is determined only by the water content of the cloud. If we put

\[ q_w=10^{-6}\ \mathrm{g/cm^{3}},\quad \eta=1.7\cdot 10^{-4}\ \frac{\mathrm{g}}{\mathrm{cm\cdot sec}},\quad \lambda=4\cdot 10^{-4}\ \mathrm{CGSE} \quad \text{and} \quad \xi=10^{-3}\ \mathrm{CGSE}, \]

then we find

\[ E\simeq 100\ \mathrm{V/cm}. \]

The correct order of magnitude of the electric-field intensity is obtained, in good agreement with the experimental results.

3. Electric field in thunderclouds

In thunderclouds the electric field has an essentially nonstationary character. To calculate the field intensity we shall use Poisson’s equation, and we shall consider the case of a cloud with a uniform ascending flux

only variations of the field in the vertical direction:

\[ \frac{dE}{dz}=4\pi\rho. \tag{39} \]

The volume density of electric charge \(\rho\) in a thundercloud is composed of three parts: a) anomalously large charges of large drops, b) equilibrium charges of small cloud droplets, c) the space charge of air ions. The distribution of volume charges with height is shown schematically in Fig. 21.

We may divide the whole cloud by height into such parts that in each of them the total charge is equal to zero. The field in each of the “plane” condensers thus formed is composed of the field produced by the charges of the given condenser and of the external field. In the diagram this is represented by a number of arrows. The greatest field strength must be at that level of the cloud where the volume charge is equal to zero.

Fig. 21. Diagram of the distribution of volume charges in a cloud.

Fig. 22. Diagram of the distribution with height of the density of the volume charge of the cloud \(\rho\) and of the electric-field strength \(E\).

In Fig. 22 the distribution with height of the density of the volume electric charge and of the electric-field strength is shown schematically.

The equilibrium charges of small cloud droplets and the space charge of ions give a comparatively small density of volume charges. The growth of the volume charge due to diffusion processes and to the spatial separation of charges during the fall of small drops proceeds comparatively slowly. A different character of the process occurs in the coagulation growth of large drops. In a short time, in a comparatively thin layer, enormous charges accumulate, which before the appearance of large drops had been dispersed over a large volume.

Therefore, for purposes of a rough estimate we may suppose that in the region of anomalously charged drops (where there occurs

active coagulation of drops) the contribution to \(\rho\) from equally charged cloud drops and air ions is relatively small*).

Then the volume density of the electric charge for this region will be represented in the form

\[ \rho=\sum_k n_k q_k, \tag{40} \]

where \(q_k\) is the charge of a drop of radius \(r_k\), \(n_k\) is the number of anomalously charged drops of radius \(r_k\) per unit volume. The summation is carried out over all sizes of anomalously charged drops located in the given volume of the cloud.

Substituting this expression into (39), we obtain

\[ \frac{dE}{dz}=4\pi\sum_k n_k q_k . \tag{41} \]

The total intensity of the electric field formed by the charges of large drops of all sizes may be regarded as the sum of “partial” intensities \(E_k\), each of which is associated with drops of a definite size.

For each of the components one may therefore write (dropping the index)

\[ \frac{dE}{dz}=4\pi n q . \tag{42} \]

Since the charge \(q\) was computed by us as a function of the radius, it is convenient on the left-hand side of (42) to pass to differentiation with respect to \(r\).

The expression \(\dfrac{dr}{dz}\) for the growth of a drop due to coagulation is given by formula (18), and the condensation term in the denominator may be neglected. If we restrict ourselves to the first term of the polynomial in the denominator of this formula and, for \(q\), use the simplified formula (30), then we obtain

\[ \frac{dE}{dr}=96\pi^3\,\frac{\eta}{\rho g}\,\frac{NQ}{q_w}\,r, \tag{43} \]

where \(N=n(v-u)\) is the number of raindrops falling through \(1\ \text{cm}^2\) in \(1\) sec., \(v\) is the falling velocity of a drop of radius \(r\).

Taking, for small time intervals, the drop-fall quantities \(Q\) and \(q_w\) to be constant, after integration we obtain

\[ E=48\pi^3\,\frac{\eta}{\rho g}\,\frac{NQ}{q_w}\,r^2, \tag{44} \]

\[ \text{*) Of course, this will distort the general picture of the distribution of the field intensity with height.} \]

i.e., the intensity of the electric field formed by the charges of anomalously charged drops is proportional to the specific charge of the drops and to the square of their radius.

For purposes of estimation one may regard all the rain drops in the given layer of the cloud as identical. Taking \(N = 0.2\ \mathrm{cm}^{-2}\mathrm{sec}^{-1}\) (which corresponds to moderate rain), \(Q \simeq 10^3\ CGSE\), and \(q_w = 1\ \mathrm{g}/\mathrm{m}^3\), we find that the electric-field intensity reaches breakdown values in the region where the radius of the drops is of the order of \(1000\ \mu\).

Since in our calculations we did not take into account the loss of charges due to the capture of ions, our value of \(r\) is an underestimate.

As we saw in Section II, drops of the corresponding size are formed at a height of 3–4 km above the base of the cloud for ascending currents of the order of \(1\ \mathrm{m/sec}\), and at greater height for higher velocities. This is in good agreement with Simpson’s experimental data\(^{32,33}\) (see Section I).

For a large drop, its growth along the path \(z\) due to coalescence with small drops is approximately given by the formula\(^{14}\)

\[ r - r_0 = \frac{1}{4} q_w z . \]

Comparing this expression with (44), we see that the field intensity in the region of active coagulation increases, roughly speaking, in proportion to the square of the path of fall of the large drops relative to the air.

The theory of the formation of thunderstorm electricity presented here does not take into account a number of important factors. However, even in this unfinished form it seems to us to describe correctly the main features of this process. It also seems essential to us that this theory describes the process of precipitation formation and the process of thunderstorm-electricity formation from a single point of view.

CITED LITERATURE

1. Bazilevich V. V. et al., Proceedings of the Research Institute of the Main Administration of the Hydrometeorological Service, ser. 1, issue 1 (1941).
2. Bergeron T., Three-Dimensionally Linked Synoptic Analysis, Publ. Central Administration of the Hydrometeorological Service of the USSR, Moscow, 1934.
3. Bergeron T., Meteorology and Hydrology No. 3–4 (1935).
4. Borovikov A. M., Proceedings of the Central Aerological Observatory, issue 3 (1948).
5. Voeikov A. I. Meteorology (1904).
6. Deryagin B. V. and Prokhorov P. S., in: New Ideas in the Field of Aerosol Research, Publ. Academy of Sciences of the USSR, 1949.
7. Deryagin B. V. and Todes O. M., ibid.
8. Zaitsev V. A., Proceedings of the Main Geophysical Observatory, issue 13 (1948), issue 19 (1950).
9. Prokhorov P. S., in: New Ideas in the Field of Aerosol Research, Publ. Academy of Sciences of the USSR, 1949.
10. Rosenberg G. V., UFN 39, 315 (1949).

1. P. N. Tverskoi, Atmospheric Electricity, Gidrometeoizdat, 1949.
2. E. K. Fedorov, Izv. AN SSSR, ser. geogr. i geof. 14, 25 (1944).
3. Ya. I. Frenkel, Theory of the Phenomena of Atmospheric Electricity, Gostekhizdat, 1949.
4. Ya. I. Frenkel and N. S. Shishkin, Izv. AN SSSR, ser. geogr. i geof. 10, 301 (1946).
5. I. I. Chestnaya, Trudy GGO, issue 7 (1948).
6. M. E. Shvets, Prikl. matem. i mekh. 13, 257 (1949).
7. N. S. Shishkin, UFN 31, 461 (1947).
8. N. S. Shishkin, Izv. AN SSSR, ser. geogr. i geof. 12, 249 (1948).
9. N. S. Shishkin, Trudy GGO, issue 7, 13 (1948), issue 24 (1950).
10. N. S. Shishkin, Inform. sbornik GUGMS No. 1, p. 47 (1951).
11. Austin P. and Bemis A., J. Meteor. 7, 145 (1950).
12. Bergeron T., Tellus, QJ Geoph. Stockholm, 1, 32 (1949).
13. Byers H. and Coons R., J. Meteor. 4, 75 (1947).
14. Findelsen W., Physik der Wolken, Berlin, 1939.
15. Findelsen W., Met. Zeits. 55, 121 (1938); 56, 365, 453 (1939).
16. Gunn R., Phys. Rev. 71, 181 (1947).
17. Israel H., Met. Zeits. 61, 1 (1944).
18. Langmuir I., J. Meteor. 5, 5 (1948).
19. Langmuir I., Science 112, 2898 (1950).
20. Marshall I., Langille R., Palmer W., J. Meteor. 4, 186 (1947).
21. Maynard R., J. Meteor. 2, 4 (1945).
22. Simpson G. and Scrase F., Proc. Roy. Soc. A161, 309 (1937).
23. Simpson G. and Robinson G., Proc. Roy. Soc. A177, 281 (1941).
24. Workman E. and Reynolds S., BAMS 30, 142 (1949).

1. As cited in the source. ↩

2. visible reference marker in the source text. ↩↩