Physics of ball lightning

PHYSICS REPORTS (Review Section of Physics Letters) 224, No. 4 (1993) 151—236. North-Holland

PH Y S IC S R EPORTS

Physics of ball lightning B.M. Smirnov Institute of High Temperatures, IVTAN, Izhorskaja 13/19, Moscow, 127412, Russian Federation Received May 1992; editor: ii. Budnik

Contents: 1. Introduction

154

Part I. Observational data 2. Observations of ball lightning 2.1. Collections of ball lightning observations 2.2. Conditions of observations 2.3. Character of appearance and decay 2.4. Probability of an observation 2.5. Authenticity and accuracy of the observational data 3. Observational parameters of ball lightning 3.1. Geometrical and temporal parameters 3.2. Glow parameters 3.3. Energy and thermal properties 4. Analysis of observational data 4.1. Correlations between ball lightning and usual lightning 4.2. Brightness of ball lightning 4.3. Energy 4.4. Correlations between size and lifetime 4.5. Mean observational ball lightning

155 155 155 155 159 162 163 165 165 167 170 172

Part II. The nature of ball lightning 5. Models of ball lightning 5.1. Experimental modeling of ball lightning as a whole 5.2. Theoretical models of ball lightning 6. Structural analogs of ball lightning 6.1. Skeleton of ball lightning and fractal clusters 6.2. Aerogel as a sparse structure 6.3. Thermal and optical properties of aerogel 6.4. Process of densification of aerogel 6.5. Aerogel as an explosive substance 6.6. Fractal fibers 7. Mechanics and gas dynamics of ball lightning 7.1. Elastic properties of ball lightning 7.2. Interaction of ball lightning and the surrounding air

179 179 179 182 184 184 188 189 191 193 197 200 200 202

172 172 174 177 178

7.3. interaction of ball lightning with air flow 7.4. Ball lightning as an acoustic source 8. Energy and transfer processes in ball lightning

204 205 206

8.1. Types of energy in ball lightning 8.2. Energy release ball lightning 8.3. Transfer of theofenergy and gas inside porous ball

206 209

8.4. lightning Time variation of heat transfer processes in ball

210

lightning 9. Radiative processes in ball lightning 9.1. Spotted structure of glowing ball lightning 9.2. Air equilibrium inside the ball lightning 9.3. Pyrotechnical composition as an analog of ball lightning 9.4. flame as model of source ball lightning 9.5. Candle Coal model of achemical with prolonged glowing 9.6. Glowing according to fractal fiber model 10. Electrical properties of ball lightning 10.1. Electrical parameters 10.2. 10.3. 10.4. 10.5.

Electrical processes of ball lightning Initial conditions of ball lightning formation Aggregation of charged solid particles Creation of the electrostatic potential of ball lightning 10.6. Ball lightning as a source of plasma 11. Properties of the skeleton 11.1. Specific weight 11.2. Surface tension 11.3. Physics of the surface tension and restructuring processes 12. Conclusions References Note added in proof

0370-1573/93/$24.00 © 1993 Elsevier Science Publishers B.V. All rights reserved

211 213 213 214 215 217 218 220 220 220 222 222 223 224 225 226 226 227 228 230 231 235

PHYSICS OF BALL LIGHTNING

B.M. SMIRNOV Institute of High Temperatures, IVTAN, Izhorskaja 13/19, Moscow, 127412, Russian Federation

NORTH-HOLLAND

B.M. Smirnov, Physics of ball lightning

153

Abstract: An up to date description of the state of the ball lightning problem is given. The properties of ball lightning have been derived from a statistical treatment of thousands of observations. The experimental modeling of ball lightning as a whole is reviewed. The analysis leads to the conclusion that ball lightning has a rigid skeleton; a spotted structure ofits glowing follows from a large difference between the radiative and mean temperatures of the ball lightning. Ball lightning is a many-sided phenomenon, and therefore has a number of analogs which are related to its separate properties and which can be modeled. The mechanical, gas-dynamical, energetic radiative and electrical processes of ball lightning are analyzed on the basis of such analogs and recent scientific information. According to this analysis, the substance composing ball lightning has a sparse fractal structure, similar to an aerogel, with the density of a gas and the behavior of a solid or liquid. The best model resembling the ball lightning structure is a knot of fractal fibers. The glowing of ball lightning is created by many thermal waves that propagate along separate fibers, use the surface energy of the structure, and form glowing hot zones with a temperature of about 2000 K. A number of models considered allow us to study the nature of ball lightning in detail.

1. Introduction Studies of ball lightning have been undertaken for about two thousand years [1]. Nowadays studies of this phenomenon have intensified [2] and recently a number of reviews and books [3—12]on this topic have been published. Two international symposia on ball lightning (in 1988 and 1990) [131helped to intensify investigations on ball lightning. Some national centers on ball lightning were created, and the International Committee on Ball Lightning was founded. All these activities brought the ball lightning studies to a higher scientific level, and attracted the attention of many scientists to this problem. Now we have a good deal of information about ball lightning observations. There are enough experiments of production of ball lightning in the laboratory, and we have a number of general concepts related to its nature. But these concepts are schematic and refer to different aspects of the nature of ball lightning. As for the experiment, it is now found at an initial stage, and hence does not provide a systematic solution of the isolated problems. The main problems concerning the nature of ball lightning that were stated by Arago [14] 160 years ago, have not yet been solved. Arago’s work and the experience of recent studies of ball lightning show that the ball lightning phenomenon contains some qualitative elements which at the present time have no analog in science. Understanding its nature should enrich the scientific knowledge and encourage the development of new directions in physics and chemistry. Thus the solution of the ball lightning problem, which is connected with a deep comprehension of the surroundings, must lead to new concepts in science. From now on one can expect rapid progress in the study of ball lightning because one has a clear idea of what is necessary for this. The aim of this review is to give an up to date account of this phenomenon and to extract the problems that require a solution. It is of interest to compare this review with the previous review of the author [9] which was published three years ago and had the same goals. The main questions of the nature of ball lightning remain, but the content of these problems has undergone important changes. It reflects both the deep comprehension of the separate properties of ball lightning, and new information related to separate physical problems. Thus now we have more realistic models describing the individual aspects of the ball lightning problem. The ball lightning problem includes, on the one hand a collection of events of ball lightning observations together with their analysis, and on the other hand models which describe separate properties of ball lightning, based on real objects and phenomena. These objects and phenomena have some properties similar to those of ball lightning, and therefore they model the corresponding aspects of the ball lightning phenomenon. This review contains the analysis of observational data and models of ball lightning. It is necessary to stress the methodological aspect of the ball lightning study. The experience gained from its research shows that it is impossible to solve this problem on the basis of a one-step approach, i.e., a brilliant idea, or a lucky experiment. The development of this problem requires at each step a comparison between the corresponding information and the physical laws. The current trend in this direction is reflected in the present review. 154

B.M. Smirnov, Physics of ball lightning

155

PART I. OBSERVATIONAL DATA

2. Observations of ball lightning 2.1. Collections of ball lightning observations The study of ball lightning has a long history [1] which has lasted for more than two thousand years. The first collection of ball lightning observations was published in the Arago book [14]one and a half centuries ago. Now we have some collections of observational data which are given in table 2.1. These data are characterized by the method with which they are processed, and eyewitness reports from the various regions are used. These data are therefore complementary. 2.2. Conditions of observations Let us consider the conditions in which ball lightning is observed. These conditions reflect the properties of ball lightning. Table 2.2 contains the distribution of the distances over which ball lightnings were observed by eyewitnesses. These data relate to observations in the USSR and Hungary. One can see a low degree of accuracy in these data. Approximately 60% of ball lightnings are observed at distances of less than 5 m from the eyewitness. Let us introduce the probability p(R) of a ball lightning being observed by an eyewitness at a distance R. The quantity 4lTp(R)R2 dR is the probability that ball lightning is observed in an interval of distances between R and R + dR. According to the normalization condition we have

J

p(R) 41TR2 dR

1.

(2.1)

Table 2.1 Collections of observational data on ball lightning Authors

Year

Country

Number of analyzed eases

References

Arago Brand Humphreys McNally Rayle Dmitriev Arabadji Grigor’ev, Dmitriev Charman Stakhanov Keul Grigor’ev, Grigor’eva Othsuki, Ofuruton Egely Bychkov

1859 1923 1936 1966 1966 1969 1976 1978, 1979 1979 1979, 1985 1981 1986 1987 1987 1991

France Germany USA USA USA USSR Holland USSR England USSR Austria USSR Japan Hungary USSR

30 215 280 513 112 45 250 327 76 1022 80 2082 2060 300 2000

[141 [15] [16] [17] [181

119] [20] [21] [5] [6,8] [22,23] [24—26] [27—29] [10,30—32] [33]

156

B.M. Smirnov, Physics of ball lightning Table 2.2 Distribution of distances over which ball lightning was observed by eye witnesses Stakhanov [6,8]

Grigor’ev [25]

fraction (%)

number of cases

fraction (%)

Egely [10,30]

Range of distances (m)

number of cases

number of cases

0—1 1—5 5—10 10—20 20—50 50—100 >100

158 331 104 102 103 107 80

16 34 11 10 10 11 8

505 476 87 95 92 62 137

35 33 6 7 6 4 9

25 119 22 21 21 5 31

total

985

100

1454

100

244

fraction (%) 10 49 9 8.5 8.5 2 13 100

Let us represent such a probability in the form p(R)

=

3/R~)’~1 n(~irR~)’(1 + R

(2.2)

,

where n and R 0 are parameters, assuming this expression satisfies the normalization condition. Table 2.3 and fig. 1 summarized the data of table 2.2 together with data obtained on the basis of formula (2.2) with suitable values of parameters. Now let us consider the other variant of a treatment of observational data. Assume that ball lightning is observed only close to the Earth’s thethat probability p( p) ofis observing 2ii~pdpsurface, p(p) isand theintroduce probability ball lightning observed itbyatana distance p. Then the quantity eyewitness at a distance in the interval from p to p + dp. The normalization condition has the form

J

p(p) 2~pdp

1.

(2.3)

~~ 1200 10

0—1

1—5

5—10

10—20 20—50 50—100

>100

Distance (m)

Fig. 1. The distribution of ball lightnings on the distances (in m) from eyewitnesses (3352 events).

B.M. Smirnov, Physics of ball lightning

157

Let us approximate this probability by the formula p(p)

2 —1

=

2

2 —n—i

n(~p0) (1 + p /p0)

(2.4)

,

where n and p0 are parameters, and where this expression satisfies the normalization condition. Table 2.3 contains one example of the distribution (2.4) with suitable values of the parameters. Table 2.4 contains the Grigor’ev data [25] for the distribution of observed ball lightnings with reference to place. These data include 1984 observed cases (see fig. 2). The frequency of occurrence of ball lightning depends on local climatological and geographical conditions. Egely [30] remarked that ball lightnings are concentrated mainly along rivers and foothills. Evidently, there is heightened probability of electrical activity in such places. But ball lightnings are seldom observed on high mountains. Berger [34] did not observe any ball lightning during many years of lightning observations in a meteorological observatory in Switzerland. Keul [23]contacted seven meteorological observatories in Austria and Germany, ranging from 1000 to 3000 m above sea level and became convinced of the absence of ball lightning observations there in the course of some tens of years. According to observational data there is a correlation between the appearance of ball lightning and usual lightning. Because thunder storms take place in the summer in the northern hemisphere, ball lightning must be observed there mostly in the summer. Table 2.5 gives the distribution of ball lightnings with respect to months in the USSR. Figure 3 contains this distribution in the USSR (taking into account the data of table 2.5) and the Egely [10, 31] data on Hungary. According to observational data there is a correlation between the appearance of ball lightning and usual lightning. Because thunderstorms take place in the summer in the northern hemisphere, ball lightning must be observed there mostly in the summer. Table 2.5 gives the distribution of ball lightnings with respect to months in the USSR. Figure 3 contains the distribution in the USSR (taking into account the data of table 2.5) and the Egely [10, 31] data on Hungary. Table 2.6 and fig. 4 give the distribution of ball lightning over the observation times in the USSR (the Grigor’ev data [25]).Table 2.7 contains the Grigor’ev data [25](1924 observational events) related to the weather conditions during which ball lightning was observed. Table 2.8 gives the corresponding data of Ohtsuki and Ofuruton [27—29].As can be seen, the USSR and Japanese conditions for the appearance of ball lightning are different.

Table 2.3 List of parameters referring to the probability distributions eqs. (2.2) and (2.4) Probability Number of cases Range of distances (m) 0—1 1—5 5—10 10—20 20—50 50—100 >100 total

Probability (%)

totals based on the data of table 2.2 688 926 213 218 216 174 248 2683

26±10 34±6 8±2 8±2 8±2 7±4 9±3 100

eq. (2.2), n R0 = 0.8

(%)

=

R0 = 1

eq. (2.4), n

16 43 12 8 7 4 9

25 42 10 7 6 3 8

28 32 9 8 7 4 12

100

100

100

=

0.2, p0 = 0.5

158

B.M. Smirnov, Physics of ball lightning Table 2.4 Distribution of ball lightning observations according to place based on Grigor’ev’s data [251 Place of observation

Fraction,

inside rooms on streets on fields in forests ashore a river, lake in sky from earth in mountains in clouds from aeroplanes

50.2 24.6 9.5 4.4 4.0 4.0 2.3 1.0

(%)

On fields 9.5% In streets 24.5%

In sky 5.1% In forests 4 4%

2.3%

Indoors 50.2% Fig. 2. The distribution of ball lightnings on the places of observation (1984 events).

Table 2.5 Distribution of ball lightning observations according to the months of the year Stakhanov’s data [6,8] Month

number of cases

May June July August September October—April

48 158 355 225 35 63

total

884

fraction (%) 5.4 17.9 40.2 25.4 4.0 7.1 100

Grigor’ev’s data [25] number of cases 117 296 823 296 69 112 1713

Total

fraction (%)

number of cases

6.8 17.3 48.0 17.3 4.0 6.5

165 454 1178 521 104 175

100

2597

fraction (%) 6.4±0.5 17.5 ±0.3 45 ±3 20 ±4 4.0 ±0.4 6.7 ±0.3 100

B.M. Smirnov, Physics of ball lightning

159

1500

e~

____ __

1200-

C ‘p

~

coo-

o

J.~Iy

Au~jst

Septeirber

Octob—Aprt

Month Fig. 3. The distribution of ball lightnings on the months of the year in which observations took place (3286 events).

~o~~ifl3iii2 0—3

3—6

6—9

9—12 12—15 15—18 18—21 21—24

Hours Fig. 4. The distribution of ball lightnings on the times of the day (2104 events).

2.3. Character of appearance and decay The behavior of ball lightning during its appearance and decay can provide useful information related to its properties. Table 2.9 contains the ways in which ball lightning can appear for the cases where the birth moment was fixed. These data use observations in the USSR and are summarized in fig. 5. The decay of ball lightning can proceed smoothly or with an explosion. In the various collections of observational data different ways are used for treating the data in the case of a slow decay. Therefore the comparison of these data is problematic. An explosion takes place in approximately 50% of the

160

B.M. Smirnov, Physics of ball lightning

Table 2.6 Distribution of ball lightning observations according to the time of the day Range of times (hr) 0—3 3—6 6—9 9—12 12—15 15—18 18—21 21—24

Probability of observations (°~~‘~

Number of cases 48 10 129 449 392 229 140 39

total

Table 2.7 The Grigor’ev data [24, 251 (1924 cases)

3.3 0.7 90

3113 27.3 16.0 9.7 2.7

1436

100

Weather conditions during thunderstorm one half hour before a thunderstorm one half hour after a thunderstorm rain cloudy clear, fine

Probability of observation (%) 61.6 6.6 8.8 7.2 6.0 9.8

Table 2.8 The Ohtsuki and Ofuruton data [27—29](2060 cases) Weather

Probability of observation

fine, cloudy rain thunderstorm other

89.1 7.6 2.5 0.8

(%)

cases (see fig. 6). Table 2.10 lists the ways in which ball lightning can decay, based on the observations in the USSR. Table 2.11 and fig. 6 include additional data. The accuracy of this information depends on both the statistics and the methods employed to process the observational data. Summarizing the information of table 2.11 one can find that the probability of explosion of ball lightning is equal to 0.52 ± 0.09; the probability of its slow decay equals 0.39 ±0.07; and the probability that it decays in fragments is 0.09 ±0.03.

Table 2.9 Ways in which ball lightning can appear when the birth moment is fixed Number of cases The birth character from the point of lightning discharge in the channel of lightning on metallic conductors after lightning discharge on metallic conductors indoors on metallic conductors from spark in air from “nothing” in clouds Total

Stakhanov data [6,8]

Grigor’ev data [25]

—

32 29 21 154 20 17 13

67

286

31 29 7

B.M. Smimov, Physics of ball lightning

161

From lightning 3 2.0%

From ~nothin( 6.8% In clouds

a.7%

On conductors 57.5% Fig. 5. The distribution on the ways in which ball lightning appeared (353 events).

Slow extinction 33.5%

Decay ki part. 7.1%

Exploalon 59.3%

Fig. 6. The distribution on the ways in which ball lightning decayed (2818 events).

Table 2.10 Ways in which ball lightning can decay Stakhanov data [6,8] fraction (%)

Character of decay

number of cases

explosion disappearance in the Earth disappearance in a conductor slow extinction decay in fragments

335

55

197

32

78

Total

610

Grigor’ev data [25] number of cases

fraction (%)

Total number of cases

fraction

43 14 19 23 11

828

48 ±5

712

41 ±5

13

493 157 100 258 123

201

11

100

1131

100

1741

100

±1

(%)

162

B.M. Smirnov, Physics of ball lightning Table 2.11 Additional data on ball lightning decay Number of cases Collection

explosion

McNally [17] Rayle [18] Charman [5] Stakhanov [6,8] Grigor’ev [25] Egely [10] Total

slow extinction

309 24 26 335 493 84

112 54 25 197 515 43

1131

903

decay in fragments — — —

78 123 —

201

2.4. Probability of an observation One of the issues of the ball lightning problem is the probability of observing a ball lightning. Though ball lightning is a rare phenomenon, this probability, when averaged over the observation place and time, it becomes appreciable. According to Rayle [18], among the 4400 NASA colleagues who were questioned, 180 saw ball lightning. Stakhanov [8] estimated the probability of observing ball lightning during a lifetime as i0~. A reliable method for estimating the lower bound of the above probability utilizes the data of Egely [10, 31], who compiled about 520 cases of ball lightning observations by 1500 Hungarians based on Hungarian news reports. There were approximately 1.5 X 106 newspaper readers. Assume that all people who observed ball lightning reported it. Take into consideration the distribution of witnesses with respect to age. Then we obtain, on the basis of the Egely studies [10], the lower bound for the probability of observing ball lightning during a lifetime as P = 2 X i0~. One more estimate may be obtained on the basis of other data [31] collected by Egely. During 1987 about 39 observations of ball lightning were reported in Hungary. Dividing this number by 1.5 million newspaper readers, we obtain the lower bound for the probability of observing ball lightning during a given year as 3 X i05 and during a lifetime as 2 x i0~. The above data for the probability of a ball lightning observation during a lifetime are collected in table 2.12. Summarizing these data we obtain, for the mean probability of an observation of ball lightning during a lifetime, P = 10_24±07.This value must be reduced to conform with the Egely limit, yielding P=

(2.5)

1022±05.

Table 2.12 The probability of observing ball lightning in the course of a lifetime Authors Rayle [17] Stakhanov [6] Egely [10] Egely [31]

Probability i0~’~ iO~ >1027

>l0~

B.M. Smirnov, Physics of ball lightning

163

Note, that these data do not agree with the Barry [2,7, 35] estimates, according which the mean probability for the appearance of ball lightning is equal to W= 109_108 km2 min1. This is approximately six orders of magnitude lower than the probability of appearance of usual lightning. Let us find a connection between the above results and this estimate. Introduce use of the distribution function 2irp dp p( p) that ball lightning is observed by an eyewitness at a distance in the interval from p to p + dp near the Earth’s surface. If T is the lifetime of a man, the above parameters are connected by the relation P~WrIp(0).

(2.6)

From the observational data (see table 2.2) we have p(O) = 1006±05 m2. Assume that the mean lifetime of a man equals r = 80 y 2 x i09 s. Thus on the basis of Barry’s data [2, 7, 35], we obtain for the probability of an observation of ball lightning during a lifetime P = 108.6±1.0 As can be seen, this value is some orders of magnitude smaller than the above estimates. The reason for such a large discrepancy is that, whereas Barry’s estimate relates to the probability of observing ball lightning, different estimates are valid with reference to the probability of observing the origin of the ball lightning. Assuming these estimates to be reliable, one can find, by comparing them, the mean probability to see a separate ball lightning, which is equal to 1064±1.5 The probability density of the ball lightning originating near the Earth’s surface follows from formulae (2.5) and (2.6), W—= Pp(0)!T =

1013±10 km2

y~

.

(2.7)

It means that the frequency of ball lightning originating near the Earth’s surface is equal 1025±10~—1 Compare the value (2.7) with the median frequency of flashes of usual lightning [36]which is equal to 5.4 ±2.1 km -2 y From this it follows that the probability of an occurrence of ball lightning is related with the corresponding value for usual lightning. Indeed, the number of ball lightnings n per single flash of usual lightning is ~.

n=4x 10±13.

(2.8)

One may conclude that ball lightning is a phenomenon which happens often, but is seldom observed.

2.5. Authenticity and accuracy of the observational data The authenticity and reliability of the observational data require a separate discussion. The authenticity of an individual observer is low. One of the problems in analyzing the observational data is the low degree of authenticity. The authenticity of an individual observation cannot be sufficient and for this reason some tens of years ago the opinion prevailed that ball lightning is an optical illusion (see refs. [16,33, 37, 38]). According to this, the strong flash of usual lightning creates a trace on the eye retina and such a trace persists as a spot for the duration of 2—10 s. This spot is perceived to be ball lightning. But with the accumulation of observational data this hypothesis loses its proponents. Indeed, each observational case involves many details, which cannot arise in the brain of an eyewitness as the result

164

B.M. Smirnov, Physics of ball lightning

of the lightning flash after-effect. Besides, there are some reliable photographs of ball lightnings (see, e.g., refs. [7,29, 117]), which confirm the reality of this phenomenon. For this reason all the authors of reviews and books about ball lightning and those who collected ball lightning observations believe that ball lightning is indeed a real phenomenon. But the reliability of an isolated event of ball lightning observation is not high. First, the eyewitness is indeed not ready to observe the phenomenon, which arises unexpectedly, hence the eyewitness is found in an excited state. This can lead to errors in the description of ball lightning. Second, in trying to use some definite schemes for the explanation of the observed phenomenon, an eyewitness can displace details of the observation. Third, such an error affects the method of determining the parameters of ball lightning. Fourth, this information often reaches the press. Due to the haste with which this sensational event is published, a distortion of the information is possible. We mention the following curious case to illustrate this [39]. The newspaper “Komsomal’skaya Pravda” of 5th July 1965 published a note entitled, “A fire guest”. It described the behavior of ball lightning of a diameter about 30 cm which was observed in Armenia. In particular, this note stated, “Having gone around a room, the fire ball passed to a kitchen through an open door and afterwards flowed into a window. In a courtyard ball lightning struck against the ground and exploded. The force of this explosion was so large, that a wattle and daub house fifty meters away from it were destroyed. Fortunately, nobody was injured.” An inquiry was lodged with the department of the Armenian Meteorological Service [39]. The answer confirmed the observation of ball lightning. The description of the behavior of the ball lightning differed in essential points from the information of “Komsomol’skaya Pravda”. The end of the answer was the following: “As for the wattle and daub house, the fate of this half ruin has no connection with the ball lightning.” Unfortunately, this history had a continuation. The information of “Komsomol’skaya Pravda” lay at the basis of an estimate of ball lightning energy [40] which was set equal to i0~kcal (one tonne of an explosive material!).*) This estimate was taken into consideration in many papers, including the books by Singer [3] and Barry [7]. Because we have a limited number of observational events at our disposal allowing us to estimate the energy of ball lightning (see refs. [7,8, 31] and table 3.9), such misinformation is unpleasant. The accuracy of the individual parameters of ball lightning is limited by human possibilities. Their limits may be estimated. For example, Charman [5] reported the observation of meteorites in the USA. Eyewitnesses were questioned after the observation was made. The time lapse during which the meteorites were observed in the sky was reported with an accuracy of about 30%, other reported parameters (colour, sound) were less reliable. Grigor’ev and Grigor’eva [26]surmounted the limitation on accuracy by questioning a large group of students. They concluded that the ball size may be determined with an accuracy of 10±0.06; the time interval, with an accuracy 10±02; and the brightness of the light source, with an accuracy 10±01. The actual accuracy of the parameters of observations on ball lightning is less than these values. It should be kept in mind that the reliability of an individual report is low, and the information about the observed ball lightning may be distorted by it. A natural way to reduce the effect of unreliable information is to gather a large number of observational events. A large statistical base would allow the reduction of errors.

*)

One can compare this value with estimates on the basis of observational data (table 3.9).

B.M. Smirnov, Physics of ball lightning

165

3. Observational parameters of ball lightning 3.1. Geometrical and temporal parameters The observational parameters of ball lightning allow us to comprehend what ball lightning is. Further we analyze the observational properties of ball lightning obtained from the treatment of eyewitness reports. Table 3.1 gives the mean diameters of ball lightning according to various data collections. By statistically averaging these data taking into consideration the number of events in each collection, we have for 3763 events D = 23 ±5 cm. Stakhanov [8] suggested the following distribution function for the diameters of ball lightning: f(D)

=

(DID~)exp(- DID0).

(3.1)

Grigor’ev [25] gives the close dependence for the distribution function. The distribution function in different collections has different functional forms. For the distribution function (3.1) we have the following connection between the mean ball lightning diameter D and the most probable diameter D0: D = 2D0. Analyzing this distribution, Dijkhuis [41]found that J~ f(D’) dD’ is approximated well by a log—normal distribution law. Table 3.2 lists some results of this analysis the geometric mean diameter, the standard deviation for the log—normal distribution and the correlation coefficient between a log—normal distribution with optimal parameters and the corresponding data; these are plotted with a rescaled V-axis where the log—normal dependence is a straight line. As its name indicates, ball lightning has usually a spherical form. The probability for the different forms of ball lightning is given in table 3.3. Usually ball lightning conserves its form during its existence. According to the Grigor’ev collection of data [25], the change in the form of the ball lightning was established in 134 cases (6%) out of 2082 observational events. In 25 cases the ball was transformed to a —

Table 3.1 Mean diameters of ball lightning from various data collections Collection

Number of cases

The mean diameter (cm)

McNally [17] Rayle [18] Charman [5] Stakhanov [6,8] Keul [22] Grigor’ev [25] Egely[10,31]

446 98 64 1005 150 1796 204

30 32 26 22 30 19 35

Total

3763

23

±5

Table 3.2 Results of the analysis by Dijkhuis

[411 of the distribution function

(3.1) for the ball lightning parameters

Collection

Number ofcases

Geometric mean diameter (cm)

Standard deviation

Correlation coefficient (cm)

McNally 118] Stakhanov [8] Grigor’ev [25] Egely [10,31]

446 1005 1796 204

28.3 16.9 14.4 23.3

0.419 0.374 0.405 0.493

0.9979 0.9988 0.9958 0.9970

166

B.M. Smirnov, Physics of ball lightning Table 3.3 Probability of the occurrence of various forms of ball lightning Number of cases Form

Stakhanov [6,8]

Grigor’ev [25]

Total

spherical ellipse tape, band no shape pear-shaped disk ring cylinder spindle

788 52

—

1836 54 52 29 7 16 9 4 5

2624 106 52 43 27 17 11 5 5

878

2013

2891

total

—

14 20 1 2 1

Total probability

(%)

90.8 ±0.6 3.7±0.3 2 ±2 1.5±0.3 0.9 ±0.6 0.6±0.5 0.4±0.3 0.2±0.2 0.2±0.2 100

band and in 15 cases the band was transformed to a ball; in four cases the ball was deformed during jumps, in 11 cases the ball lightning was stretched along the direction of a conductor. In addition to the Grigor’ev data [25], we have the following information: in 226 cases (11%) the ball lightning had a transparent membrane, in 119 cases (6%) it had a tail, and in 143 cases (7%) there were moving points or filaments inside the ball lightning. Let us introduce the probability P(t) that ball lightning, which is formed at time t 0, does not decay up until time t. According to the analysis of observational data the function P(t) is not the exponential function. The best analysis of this function is made by Grigor’ev et al. [25]. The probability of ball lightning conservation is given in the form P(t)

=

A1 exp(—tit1) + A2 exp(—tit2)

+

A3 exp(—tIt3).

(3.2)

Table 3.4 contains the parameters of this formula according to various observational data. Due to the complicated form of the probability of ball lightning conservation in time, P(t), we cannot determine exactly the mean life time of ball lightning. Let us introduce the following mean times of ball lightning:

ri

Jt~_~~ dt, r2

P(r1)= lie,

~

P(r4)112.

(3.3)

Table 3.4 Parameters of eq. (3.2) obtained from various data collections Data

Number of cases

A

McNally [17] Rayle [18] Stakhanov [6,8] Grigor’ev [25]

445 95 982 437

0.86

~ (s)

A,

r, (s)

3.5

0.14 1.0 0.43 0.27

44 14 54 30

—

—

0.57 0.59

11 3.0

A

r7 (s)

—

—

—

—

—

—

0.14

315

Based on 256 observed cases in which the origin of hall lightning and its decay were established, and on 181 cases, when ball lightning appeared after the flash of usual lightning and its decay was established.

B.M. Smirnov, Physics of ball lightning

For the exponential law of ball lightning decay, P(t)

‘-~

69T

167

exp(—tIT), we have r1

=

=

r3 and r4

=

1. Table 3.5 contains the values of the mean times according to different observational data.

O.

Usually r4 is introduced as the mean life time of ball lightning. According to the data of table 3.5, it is = 8 x 10±03s. It is close to the mean value T2 = 9 X 10±03 s. Thus one can conclude now that the mean life time of ball lightning is r

8

X

10~°~ s.

(3.4)

Note that according to the analysis of Dijkhuis [41]the cumulative probability P(t) is approximated well by the log—normal law. Usually ball lightning is moving uniformly in a horizontal direction. That is established in 53% of the Rayle data [18] (98 cases), in 75% of the Stakhanov data [6, 8] (928 cases) and in 75% of the data of Grigor’ev et al. [25](1743 cases). Ball lightning is observed to move along conductors in 20% of the cases according to the McNally data [17] (513 cases), in 16% of the cases according to the Rayle data [18] (98 cases) and 4% of the cases according to the Grigor’ev et al. data [25](1743 cases). The mean ball lightning velocity is of the order of 1 m/s [5, 25]. Furthermore, according to the Grigor’ev data collection [25], in 45 observational cases (2.6%) ball lightning moved around obstacles, in 82 cases (4.7%) it fell from clouds and in 7 cases (0.4%) it rose toward the clouds. 3.2. Glow parameters Table 3.6 lists the distribution of ball lightning with respect to colour according to different collections of observational data. The last column presents the probability of a given colour of ball lightning, obtained on the basis of the summarized data and the Stakhanov data (in brackets). As can be seen, the summarized data are close to the Stakhanov data (except in the case of colour mixture). Table 3.7 compares the summarized data of table 3.6 and the Grigor’ev data [25]containing the same number of observational cases. There is no agreement between these. Future investigations must show the reason for this divergence. The summarized data of table 3.7 are shown in fig. 7. The distributions of ball lightning with respect to brightness according to the collections of the ball lightning observations of Stakhanov [8] and of Grigor’ev et al. [25] are given in table 3.8. Figure 8

Mean decay times r

Table 3.5 r4 of eq. (3.3) from various data collections T’~

Collection McNally [17] Rayle [18] Stakhanov[6,8] Grigor’ev [25] Egely [10] total mean

Number of cases 445 95 982 437 152

. . . Table 3.6 Distnbution of ball lightning colour from various data collections

(s)

Number of cases

r~

r~

T3

T4

Colour

12 14 30 40 38

4 14 17 5 9

4.5 14 22 9 18

3 10 14 4.5 7.5

white red orange yellow green blue, violet mixture

2111 30

9

13

~ The accuracy of the r1 value is 10~02;in other cases it is

8

3.

io’°

total

[171 [18] 44 48 50 40 3 42 84 311

27 7 46 37 10 25

[5]

—

15 5 12 20 2 5 9

152

68

[10,30] 55 56 7 43

Fraction [6,8]

Sum

18 26

244 180 113 246 12 111 30

385 296 228 386 27 201 149

205

936

1672

—

(%) 23(26) 18(19) 14(12) 23(26) 2 (1) 12(12) 9 (3) 100

168

B.M. Smirnov, Physics of ball lightning Table 3.7 Comparison between the data of table 3.6 and the Grigor’ev data Sum of data in table 3.6

Colour white red, pink orange yellow green blue, violet mixture total

Number ofeases 385 296 228 386 27 201 149

Grigor’ev data [25]

Fraction

(%)

23 ±1 18 ±1 14 ±1 23 ±1 1.6±0.3 12 ±1 9 ±3

1672 Given to statistical accuracy only.

100

Number of cases

Fractiona)

247 297 633 307 22 230 67 1803

(%)

14 ±1 16 ±1 35 ±1 17 ±1 1.2±0.3 13 ±1 3.7 ±0.5

Total fraction

(%)

18 ± 5 17 ± I 25 ±10 20 ± 3 1.4± 0.3 12 ± 1 6 ±3

100

100

contains the summarized data. The ball lightning brightness is compared with the power of an equivalent electrical bulb. The accuracy of these data does not increase as the number of observations increases, and is estimated to within a factor 2—3. One of the questions of ball lightning luminescence is whether or not ball lightning is an optically transparent source. Some information may be obtained by photometry of ball lightning track. Specifically in this respect we consider the example of recording a ball lightning track [42, 9], which was photographed by Derjugin [42] during a thunder storm in 1958. Figure 9 shows the results of photometry measurements of ball lightning along the trace. As is seen from fig. 9, the temporal dependence of the radiation intensity of ball lightning is irregular and the main part of the radiation is emitted during a time interval short in comparison with the lifetime of the ball lightning. Let us analyze the transversal photometry of ball lightning photography [42]. Consider two limited cases with respect to the optical density of ball lightning. In the first case ball lightning radiates from the surface (optically thick system) and in the second case it radiates from the entire volume (optically thin system). We assume that ball lightning is homogeneous and that it moves uniformly. Let the radiation be projected on a given point of the film from the impact parameter p (p is the minimal distance from Yellow 20.2%

and viole

Orange 23.1%

—~

11.4%

Mixture 5.3%

N~’White

20.9%

Red and rose 17.7% Fig. 7. The distribution of ball lightnings on colours (4112 events).

B.M. Smirnov, Physics of ball lightning

169

Table 3.8 Brightness distribution of ball lightning based on the Stakhanov and Grigor’ev data collections Stakhanov data

[81

Grigor’ev data [25]

Power of equivalent bulb (W)

Number of cases

Fraction of total numbers

<10 10— 20 20— 50 50—100 100—200 200—500 >500

55 83 109 140 150 39 21

0.092 0.139 0.183 0.234 0.251 0.065 0.031

Total

597

1

Fraction of total numbers

89 103 209 314 376

0.067 0.078 0.158 0.238 0.285

230

0.174

1321

—2

—1

Number ofcases

—3

~

—~

0.40 0.35 ~

~

~1O

10-20

20—50

50—100100—200 ~20O

Brightness

,

W

Fig. 8. The distribution of ball lightnings on brightness. 1, ref. [8]; 2, ref. [251;3, ref. [33]; 4, eq. (4.3) with n

Distance along the trace Fig. 9. The intensity of the ball lightning glow as a function of the distance along its trace.

=

1.3, J

020W.

170

B. M. Smirnov, Physics of ball lightning

the center of the ball lightning to the whole complex of the ball lightning points which are projected on a given region of the film). In the case of an optically thick system the film blackening is proportional to the exposure time, 2(R~ p2)H2iv, where R0 is the ball lightning radius and v is its velocity. The 2, where the blackening at the track center is unity. In relative is equal to (1 R~ip~)U the case film of anblackening optically thin system the relative film blackening equals 1 R~Ip2.If the assumptions of the considered model are satisfied, the ratio of the observed and model blackening y should equal unity for any impact parameter p. Statistical treatment of the above data for the first model results in y = 0.81 ±0.16 and for the second model in y = 0.96 ±0.17. Thus we conclude that the model according to which the ball lightning radiates from the whole volume is favoured, but we cannot exclude the model with radiation from the surface. The spread in the data does not allow us to draw a definite conclusion about this event. It is important to add that it is impossible to judge the optical density of different ball lightnings on the basis of one event. —

—

—

3.3. Energy and thermal properties The estimate of the energy of ball lightning collected in table 3.9 were taken from books by Barry [7] and by Stakhanov [8], and papers by Egely [31]and Dmitriev et al. [431. These estimates were made on the basis of an analysis of separate observational cases. The treatments of the energy densities of ball lightning are similar [2,35]. Let us elaborate on two cases of ball lightning observation when a thermal action of ball lightning took place and this action was analyzed. In the summer of 1977 ball lightning was observed in Frjazino (near Moscow, USSR) by a group of pupils and their teacher. Red ball lightning with a diameter of approximately 5 cm approached a window frame from outside. A small hole with glowing edges having

Table 3.9 Estimates of the energy of ball lightning Character of the energy released water heated in a barrel splintering of a log burning a woman formation of 0~and NO, in the ball lightning track melting of asphalt trail of scorched grass heating of a wire bending of an iron pipe into a loop burning of a hole in a metallic pipe metal evaporation of a gun ramrod splintering of a log melting of a hole in glass fusion of a garden tap metal evaporation on an airplane antenna fusion of a metallic hook destruction of a brick lifting off roof explosion indoors knocking out a stone melting the ground

Energy (kJ) (1—3) 150 0.4 0.5

X

—

1700 150 80—100 150—200 2 90—120 20 5 20—120 0.7 10— 20 10 1500 200—500 10~

10~

Energy density (J/cm3)

References

(2—6) 85 1 0.4 100 900 18 10—12

[7] [7] [7] [7] [7] [7] [8] [8] [8] [8] [8] [8] [8] [8] [8] [8] [31] [31] [31]

— —

10—15 — —

9—60 —

5—10 —

360 15 500

X

i0~

[431

B.M. Smirnov, Physics of ball lightning

171

red colour was formed in the window glass. Then the diameter of the hole increased up to 3—4 cm and after that ball lightning blew up and disappeared with a loud explosion. It left a hole with a diameter of 5 cm in the window glass (of thickness 2.5 mm) without melting its edges. The time duration of this event was about 5 s. This case was analyzed by Kolosowskij [44] who carried out a model experiment by irradiating a glass with a CO2 laser. Two situations can arise. In the first, the specific laser power is large enough. Then a hole is melted in the glass, and after switching off the laser, a smooth crack is formed around the heated region. The edges of the hole show no signs of mechanical tension. This hole is similar to the hole caused by the action of the ball lightning. The second situation corresponds to low laser power. Then no hole is melted, but after switching off the laser, holes of irregular form with sharp splintering edges are formed. The author observed the same effect by using a glass-blowing burner to heat thin glass. The experimental modeling using a C02-laser [44] leads to a heating energy in the above case of 20 kJ. The calculation shows that this energy allows one to heat a comparable region of the glass (for a hole with a diameter of 5 cm the diameter of the heated region is 3.5 cm) up to the melting temperature. After considering the available information on the energy of ball lightning, this case testifies in favour of a thermal action of the ball lightning.

Another case was analyzed by Dmitriev et a!. [43]and relates to the ball lightning observed in Khabarovsk (USSR) in the summer of 1978 during a heavy rainfall. An orange ball lightning with a diameter of 1.5 m arose over a local cinema and was observed for the duration of 1 mm. After its explosion electric wiring was destroyed at a distance of 150 m and despite the heavy rain the ground was charred and melted in 3). a region a diameter 1.5 m pieces and a depth of 20—25 of The this A slagwith consisted of separate of irregular formcmof (the a sizevolume 5—6 cm. region was region about 0.4 m soil in this included a slag soil and a glass of an ectoplasm structure. The authors [43] performed experimental modeling and calculations for this observational case. The best results gave a high-frequency gas discharge (the frequency is equal to 108_lOs Hz and the wavelength is 0.3—3 m). The estimated energy which causes the soil to heat up to melting and water inside it to evaporate, is approximately equal to i09 J. This observational event is also compatible with the thermal action of the ball lightning. It possesses the characteristics of such an action. Note that the energy released during the decay of ball lightning may depend on geophysical conditions. According to the Egely studies [10,30], 90% of the ball lightnings in Hungary where an amount of energy higher 1 kJ was released, relates to places which are found in mountains or at distances of less than 10 km from mountains. Mountains occupy 30% of the territory of Hungary. Egely states that in Hungary lightnings on mountains have more energy than on the plains. Table 3.10 contains the Stakhanov data [6, 8] of ball lightning observations when eyewitnesses had a Table 3.10 Summary of the Stakhanov data which relate to heat sensation Distance of observation (m)

Number of cases of heat sensation per total number of observed eases

<1 1—2 2—5 >5

25/294 8/131 20/379 9/676

172

B.M. Smirnov, Physics of ball lightning

heat sensation from the ball lightning. The first quantity is the number of events when a heat sensation has taken place, the second quantity is the total number of observations for a given range of distances. Grigor’ev et a!. [25]reported a heat sensation in 64 out of 383 cases, in which the eyewitnesses analyzed the heat action of the observed ball lightning.

4. Analysis of observational data 4.1. Correlations between ball lightning and usual lightning The analysis of the conditions controlling the origin of ball lightning shows that the appearance of ball lightning usually relates to atmospheric electricity. According to european data collections [6,8, 10, 15, 25] (see also table 2.7), (70—80)% of the observations of ball lightning take place in thunderstorm weather. But according to japanese information [27—29](see table 2.8) most observations relate to clear weather. In spite on these contradictions a correlation between ball lightning and usual lightning exists in both cases. To demonstrate that let us introduce the correlation coefficient k = ((X

-

X)(Y

-

Y)) I[K(X

-

X)2) K(Y

-

Y)2~]U2

(4.1)

where X is the distribution function for ball lightning and Vis the same distribution for usual lightning; X, V are their mean values; averaging ( ) takes place on the distributions of these values. If X and V are random distributions the correlation coefficient is equal to zero. If these values are connected by relation V = AX + B, the correlation coefficient is equal to unity. The values of the correlation coefficients are given in table 4.1. The first column relates to the distributions over months, the second one uses the probability of observing these phenomena at certain times during the day. As can be seen, there is some correlation between observations of ball lightning and usual lightning. The average correlation coefficient is k = 0.84 ±0.04. This value confirms the connection between ball lightning and the phenomenon of lightning in the atmosphere. From this it follows, in particular, that ball lightnings in Europe are observed mostly in the summer. 4.2. Brightness of ball lightning Let us analyze the observed brightness of ball lightning (see table 3.8). For a treatment of these data let us introduce the distribution function of ball lightning over brightnesses in the form f(J) = (nJ~’/J~) exp[—(JIJ 0)~], Table 4.1 Correlation coefficients between ball and usual lightning Correlation coefficient Data

seasonal distribution

Stakhanov [6] Grigor’ev et al. [25] Ohtsuki and Ofuruton [27,281

0.83 0.79 0.86

time of day distribution —

0.88 —

(4.2)

B.M. Smirnov, Physics of ball lightning

173

where f(J) dJ is the probability that the ball lightning brightness is found in an interval from J to J + dJ; J0 and n are parameters. The probability that the brightness of ball lightning lies in an interval between J~and ~k

~

5] exp[—(Jk/JO)5]. P(J~,~k) = exp[—(J~/J0) Let us find the correlation coefficient (4.1) where X, =

(4.3)

—

P

0b~(J,, J~~1) is the observational value of the probability that the ball lightning brightness is concentrated in an interval between J~and J~ + 1; the robs are the summarized data of table 3.8; Y~= P(J1, .J~÷1) is this probability expressed through the distribution function (4.2). We choose the parameters n and J0 so that they correspond to the maximum value of the correlation coefficient. These parameters are included in table 4.2 and give the mean brightness of ball lightning, J = 1500 ±200 lm. Let us find the connection between the distribution functions f(J) and p( p) in the case in which these quantities are correlated. Use a simple model based on the assumption that ball lightning can be observed if illumination from ball lightning exceeds a critical value j0. Then the probability of a ball lightning observation is proportional to the time during which holds. Ballobservation lightning of 1 /2 this andcondition the duration of its is brightness J can at a If distance than r0 = (J/j0) function of the ball lightning over the proportional to be r observed (r~ p2)”2. F(J) islessthe distribution brightness, the observed distribution function is ~

—

F(J) = const X J312f(J).

(4.4)

The probability for an eyewitness to observe ball lightning passing by and having impact parameter p is p(p)

=

const

X

f

(J1j

2)1’2F(J) dJ = 0

—

f(1

~

p

l~p2

—

p2Ix2)~’2f(j 2) 0x

p

Use the simplest dependence (4.2), f(J) = exp(—JIJ

-~

p 2 ln(p 1 0/p),

2 we have the

0)/J0. Introducing p0 = (J0/J0)”

following asymptotic expressions for p(p): p(p)

(4.5)

~.

2 exp(—p2/p~), p ~ p p

‘~

p0

p(p)

p

0.

(4.6)

As can be seen, in the first limiting case, the data of table 2.2 are described well. Note that values of 1000 Im correspond values of p0 of the order of tens of meters. -~

Table 4.2 Parameter values for which the correlation coefficient is maximum Parameter

Stakhanov [8]

Grigor’ev et al. [25]

Total

total number of cases n ~ (W) J (W)

597 1.30 94 87

1321 1.36 133 122

1918 1.33 125 114

174

B.M. Smirnov. Physics of ball lightning

4.3. Energy Though there are some thousands of documented observations of ball lightning, the energy of the action of ball lightning can only be estimated in some tens of these events (see table 3.9). One can divide these events in two types. In the first, the action of ball lightning has an electric character. Then atmospheric electricity is the source of the released energy, and ball lightning causes an electrical breakdown in the atmosphere. In the second case the internal energy of ball lightning is used. Then a slow transformation of energy takes place and, probably, this energy is chemical. The corresponding data are included in table 3.9, and in tables 4.3 and 4.4. Let us treat these data in the traditional way (see refs. [9,46]). Assume that the probability that the energy of ball lightning is found in an interval from E to E + dE is proportional to dEIE. Assume also that the energy of ball lightning is limited in the range Emin < E < Emax~ Then we have for the probability that the ball lightning energy exceeds E, P(E) = A

—

B log E;

B = log(E~~~IE~j5), A = B log

(4.7)

Emax.

First let us process the data of table 4.3. We must discard nine values from this table because of their low energy. Then if the kth event corresponds to energy Ek, the probability that the ball lightning energy exceeds Ek is equal to (2k 1)/18. The results of this procedure give the parameters of formula —

(4.7), 09±02 ~

mm

‘

max —

i11 E The most probable energy (its probability is equal to 1/2) is E —

112

=

100.8±0.2kJ

3. = E1,2I(4irR~i3)= ~ The average internal energy is

E=I

and its density equals (4.8)

J/cm

EdP= Emax[lfl(EmaxIEmmn)]~’ =5OkJ x

(4.9)

10±0.2.

Table 4.3 Release of the internal energy of ball lightning Means of energy release burning a woman formation of 03 and NO 2 in the ball lightning trace burning a hole in a metallic pipe metal evaporation from a gun ramrod melting a hole in a glass fusion of a garden tap metal evaporation from an airplane antenna fusion of a metallic hook heating a wire

Energy (kJ)

References

0.4

[7]

0.5 150—200

[7] [8]

2 10— 20 5

[8] [8] [8]

20—120 0.7 150

[8] [8] [8]

Table 4.4 Events of electric energy release by the action of ball lightning Means of energy release Energy (kJ) References heating a water in a barrel splintering of a log bending an iron pipe in a loop splintering of a log destruction of a brick lifting off a roof explosion indoors knocking out a stone

1000—3000 150 80— 100 90— 120 10— 20 10 1500 200— 500

[7] [7] [8] [8] [8] [31] [31] [31]

B.M. Smirnov, Physics of ball lightning

175

From the data of table 4.4 we obtain for the electric energy of the atmosphere which is released as a result of the breakdown of the air by the action of ball lightning, Emmn

=

1008±02kJ

,

Emax

=

103.5±0.2kJ

The most probable energy equals E,,2 = 102.2±0.2kJ. Because this is the energy of an external source, it may assume, in principle, any value. We do not include in table 4.4 an estimate by Dmitriev et al. [43]; these authors investigated an event of a ball lightning observed in Khabarovsk and obtained an amount of released energy approximately equal to 106 kJ. This value significantly exceeds the data of table 4.4, even though it was obtained correctly. Figure 10 contains the results of processing in this way data of tables 3.9 and 4.3. +

L~’o

0.00

0.20

Bw,y

0

0.40

0

St*Jw,m

0.60

E~ly

0.80

1.00

The probability

+

8w,,,

0

St~w,ov

100

0.00

0.20

0.40

0.60

0.80

1.00

The probability Fig. 10. The ball lightning distribution as a function of energy storage. (a) Total energy. (b) Internal energy. Data from Barry [7], Stakhanov [8], Egely [32].The solid line is the average of these estimates.

176

B.M. Smirnov, Physics of ball lightning

Note that the energy released in a decay of ball lightning may be dependent on geophysical conditions. According to the analysis of Egely [30], 90% of the ball lightnings in Hungary with an amount of released energy higher than 1 kJ favour places found in mountains or at distances of less than 10 km from mountains. Mountains occupy 30% of the territory of Hungary. Egely states [30]that in the mountains of Hungary lightnings have larger energy than on the plains. On the basis of the above results one can arrive at an interesting estimate. Assuming that ball lightning is a secondary phenomenon of the usual lightning, let us estimate the portion of the energy that goes in the origin of ball lightning. The mean power of ball lightnings P is determined by the formula P = EWS = 10~’~ 1.2 kW, where E = 50 x 10±0.2kJ is the mean internal energy of the ball lightning; W is the probability that ball lightning will originate per unit time per unit area (eq. 2.7) and S = 5.1 x 108 km2 is the area of the Earth. The mean power of usual lightning is of the order of 5 x i07 kW [47—50].Thus assuming that ball lightning is a secondary phenomenon of usual lightning, we have that for ball to occur the expended amount of energy is 10353n12 of the energy of usual lightning. The above result allows to estimate the light output (or light efficiency) of an average ball lightning that is the ratio of the brightness J to the power E/T at the center of the ball lightning (E is the internal energy, T is the lifetime). In the first estimate we choose each of these parameters from the relation P(x) = 1/2, where P(x) is the probability that this parameter exceeds the value x. Then we obtain = .JT/E = 10±08lm/Wt, including only the statistical error. In the second estimate we have

=

J

dP j(~~

=

EminAi[li1(Ernax/E~n)]2~1’~=

0.6 x 10±0.8lm/Wt.

From this follows the light output of the average ball lightning, ij = 0.8 X 10±08 lm/Wt, where only the statistical error has been taken into consideration. Figure 11 gives the light output of different light sources.

2~L~~/w~

iSOO

2000

~.5o0

3Ooc,

Fig. 11. The light output of different sources as a function of their temperature; 1. ball lightning; 2. candle; 3. pyrotechnical compositions; 4. electric lamp. The solid line refers to the black-body radiator.

B.M. Smirnov, Physics of ball lightning

177

4.4. Correlations between size and lifetime There is a connection between the life time of ball lightning and its diameter. This was first noted by Stakhanov [8] and later confirmed by Grigor’ev et al. [25,26]. Table 4.4 contains the Stakhanov data for the probability P(t) of ball lightning conservation up to time t. Let us treat these data by using eq. (3.1) and the Stakhanov data of table 4.4. In the considered case formula (3.1) takes the form P(t)

=

A1 exp(—tIt1) + (1

—

A,) exp(—t/t2);

t1

=

11 s,

t2

54s.

=

From this follows T1—A1t1+(1A,)t2.

(4.10)

Let us assume that A1 depends on the diameter D of the ball lightning. Choose the functional form A,

=

exp(—D1D,).

(4.11)

By averaging this expression2.with the the distribution (3.1) we4.5 obtain thewe mean the Taking Stakhanovfunction data from tables and 4.6 havevalue D, = of 34cm. coefficient A1 = (1 + D0/D1) The results are given in table 4.7. In making the comparison note should be taken of the limited accuracy of the data of table 2.6. Figure 12 contains the values of the lifetime r 1 obtained on the basis of formulae (3.2), (3.3) and (4.12), using the data of Stakhanov [8], of Grigor’ev et al. [25], and of Bychkov [33].

Table 4.5 Parameters from the Stakhanov data Probability of conservation P(t) Range of D (cm)

Number of events

t=0

t=20s

t50s

0—10 10—30 >30

246 548 211

1 1 1

0.22 0.36 0.58

0.08 0.16 0.32

Table 4.6 Comparison of the Stakhanov data with eqs. (3.1) and (4.11) From the data of table 4.5

From eqs. (3.1) and (4.11)

Range of D (cm)

A,

T4

(s)

A1

~ (s)

0—10 10—30 >30

0.9 ±0.1 0.69±0.08 0.23 ±0.03

8±1 11±1 25 ±2

0.83 0.58 0.29

8 13 23

178

B.M. Smirnov, Physics of ball lightning

__

50~ ‘t~3~~

I

80

~7o

3°T~

0

50

—-

20

30

40

50

CO

~0

80

SO

100

Fig. 12. The connection between diameter and average life time of ball lightning; the solid line corresponds to eq. (4.3);

•, ref.

[33]; 0 ref. [52].

4.5. Mean observational ball lightning

The parameters of ball lightning obtained through eyewitness reports form the basis for understanding what ball lightning is and which properties it has. After averaging the observational parameters of ball lightning, one can construct ball lightning by using the mean values of these parameters. Let us call this mean observational ball lightning or mean ball lightning. The mean ball lightning is useful for an analysis of the processes related to its nature, origin or evolution. Comparing the parameters corresponding to some physical processes with the equivalent parameters of mean ball lightning one can check the reality of these processes for ball lightning. Such analysis that uses observational parameters of ball lightning allows to understand the individual aspects of its nature. Some parameters and properties of the mean ball lightning are presented in tables 4.7 and 4.8. Table 4.7 contains the mean parameters of ball lightning. Tables 4.7 and 4.8 show the total number of observational events which are contained in the aforementioned collections of observational data. An Table 4.7 Parameters of the mean ball lighting Parameter diameter life time

Value

Number of cases

23 ±5cm 3763 = 9 X iO~°~ s 2111 = 8 x iO°~ s brightness 1500 ±200 Im 1918~ 2kJ 18 internal energy”~ 7x10’° Observations with reference to USSR conditions. b) Energy which is released as a result of ball lightning decay. The mean internal energy of ball lightning equals 10>< 10~°’ kJ.

B.M. Smirnov, Physics of ball lightning

179

Table 4.8 Mean probability associated with various properties of ball lightning Property, process

Characteristic, parameter

Probability,

form

spherical form

91

observation

observation during a life time

colour

birth place of observation

1

X 10~°

Number of cases 2891~ --2000

18 ± 5 17 ± 1 25 ±10 20 ± 3 1.4± 0.3 12 ± 1 6 ±3

3479

explosion slow decay decay in fragments

52 39 9

2291

indoors outdoors

60 40

± 10

indoors outdoors

50 50

±

white red, pink orange yellow green blue, violet mixture

decay

0.6

±

(%)

9 7 ±3 ±

±

353’~

±10

±

5 5

1984’~

Observations with reference to USSR conditions.

‘~

asterisk marks the data of the USSR observations. The life times are introduced on the basis of the following expressions: T2 = dP/dtj,~i’0, P(r4)

=

1/2,

where P(t) is the probability that ball lightning does not decay during a time t after its origin. Table 4.8 gives the mean probabilities for various properties of ball lightning.

PART II. THE NATURE OF BALL LIGHTNING

5. Models of ball lightning 5.1. Experimental modeling of ball lightning as a whole If we were able to reach an understanding of the nature of ball lightning, we would in principle be in a position to reproduce this phenomenon. The history of research on ball lightning reveals numerous attempts to reproduce ball lightning in its entirety under laboratory conditions. The goal of such experiments was to check and demonstrate some of the hypotheses concerning the nature of ball lightning. Some of the attempts to reproduce ball lightning have been successful, resulting in the formation of glowing regions in the air. There is a detailed description of some experiments in the books by Singer and Barry [3, 7]. We will discuss some ball lightning experiments below, in order to

B.M. Smirnov, Physics of ball lightning

180

illustrate some general trends in modeling ball lightning, and in order to view them from a modern standpoint. Experimental modeling of ball lightning offers some experience which is useful for the analysis of this phenomenon. In spite of the different hypotheses on the nature of ball lightning, any experimental model uses gaseous discharge for gas excitation because the gaseous discharge is an easy and feasible way to enclose energy into a gas. The first experiments on the creation of the glowing ball were performed at the end of the last century by Tesla. After constructing the power discharge setup with a discharge frequency of 4.2 kHz, Tesla observed, under certain conditions, glowing balls with diameter of 2—6 cm. Unfortunately, these studies resulted in brief publications. Since the results of the experiments are not known in detail, it is difficult to evaluate them. Detailed investigations of glowing ball production in the discharge were performed by Babat [53] in 1942. He used an ultrahigh frequency discharge with a frequency of 1—100 MHz and the discharge power went up to 100 kW. When the pressure reached the order of 10 Torr a burning ball was observed which did not touch the walls of the discharge cavity. The Babat experiments were repeated more extensively by some scientists many years later. Kapitza solved the problem of the production of a glowing ball at atmospheric pressure [54, 55]. He switched on the ultrahigh frequency discharge in helium at a pressure of some atmospheres. The glowing part of the discharge did not touch the walls. The addition of an organic admixture increased the glowing brightness. The Kapitza experiments are consistent with proposals concerning the plasma nature of ball lightning. By estimating that the plasma of the ball lightning model must decay quickly, Kapitza decided that the energy in the plasma must have an external source. And his experiment demonstrates the possibility of the existence of a glowing plasma ball with an external enclosure. So the Kapitza hypothesis and the experimental result are logically compatible. However according to subsequent investigations [56], the realization of such ball lightning is hardly probable, because the lightning radiation in the ultrahigh frequency range is too small. For example, according to measurements [56],the microwave radiation of lightning is observed during 0.1—0.4s after the lightning flash and the power of the microwave radiation is of the order of 109_1O8 W at a distance of 1 km from the lightning. This power is too small to support a microwave discharge. Now the Kapitza experiments are set forth by Japanese scientists [57]. Another way of producing a glowing ball in a radio-frequency discharge at atmospheric pressure was used in the Powell and Finkelstein experiment [58].The discharge at atmospheric pressure was initially formed by arc and subsequently a radio-frequency discharge with frequency 75 MHz, and a generator power of 30 kW was used. The discharge was maintained in an open glass tube and it was possible to change the luminous region by moving the electrodes. After switching off the discharge, the luminous region became spherical and then it decayed into fragments within seconds; in the open air the decay took place twice as fast as in the glass tube. The detailed investigation of the plasma spectrum radiation was carried out. Though the lifetime of the observed luminosity is remarkably less than the ball lightning lifetime, it is remarkably greater than the usual plasma decay time at atmospheric pressure. The authors attributed such an anomaly to the presence of a large amount of metastable atoms and molecules. The above experiments demonstrate that it is possible for gas discharge plasma to be produced as a spherical luminous form, although on the basis of their properties such objects are distinct from real ball lightning. In the experiments of Andrianov and Sinitzyn [59],the authors used the proposal that ball lightning is formed from the evaporated material after usual lightning. For such a model the authors used the erosion discharge impulse discharge that creates a plasma from the evaporated material. The energy storage in these experiments was 5 kJ, the electrical tension 12 kV, the discharge condenser capacity was —

B.M. Smirnov, Physics of ball lightning

181

80 p~F.The discharge was directed toward the dielectric material, the maximum electric current was 12 kA. At first the discharge region was separated from the atmosphere by a thin diaphragm. It decayed including the discharge and the erosion plasma was splashed out in the atmosphere. The luminous region assumed a spherical or torroidal form, and optical plasma radiation was observed after approximately 0.01 s. The plasma glow was recorded for the duration of less than 0.4 s. These experiments reaffirm that the decay of luminous plasma in the atmosphere is remarkably less than the ball lightning lifetime. Now investigations of erosion plasma are being carried out by Avramenko [51, 166]. New studies show that erosion plasma may model some properties of ball lightning. Among the experiments modeling the chemical nature of ball lightning the most interesting and consistent are Barry’s experiments [7,60]. A six-sided plexiglas box with dimensions of 50)< 50 X 100 cm contained atmospheric air with an admixture of hydrocarbon gas. This mixture was ignited by a spark. The distance between the electrodes was 0.5 cm, the electrical tension 10 kV, the duration of the discharge impulse i03 s, the storage electrical energy 250 J. If we have propane as the hydrocarbon gas, the ignition limit corresponds to its concentration of 5%. The combustion ceases at a propane concentration of about 2.8%. But at concentrations of 1.4%—1.8% after the spark discharge a yellow—green luminous ball of some centimeters across is formed. It undergoes rapid motion in the box and is quenched after 1—2 s. This phenomenon resembles ball lightning with respect to its properties and may be considered to be the analogue of ball lightning. Evidently, complicated organic compositions were formed under the chosen conditions (see refs. [7, 52, 61]), and such a process was responsible for the observed glowing. These compositions including hydrocarbons are condensed at room temperature. They formed the aerosols and concentrated them in a small region of space. The initial spark produces enough complicated compositions; a small region in which propane is concentrated in the luminous ball shows the competition of different chemical processes in the experimental conditions. The Barry experiments under consideration are the laboratory modeling of ball lightning. At present the Barry experiments are performed by Japanese scientists [28,62]. In the experiments by Ofuruton and Ohtsuki [28,62] a similar scheme was used for the excitation of a combustion mixture, but the conditions of the Barry experiments were extended and the description of the experiments was more detailed. The experiments took place in a glass chamber with dimensions 734 x 372 x 429 mm. The electrodes were made of copper rods. The electric energy storage did not exceed 350 J. The experiments took place in methane—air and ethane—air mixtures at atmospheric pressure and room temperature. Further, cotton fibers were added to air and the air—ethane mixture. The evolution of the process was recorded by video cameras. The best results of fire ball formation were obtained at a combustion gas concentration near the ignition limit. At the first stage the colour of the formed fire balls was green, and subsequently the colour changed. The diameter of fire balls was up to 6 cm. The lifetime of the fire balls was 0.3 s in the air—methane mixture with the methane concentration at 2%. In the air—cotton system the lifetime of fire balls was 0.8 s and in the air—ethane—cotton mixture it increased up to 2s. The authors [62]note the low reproducibility of these experiments. Thus the above experiments show that fire balls may be formed in chemically active gaseous mixtures which are excited by an electric discharge. Another way of obtaining fire balls relates to short circuits in electrical systems releasing a large amount of energy. The basis of these studies was accidental short circuits in electric systems of USA submarines [63, 64]. As a result of this process, green fire balls arose near the circuit breaker electrodes. They had a diameter of approximately 10—15 cm, a lifetime of about 1 s and flew into the engine room. According to Silberg [63, 64] estimates of the released energy during a short circuit were 0.4—4 MJ. Such experiments have been undertaken to cause fire ball formation. According to Golka’s report

182

B.M. Smirnov, Physics of ball lightning

[65], Tuck has made a series of experiments in the Los Alamos National Laboratory up to 1971 by using the electrical equipment of submarines. After about 30 000 shots over a 2.5 year period he had four photographs of fire balls. Golka [65]used the locomotive electrical system for this purpose, Dijkhuis [66, 67] constructed an electrical system which was a modification of the system of the submarines. He used a copper film as electrodes and as a result of some short circuits fire balls were obtained with a diameter of 10 cm with a lifetime of 1.3 s. The input energy was 0.5 MJ. These experiments led to the accidental formation of fire balls. But their low reproducibility and limited diagnostics are comparable to the preliminary studies. Among the experimental studies of ball lightning the papers of Corum and Corum [68,69] rank highly. As a matter of fact, the authors renewed the Tesla experiments by using contemporary experimental equipment. Because the Tesla investigations are not published in detail, this problem was not simple. In refs. [68, 69] the Tesla RF (radio frequencies) generator was used. The slow wave helical resonator was magnetically coupled to a spark gap oscillator (the power is 70 kW and the maximum voltage is 2.4 MV at a frequency of 67 kHz). The actual average power delivered to the high voltage electrode was approximately 3.2 kW. It is able to produce a 7.5 m RF-discharge. Photographs and video cameras were used in this experiment for the detection of the phenomenon. Along with glowing RF-discharge, the formation of fire balls sometimes was observed. The diameters of fire balls changed from a few millimeters up to several centimeters and their lifetimes ranged from one half to several seconds. The colours of the fire balls ranged from deep red to bright white. This study is of great interest both for the employment of contemporary experimental methods, and the reproducible production of fire balls providing us with a new method to investigate ball lightning experimentally. In addition to the attempts to obtain fire balls in the laboratory, there are many cases of observation of such objects formed as a result of human activity. As an example we provide information from the Koldamasov report [156] on the appearance of fire balls in an engine plant in Kuibyshev during the expansion of dielectric liquid from the nozzles. The accumulation of electric charge on the nozzles created a corona discharge near them. Rapid switching of a system led to separation of the corona discharge from the device; it was transformed in a ball and flew in the air during a time interval of 3—5s. In analyzing the experiments on the modeling of ball lightning as a whole, we need to bear in mind that these experiments also reflect a general attitude toward the problem. On the one hand, the modeling is sometimes successful in producing glowing formations reminiscent of ball lightning. This success demonstrates that the phenomenon of ball lightning is governed by natural processes in excited air. On the other hand, the poor reproducibility of the experiments in giving rise to glowing structures and the difficulties in controlling the experiments have made it impossible to extract additional information about ball lightning from these experiments. The experience gained from these studies demonstrates the complexity of the phenomenon of ball lightning, and even the success in modeling it as a whole does not bring us closer to an understanding of the nature of this phenomenon. Consequently, it is more profitable to model individual aspects of this phenomenon and to analyze its individual properties with the help of physical systems, for which these properties of ball lightning are reproduced. In what follows we will pursue this line of approach. 5.2. Theoretical models of ball lightning There are many theoretical models aiming at the total description of the ball lightning phenomenon. As an example, one can cite the paper of Likhosherstnykh [70], “138 models of the nature of ball

B.M. Smirnov, Physics of ball lightning

183

lightning”, which summarizes the results of a debate among a number of Soviet readers of one popular journal in 1982, on the nature of ball lightning. One might classify the theoretical models dealing with ball lightning in its entirety in two categories. The first category is based on new physical effects, the second one uses known physical systems which are found under definite conditions. Let us analyze some examples of the first group of models. An example of such a model is an interesting hypothesis of ball lightning as a non-ideal plasma [71,72]. As a matter of fact, it states that non-ideal plasma (or plasma with strong coupling) has a steady state, and this state is ball lightning. But according to recent notions, non-ideal plasma is a complex system which contains, in addition to charged particles (electrons and ions), also atoms. Therefore it is matter at extreme conditions (high temperature and pressure). Recent evidence about such an object does not confirm the existence of steady states of systems at extreme conditions. Physical effects, which can create a steady state of matter at extreme conditions, are in principle possible. An attractive concept was developed by Dijkhuis [73—75] who assumed that in a dense plasma, electron pairing takes place as in a superconductor. If this effect exceeds other interactions in a dense plasma, a steady state exists in the plasma. This plasma, which is created by the action of strongly impulsed currents, is ball lightning. Such a program for investigations was developed by Dijkhuis (see preceding paragraph) but there are doubts about feasibility of this concept. A number of theoretical models are based on atmospheric phenomena which allow us to generate an amount of energy in some region of space. The glowing of this region is a result of such an action. Handel [76, 77] assumed the existence of a maser effect in the atmosphere as a result of rotational excitation of water molecules under the influence of atmospheric phenomena. This effect was called the caviton effect. Maser radiation can be concentrated in small regions of space. It produces excitation in such regions and causes glowing. Another variation of the energy concentration under the influence of water vapor in the atmosphere was developed based on the assumption that this vapor is condensed and the released energy is concentrated in a small region of space [78]. Besides ball lightning models, these concepts are useful for the research of atmospheric phenomena. Besides these models, nuclear and cosmic hypotheses of ball lightning are found (e.g., refs. [79—83]). Some of these models are considered in the books by Singer and Barry [3, 7]. As an example of such a model, we analyze the model of the Dirac monopole which was developed by Korshunov [84]. It is shown that if the Dirac monopole exists, it can explain an observational ball lightning. The Korshunov analysis is sufficiently thorough, however it is noteworthy that the existence of the Dirac monopole under consideration must lead to more essential atmospheric phenomena than ball lightning. While considering hypotheses of ball lightning based on new physical effects, one should note that if the concepts, on which they are founded, are correct, the hypotheses are valid. But usually the assumptions are dubious, and checking them or demonstrating them is rather difficult. This question will be left open for a long time. One more peculiarity of the above hypotheses is their remoteness from real systems. Therefore descriptions based on these hypotheses are schematic, and it is impossible to compare a hypothetical ball lightning with an observational one with reference to their numerical parameters, such as size, lifetime, brightness, etc. Another group of theoretical models uses known physical systems as a basis for these models, but the conditions of their existence may be different from the usual ones. Most of these models take the plasma as a basic form to explain the existence of ball lightning (see ref. [7]). But the conditions for this plasma to have a spherical form differ from the air conditions in which ball lightning is observed. For example, Shafranov [85]has shown the existence of a spherical fusion plasma, but such a phenomenon may be observed in special external fields and at low plasma pressure, that has no relation with real ball lightning.

184

B.M. Smirnov, Physics of ball lightning

Some conditions for the existence of ball lightning are open to discussion. For example, this problem depends on the assumption of the presence of external fields. If we assume that ball lightning is governed only by an internal source of energy, a plasma cannot be a basis of ball lightning. My experience as a specialist in plasma physics is. that laboratory plasma not supplied by a source of energy decays at atmospheric pressure within a time less than iO~s as a result of processes of collision and recombination of the plasma particles. The same view was held by Kapitza (see preceding paragraph) in his research. At least one half of the observational ball lightnings which are found indoors have an internal source of energy. If all ball lightnings have an internal source of energy, the above model of ball lightning is invalid. But even in such cases these models are useful. Indeed, ball lightning, as a source of a plasma, can cause an electric breakdown in the air during thunder storm weather. The processes of interaction between ball lightning and external fields are the same as in the above ball lightning models. Thus, though we must reject most models of ball lightning, these models can be useful. Besides providing concrete information related to ball lightning processes, they convince us of the complexity of this phenomenon. Such a problem cannot be solved straightforwardly. Understanding of the ball lightning phenomenon demands a detailed study of its various aspects. The negative experience obtained from experimental and theoretical models of ball lightning necessitates a method of investigation of this problem which consists in studying separately its various aspects. This method is based on the assumption that ball lightning is controlled by known physical laws. Working from the logical closure of the world around us, we can then find other physical objects and phenomena in which these laws are manifested. We consider these systems as some analogs of ball lightning, and use them to model some properties of ball lightning. Analyzing such systems we can find an optimal way for understanding of the nature of ball lightning. Below we will keep this line of approach.

6. Structural analogs of ball lightning 6.1. Skeleton of ball lightning and fractal clusters One of the central questions concerning the nature of ball lightning connects with its structure. Because ball lightning can float its substance must not be condensed, i.e., the material composing ball lightning is not a dense liquid or solid. Another important fact for analyzing the structure of ball lightning is the conservation of its form and size during its lifetime. (At any rate, this holds for most of the observational cases.) It means that there are internal forces inside the ball lightning similar to surface tension. These forces maintain the stability of the shape of the ball lightning. From this it follows, in particular, that the substance of which ball lightning is made cannot be found in the state of a gas or dust. Let us prove it. Let us consider the behavior of dust in heated atmospheric air. Dust particles move with heated air by separate vortices. The cold air outside goes between neighboring vortices and the volume initially heated expands. The flow of molecules of cold air inside the initial volume equals [8611= O.lvN, where N is the number density of molecules; v is the velocity of hot air in the vortices; 0.1 is a numerical coefficient [86]. From this we have as a typical time of expansion of the hypothetical ball lightning T = 1OR/v, where R is the radius of ball lightning. Estimates [87, 11] give for typical conditions v io~cm/s, i.e., the expansion time is T —-0.1 s. A typical time duration for the change of the form of —-

B.M. Smirnov, Physics of ball lightning

185

ball lightning is of the same order of magnitude. Thus we conclude that the substance of ball lightning cannot be found in the state of a gas or dust. Besides, we have no knowledge of the processes which take place when plasma states or systems of spherical form exist long enough without an external source of energy and conserve their form during times of order of the lifetime of a typical ball lightning. Thin films are destroyed under the conditions of the existence of ball lightning. Therefore the most natural structures cannot ensure the observational properties of ball lightning. Thus, one can conclude, on the one hand, that ball lightning has a rigid skeleton, and, on the other hand, that its specific gravity is comparable with the specific gravity of air. Therefore, ball lightning is an extremely porous system, and the pores occupy the main volume inside it. Note that the conclusion about the existence of a rigid and light skeleton of ball lightning relates to the internal way of energy storage only. There are spherical plasma formations which are maintained by external fields and currents. The concept of a rigid skeleton of ball lightning has a long history. For example, back in 1972 Zaitsev [88] asserted the following: “The appearance of ball lightning begins with the formation of threedimensional cellular structures.” However, this concept was formulated on a fairly solid basis by Alexandrov et al. [89]in 1982. The authors called this structure a threadlike aerosol structure. The basis of this proposal was experiments [90] on the explosion of metallic wires in vacuum by the action of electric currents which go through the wire. Under certain conditions a relaxing metallic vapor formed trace structures fastened to the walls of a vacuum camera, and could be found in this state for the duration of 1—2 days. The authors of ref. [90]estimated the transverse dimension of these constructions as —lOnm. A more detailed study of the structures which are formed as a result of an explosion of metallic wires in vacuum showed [91] the fractal structure of the resulting systems which was called fractal aggregates or fractal clusters. From this one can conclude that ball lightning has a rigid skeleton with fractal structure [92]. A fractal aggregate, or fractal cluster is an element of such a structure. It is a system of bounded individual macroscopic particles. Though this object came into physics not long ago, it has been thoroughly studied. There are some books and reviews [93—104,223] related to this topic. Further, we consider only those properties of fractal structures which are connected with the problem under investigation. Fractal clusters are formed as a result of the fact that hard macroscopic particles join and move in space according to a definite law (e.g., diffusion). Fractal clusters have a characteristic branching structure. One of the main parameters of fractal clusters is their fractal dimension. This value can be determined in two ways. According to the first of these, let us take a point in a fractal cluster with coordinate r’ and draw a vector r from it. Construct the correlation function [105] C(r) = (p(r’ + r)p(r’))/(p(r’)),

(6.1)

where p(r) is the density of the cluster equal to unity at an occupied point and zero at a point not occupied by the cluster material; angular brackets mean averaging over the positions of cluster points. The dependence of the correlation function on a distance r has the form C(r) = const/r°° .

(6.2)

B.M. Smirnov, Physics of ball lightning

186

In the case when a cluster is a continuous medium we have a = 0 for r ~ R, where R is the size of the cluster. The fractal dimension is introduced on the basis of the relation (6.3) where d is the dimension of the space in which the cluster is inserted. For a continuous object D0 = d. Consider another way of determining the fractal dimension of a cluster. Let us draw a sphere of radius r with the centre at a point of the cluster. The mean mass m of the cluster material averaged over the cluster points depends on its radius according to the law Km(r)~ r’~ ~

(6.4)

.

Naturally, in the limit of the accuracy the fractal dimension D~is the same as the quantity Da in formula (6.3). Formula (6.4) means that the number of particles in a cluster of radius r is determined by the relation n

=

(r/r0)’~ ,

(6.5)

where r0 is the particle radius. Relations (6.2) and (6.4) two practical ways for determining the fractal dimension of fractal clusters. Table 6.1 gives the fractal dimensions of fractal clusters which are formed as a result of the evaporation of wires in vacuum after an electric current has passed through them. There are other ways in which fractal clusters can form in a gas as a result of the evaporation of a surface [106—116]. Note that the size of particles in a fractal cluster is usually small, of the order of some nanometers. Indeed, the time of formation of a cluster as a result of the joining together of hard particles depends strongly on the size of the particles. For example, let us consider the formation of fractal clusters in atmospheric air. Solid particles join to form small clusters, and small clusters join to form larger ones. Assume that particles and clusters have Brownian motion in space. Table 6.2 contains the sizes of particles for which the process of formation of a fractal structure must be completed within 1 s. The assumption is made that the probability of joining two contacting particles equals unity. The parameter Z is the ratio of the mean specific gravity of the material in space to the specific3)gravity and Dofisair the(assume fractal the specific gravity of the material to be the same as that of glass, p0 = 2 g/cm dimension of the formed fractal cluster. The data of this table show that usually only fractal clusters composed of small particles may be formed. The theory [118] shows the instability of fractal clusters of large size. As the cluster size increases its density decreases. This leads to a decrease of the mechanical and thermal strength of the cluster. According to the theory, the ratio of the radius of a fractal cluster to the radius of the particles, of Table 6.1 Fractal dimensions of clusters formed as a result of material evaporation [91] Material Fe 1.69

± 0.02

Fe

Zn

1.68 ±0.01

1.67

1.56±0.02

1.50±0.04

±0.02

Zn

SiO,

1.68 ±0.02

1.55 ±0.02

1.60±0.04

1.55±0.06

D 5

1.52±0.04

B.M. Smirnov, Physics of ball lightning

187

Table 6.2 Sizes of particles of fractal clusters r 0 (expressed in nm) that are formed in air within is D Z

1.7

1.8

1.9

2.0

2.1

0.1 0.3

2.7 6.3 16 370

1.4 3.6 10 25

0.7 1.9 5.6 15

0.3 0.8 2.8 8.4

0.1 0.3 1.2 4.1

3

which the cluster is composed, cannot exceed the value l0~—1O~. It corresponds to a low density of a cluster. For example, a fractal cluster of iron 3from table 6.1 with radius 10 urn radius according to formula (6.4). It and is ofa aparticle lower of order of 4magnitude nm has athan mean density p = l038±02g/cm the atmospheric air density, and allows us to understand the reason for the instability of large fractal clusters. One can construct a large sparse system by joining fractal clusters in one large cluster. Such a system has fractal properties at small distances and is homogeneous on the average at large distances. On the basis of the analysis of refs. [9, 92, 120] we conclude that ball lightning has such a structure, i.e., it has a rigid skeleton consisting of bound hard particles. This skeleton has fractal structure at small distances and is homogeneous on the average for large distances. The macroscopic systems with fractal structure that exist in the real world, can be considered as analogous to ball lightning. We analyze such systems below. 6.2. Aerogel as a sparse structure Thus ball lightning has a rigid skeleton consisting of bound hard particles. There are systems of a similar nature, one of these systems is aerogel (see refs. [119—121, 223]). Further we analyze properties of aerogel which are of interest from the view point of the problem of ball lightning. Aerogel is a peculiar physical object consisting of rigidly connected macroparticles. This rigid skeleton occupies only a small part of an aerogel volume the rest is taken up by pores. Hence the prefix “aero” characterizes the low specific gravity of this object. The very first samples of a silica aerogel, obtained by Kistler [122—124] over fifty years ago, exhibited specific gravity as low as 20 g/l. At present there are aerogels of approximately ten oxides. A small piece of an aerogel sample which contains a large number of individual macroparticles is a fractal cluster. The fractal structure of an aerogel is determined by the way it is formed from individual particles. At the first stage solid particles of an aerogel material are formed in the solution. Subsequently they join in clusters whose sizes grow until these clusters occupy the entire volume. Then a porous substance is formed an aerogel with fractal properties and sizes in the range —

—

—

(6.6) where R~is the correlation radius which equals the maximum size of the pores or the typical cluster size when the clusters occupy the entire volume. For r 2~’R~an aerogel is homogeneous on the average. According to one of the properties of fractal clusters the mean mass density of the substance inside a sphere of radius r is

188

B.M. Smirnov, Physics of ball lightning 3-D p(r) =

p0(r0Ir)

(6.7) where p0 is the density of the cluster material; r0 is the mean radius of the particles and D is the fractal dimension of a cluster. This relation is valid in the range (6.7) and gives for the maximum radius of the pores in an aerogel, ,

I /(3—D) R~—r0(p0Ip) —

(6.8)

.

The fractal dimension of aerogel may be found by different methods which account for the distribution of the sizes of the aerogel pores. A determination of this distribution for silica aerogel according to measurements [125] yields [120] D = 2.3 ±0.1. The traditional method uses scattering of X-rays, electrons, neutrons at small angles. Another method of obtaining the fractal dimension of aerogel is based on the vibrational spectrum of aerogel in the range of wavelengths r0 ~ A ~ R~.The sum total of such measurements [126—130]yields a fractal dimension in the interval 2.3—2.4, though some of these results are contradictory. These measurements allow the simultaneous determination of the correlation radius R~.Note that values of R~obtained from scattering at small angles are lower than values obtained from the vibrational spectrum of aerogel. For example, small angle scattering and the vibrational spectrum of silica aerogel was studied in ref. [129]. The fractal dimension is D = 2.33 ±0.05 according to both methods; the correlation radius from small angle scattering is R~= 14 nm, and from the vibrational spectrum, R~= 52 nm. Formula (6.8) gives in this case R~= 40±10 nm. Consider a simple model of aerogel consisting of spherical particles of approximately the same radius with bonds at the points where the spheres nearly touch. In this case the specific area S of the internal surface of the aerogel can be expressed in terms of the particle radius r0 by the formula S—3/p0r0,

(6.9) 3) where material. In particular, silica aerogel 2.2 g/cm at r p0 is the mass density of the 2/g.aerogel In most real samples of silicafor aerogel S lies in (p0 the= range 300— = 2 nm we obtain S = 700 m parameters of silica aerogel samples obtained and analyzed in ref. 15000 m2/g. Table 6.3 contains typical [131] (cylindrical samples of diameter up to 8 cm). The first three columns are taken directly from ref. [131], where the specific internal surface area was obtained from nitrogen adsorption and the particle radius was found by the TEM (transmission electron microscopy) method. For the simplest model of aerogel, which is a system of separate solid particles of the same radius r 0, the number density of particles N’ satisfies p

N’~4~rr~/3.

(6.10) Table 6.3 Microstructure parameters of silica aerogel samples [131]

3) Mass density (g/cm 0.03 0.05 0.15 0.16

Specific internal surface (m2/g)

Particle radius (nm) TEM photography eq. (6.9)

Maximum pore radius (nm) (eq. 6.8)

1590 1080 520 740

4 5 10 4

120 100 55 35

0.9 1.3 2.6 1.8

B.M. S,nirnov, Physics of ball lightning

189

Because of the large area of its internal surface, the aerogel has a large surface energy since a considerable amount of the molecules is found on the surface of the particles. A decrease in the internal surface leads to a release of energy, because some molecules move inside the particles and form new chemical bonds. For silica aerogel, with the typical specific area S = 1000 m2/g, formula (6.9) gives for the mean radius of the particles r 0 = 1.5 nm. A particle of aerogel includes approximately 300 molecules, and 230 of these are found on the surface of the particle. Assume that the interaction inside the particles takes place only between neighboring molecules. 2 = (AN 2’3, where A is The area per molecule which is found on the particle surface equals a 0/p0) the molecular weight of the molecule; N 0 is the Avogadro 2number; p0 is theiXH density the particle ~HI2, where is theof enthalpy of material. The surface energy per unit area equals r = a gasification of the molecules. For SiO 2 we specific have L~H = 133 7 kcal/mol from this 2. For the typical area of ±the internal [132,147], surface ofand aerogel, follows ~ = 3.7 ± 0.2 JIm corresponds to a specific surface energy of 3.5 kJIg, which equals the specific chemical energy of gunpowder. As gunpowder aerogel is an explosive substance, but aerogel can have such properties at high temperatures. At low temperatures aerogel is heat-resistant and a chemically passive material just like ceramics and glasses. 6.3. Thermal and optical properties of aerogel The thermal conductivity of an aerogel is relatively small. The thermal conductivity coefficient of silica aerogels at room temperature is less than that of atmospheric air. Below we find the kinetic coefficients of a sparse aerogel. The main transfer mechanism inside a rare aerogel is determined by collisions of air molecules with the internal surface of the aerogel. Assume that an aerogel consists of bounded macroparticles of the same radius r 0. Then the free path length of a molecule inside the aerogel is A=lI(irr~~N’)=4I,~S,

(6.11)

where N’ is the number density in space of macroparticles which constitute aerogel; S is the specific area of the internal surface of aerogel, j~isinternal its density. aerogel density 3 and with a typical area of the surfaceFor S =example, 900 m2/g for we have A = with 2.2 Xa106 cm. = 0.05 The free g/cm path length due to collisions between molecules is larger than this value (for example, for atmospheric air and T = 1000K this value equals A = 5 x i05 cm). Thus, for real aerogel (~ >0.01 g /cm3) one can neglect transfer processes due to molecule—molecule collisions. On the basis of classical formulae [133,134] making use of relation (6.9) for the mean free path of the molecules, we obtain the following expressions for the thermal conductivity coefficient K and the diffusion coefficient D: K =

(4/9V~)(2T/m)”2An,

D = j1g~”2(2T/m)1’2A.

(6.12)

Here m is the mass of the molecule; n is the total number density of the molecules; the expressions for K and D relate to the Chapman—Enskog approach [133,134]. According to formulae (6.12) we have the following expression for the thermal conductivity coefficient with respect to the considered mechanism of heat transfer

B.M. Smirnov, Physics of ball lightning

190

K

=

K1(p,Ip),

(6.13)

3 and K with parameters for a silica aerogel, p1 = 0.05 g/cm 1 = 0.78 mW/mK. Another mechanism of the heat transfer relates to the heat transfer through an aerogel lattice [135—140].This mechanism is important at middle and high aerogel densities. Choose the following dependence for the thermal conductivity coefficient in this case: K

= ,~2(~/p1)~~

(6.14)

3 [135,136]; K = 11 The measured values of this coefficient are: K = 4 mW/mK for j~ = 0.105 g/cm mW/mK for j~=0.27gIcm3[135,136]; K =8mW/mK for j~=O.105g/cm3[137]; K = 13.lmW/mK for = 0.109 g/cm3 [138]; K = 19 mW/mK for ~ = 0.14 g/cm3 [139]. The exponent g equals 1.6 according to ref. [120] and 1.8 according to ref. [140]. Choosing g = 1.7 in eq. (6.11) we obtain for p, = 0.05 g/cm3, K

1.8 x i0~03 mW/mK. Taking into consideration that both mechanisms of heat transfer are independent, we may rewrite the total thermal conductivity coefficient in the form 2

=

K

=

(6.15)

K1(p0Ip) + K2(~/~0)~.

The dependence of the thermal conductivity coefficient on the silica aerogel density for room temperatures is given in fig. 13. The thermal conductivity coefficient as a function of the aerogel density according to formula (6.15) has a minimum. Note that the minimum value of the thermal conductivity coefficient at room temperature and atmospheric pressure is approximately four times lower than that of air at room temperature (Kair = 26 mW/mK [141,142]). A new mechanism of thermal conductivity due to the evaporation and condensation of molecules of the structure takes place at high temperatures. This mechanism is remarkable at relatively low vapour densities because each molecule transfers the binding energy, and the density gradient is relatively large. In fig. 14 this mechanism is compared with the above one.

20~

A

18 * E

16 14 V

12 10 8 C

3

6 4

22 2 0

50

.

~

~—

~O0 Densty

,

g / I

Fig. 13. The thermal conductivity coefficient of silica aerogel. Solid line: eq. (6.15). Experiments: ref. [139].The solid curve is constructed on the basis of eq. (6.15) and the experimental data.

•, refs. [135,136]; U ref. [137];V ref. [138];A

B.M. Smirnov, Physics of ball lightning

191

Silica aerogel is a transparent material. *) For example, the specific optical density of silica aerogel at a wavelength of 0.55 p~mequals 11 ±3 cm2/g according to measurements [142]. It corresponds to the absorption of half the incident light in sample 1 of table 6.3 after passing through 2 cm of material. The specific gravity of aerogel places is somewhere between a solid and a gas. This also holds for the refractive index n which is close to unity but still larger than the index in gases. According to measurements [143—145],it is approximated by the form n

—

1

=

(0.210 ±O.OO2)p,

(6.16)

where p is the aerogel density expressed in g/cm3. Note that in pure silica glass the quantity (n equals 0.207 cm3/g.

—

l)/p

6.4. Process of densijication of aerogel Along with a high absorption ability aerogel has a high thermal stability. Already in his earliest studies of aerogel [122—124]Kistler noted that silica aerogel samples do not change up to temperatures of 700°C,whereas at 900°Ctheir porosity decreases. According to the data of table 6.4, silica aerogel samples heated up to 800°Cdo not undergo a change in internal structure. At higher temperatures the area of the internal surface of a skeleton becomes less and the porosity decreases, with the subsequent increase in the aerogel density. The process of change of the internal surface area of a porous system is called the process of structure densification. An increase in the size of the particles of the structure leads to a release of energy, because some molecules move inside the particles and form new chemical bonds. Let us introduce the time of the process of densification on the basis of the equation (6.17)

dS/dt= —S/T(T), 10 C 0

o I-

o .2 .1, 0.1 I I I I I 18001820184018601880190019201940196019802000

Temperature, K Fig. 14. The ratio of the thermal conductivity coefficients of silica aerogel are created by transporting the evaporating molecules and air molecules inside the aerogel. °~ It has been asserted (see ref. [146])that a large contribution to light scattering by silica aerogel is made by chlorine atoms incorporated in the aerogel. These atoms are produced by the dissociation of hydrogen chloride which is added in small quantities to the solution as a catalyst. This means that the transparency of an aerogel depends on conditions of its production.

192

B.M. Smirnov, Physics of ball lightning Table 6.4 Silica aerogel densification at heating to 1250°C.The duration of heating is 12 mm [131] 3)

Temperature (°C) 300 800 1150 1200 1210 1225

Gel density (g/cm 0.16 0.16 0.27 0.66 1.00 1.41

Specific internal surface area (m2Ig)

Particle radius (nm) TEM photography

740 780 530 160 76 36

4 4 5 6 12 20

S is the specific area of the internal surface, T is the temperature. One can estimate the time of aerogel densification T(T) on the basis of the data of table 6.4. Figure 15 shows the results of the treatment of the data of table 6.4 making use of eq. (6.17). At temperatures below the melting point of the aerogel material the mechanism of the growth of large particles and evaporation of small particles is the following. As a result of radiation of the system, the temperature of the aerogel particles is smaller than the temperature of the gas inside its pores. Then the energy release of the large particles of the aerogel is higher and their temperature is lower than that of small particles. Therefore the specific flux of evaporation from large particles is smaller than from small particles. Thus large particles are increased as a result of the processes of evaporation and condensation, and small particles are decreased. As a result the specific area of the internal surface of aerogel is decreased. The time duration of the process is inversely proportional to the evaporation flux j, where

1IT(T)

=

Ca2j.

(6.18) • 1420

Mulder 1440

Exper. 1460 1480

Appro 1500

1520

C

10

10

‘P.

E E P1—

\.\ “ •

2 I 1420 1440

I

I

I

1460

1480

1500

1520

Temperature, K Fig. 15. Times of the aerogel densifleations: U experiment [131];solid line approximates these data; dotted line is eq. (6.19).

B.M. Smirnov, Physics of ball lightning

193

Here C is a numerical coefficient; a2 is the area occupied by one molecule on the particle surface, i.e., a = (AN 3(A is the molecular weight of the molecule; N 0Ip0)’~ 0 is the Avogadro number; p0 is the material density). Assuming that the accommodation coefficient of the particle surface is unity, we have for the specific flux of evaporation j = N(T)t314, where N(T) is the molecular number density corresponding to the pressure of the saturated vapor at this temperature; [3is the mean velocity of the molecules. One can find the value of the coefficient C by comparing eq. (6.18) with the data of table 6.4 related to silica aerogel at temperatures 1150—1225°C. Such a comparison gives C = 2 x 10_40±02 and formula (6.18) has the following form for silica aerogel: T

= 8 x 1018 exp(s~HIT),

(6.19)

4 K [132,147] is the enthalpy of SiO and L~H= 6.7 x i0 2 gasification. Figure 15 contains, along with experimental values of the densification times, the results of calculations on the basis of formula (6.19). Let us consider another mechanism of densification of the structure when the particles are liquid [148]. The rate constant of joining two small liquid drops of radii r1 and r2 is 2ir(r 2, (6.20) k = [8T(m, + m2)Iirm1m2]” 1 + r2) where m 1 and m2 are the masses of the particles with radii r1 and r2, respectively. Joining213. these drops Using the leads to a decrease of the total area of the particle surface, z~S = 4,r(r~ + r~) 4ir(r~ + r~) simplest distribution function on the masses, f(m) exp(—m/th), where th(t) is the average mass of the particles, yields the value of T in eq. (6.17), where 1 is expressed in seconds

—

2IN’ (6.21) 0I(Tr0)]~ Solving eq. (6.17) and taking into account that during the process N’r~= constant, we obtain for the =

0.67[p

.

specific area of the internal surface of the particles, S(t), S = S~(1+ 2.5v 4, v~,= 1 Ir N 2, (6.22) 0t)° 0k0 = 1.5N0[Tr01p0]” where N 0 and k0 are the number density and the rate constant at zero time; S~is the initial specific area of the internal surface of the particles.2/g Note large rate ofp this process. silica v~ aerogel with a andthe mean density = 0.01 g/cm3 For we have = 6 X i07 s~. specific area the silica surface S = 1000 m (6.19) gives at the melting point T(T = 2000 K) = 0.003 s. As Note that for of solid aerogel formula can be seen, melting of the structure increases the time of densification by several orders of magnitude. 6.5. Aerogel as an explosive substance As a physical object aerogel is a rare structure with a high internal energy (see fig. 16). This energy can be released only at high temperatures. Because of the storage of high energy, its release is accompanied by explosion. Figure 17 contains threshold parameters of the thermal explosion of silica aerogel. It is supposed that a region of radius R inside the aerogel is heated up to a certain temperature. At the threshold temperature of this region a thermal wave propagates from the region. The wave uses

B.M. Smirnov, Physics of ball lightning

194

~÷ 2/g). Each drop contains n Fig. 16. Joining of two liquid silica drops having diameter 2.7 nm (and specific surface area 1000 m = 146 of these are found on the surface. After these drops are joined, the released energy goes to the evaporation of n process of condensation of these molecules leads to the release of surface energy.

= =

234 molecules and 22 molecules. The

the internal energy of aerogel for its heating. Thus a thermal explosion of aerogel takes places under these conditions. In this and further calculations we suppose that the aerogel thermal conductivity is determined at small temperatures by air molecules which are found in the aerogel pores, and the free path length of the molecules is determined by their collisions with the aerogel skeleton. At high temperatures the thermal conductivity of the aerogel is determined by the evaporation and condensation of the aerogel molecules. The transition from one mechanism to the other takes place at a temperature of 1880 K for silica aerogel (see fig. 14). Release of the surface energy of an aerogel causes a thermal explosion with the propagation of a thermal wave. Let us estimate the parameters of the thermal wave propagating in silica aerogel. The melting point of silica aerogel equals 1993 K and the boiling point equals 2250 K (at this temperature the pressure of saturated vapor equals 1 atm). Assuming that the mean density of SiO exceeds 3), we neglect the thermal capacity of air. The2 released considerably the air (p ~melt) 0.001the g/cm energy continues to density heat (and structure and to cause evaporation of the molecules. The contribution of the second channel is small at the melting point (—1%) and both channels are equalized at the boiling point. Figure 18 gives the temperature behind the front of a thermal wave in the considered case. We assume for these estimates that the thermal capacity of SiO 2, which equals 0.75 J/g K [132,147] at room temperatures, does not change as the temperature changes. Now let us estimate the velocity of the thermal wave by the traditional way on the basis of the Zeldovich—Frank—Kamenezkij theory [149,150]. The equation for the temperature of the thermal wave has the form 2TIäx2. (6.23) pc~aT/at = rplr + K d 10 N.

0.1 1500

I

I

I

1600

1700

1800

Temperature, K Fig. 17. The threshold of the thermal explosion of silica aerogel.

1900

B.M. Smirnov, Physics of ball lightning

195

2300

/~

‘1800 600

800

1000

1200

Specific area

,

1400

1600

m2/g

Fig. 18. The temperature behind the thermal wave front in silica aerogel.

Here x is the direction of propagation of the wave; ci,, is the specific heat capacity of aerogel; r is the specific internal energy; r is the time of aerogel densification; K is the thermal conductivity coefficient. Supposing that the wave propagates to the right, we have that the wave parameters depend on the combination x Ut, where u is the wave velocity. Then the equation has the form —

—udTldx=f+ ~d2T/dx2,

f(T)= rI(c~r)=Tm/T(T),

(6.24)

where Tm is the temperature behind the wave, ~ = K /(c~p) is the thermal diffusivity coefficient of aerogel. Using the Zeldovich—Frank—Kamenezkij method, introduce the quantity Z = d TI dx. Equation (6.24) is transformed to the form

—

—uZ+ ~ZdZIdT+f(T)=0.

(6.25)

If the temperature behind the wave Tm is lower than the melting point of aerogel, one can use the traditional way of determining the wave velocity [149,150]. In the temperature region Tm T T2/Ea (Ea is the activation energy of the process of energy release), one can neglect the last term of the above equation and its solution is —

z= ufdT’/~,

(6.26a)

where T 0 is the initial temperature. In a region Tm equation, and we have

2

z

=

2

J

f(T’) dT’I~.

Joining together

~-

—

T

°~

T

— T0 one can neglect the first term of that (6.26b)

these solutions yields the Zeldovich—Frank—Kamenezkij formula for the wave velocity,

196

B.M. Smirnov, Physics of ball lightning Tm

u=(2J

Tm

(f

f(T’)dT’/~)

(6.27)

dT’I~).

—

T~ satisfies the relations Tm T~~ T,~ T0, Tm T~ T2 IEa~and the integral does not depend on this value. Because ~ T -1/2 and r( T) exp(— E~ / T), we have for the velocity of the thermal wave for the values of Tm under consideration, —

—

~‘

~

u

=

(6.28)

(8Tm~(Tm)/9ET(Tm))”2.

The wave velocity in silica aerogel, as a function of the temperature behind the wave Tm~is given in fig. 19. The first discontinuity of this curve takes place at a temperature T 1 = 1880 K where the mechanism of thermal conductivity changes (see fig. 14). At higher temperatures there is a sharp temperature dependence of the thermal conductivity coefficient, and the wave velocity depends slightly on the maximum wave temperature. The second discontinuity of the wave velocity takes place at the melting point T2 = 1993 K, where the mechanism of the process of structure densification changes. Let us find the asymptotic value of the wave velocity for high temperatures T T2 ~ IXT = T~IEa (for silica aerogel z~T=60 K). Then for T< T1 the solution of eq. (6.25) is given by formula (6.27). The contribution to the value from the domain T1 < T < ~‘2of Z is small because ~(T) exp(—EaIT). In the region T> T2 one can neglect the last term in eq. (8). Then for temperatures T T2 t~Twe have —

-=

—

~‘

112, Z= {2V(T2)TmT~/[Ee~(T2)]} where ~(T 2) is the frequency of the densification process at the melting point. Evidently, ~(T2) = and joining the solutions gives 2(Ti)~”~2

U=~~\ 9Ea~(T T2 I8TmVo~

2) )

(6.29)

100~

0.01 1700

I

1800

1900

2000

2100

Temperature, K Fig. 19. The velocity of a thermal wave inside silica aerogel.

2200

B.M. Smirnov, Physics of ball lightning

197

At high temperatures the asymptotic value of the wave velocity is u specific area of the internal surface S = 1000 m2/g (Tm = 2400 K), u

—

=

T~2.For silica aerogel with the 80cm/s.

6.6. Fractal fibers In the experiment by Lushnikov et a!. [151]the formation was discovered of fractal fibers as a result of the laser irradiation of metallic surfaces in atmospheric air and some inert gases (see fig. 20). These fibers were formed effectively only in an external electric field and grown in space. Later they became attached to the electrodes. The diameter of the fiber was 30—40 p~m,the length was several centimeters. Some tens of fibers were formed simultaneously. The internal structure of fractal fibers is similar to that of aerogel. The following processes take place during the formation of fractal fibers. There is a strong evaporation of atoms and molecules from the surface under the action of laser radiation. The temperature near the surface is up to i04 K, the pressure of an evaporating gas is 10—100 atm. The expansion of evaporating atoms in space causes their cooling and leads to condensation of atoms on ions as nuclei of condensation. Simultaneously with condensation, coagulation of the formed liquid particles takes place. When the particles become solid, their size lies in the range 5—20 nm. Then the solid particles are joined in fractal clusters (fractal aggregates). Up to this stage this process has been studied thoroughly (see section 6.1), and fractal clusters may be produced by different actions on a surface; e.g., as a result of the electric explosion of metallic wires [91] or their heating [106—111].The laser excitation [112,113] is only one of the methods of the production of fractal clusters.

The presence of an electric field is in principle required for the formation of fractal fibers. In an

~ •

..s~.

.=.~.

.‘p.,

~

.

~

~

~

..-

I

~

~

i

iOO)A~

Fig 20 The structure of a fractal fiber 1151] for different scales.

I

B.M. Smirnov, Physics of ball lightning

198

electric field fractal clusters are polarized and their electric interaction leads to aggregation. Because this process proceeds in an electric field, the patterns formed are anisotropic (have the form of fibers). Aerogels which are formed in solutions, have an isotropic structure. But an aerogel and a fractal cluster consist of the same elements fractal clusters. Thus the production of fractal fibers demands that two conditions are met; a strong action on a surface that causes evaporation of molecules and the presence of an electric field. Evidently these conditions may be satisfied by different actions on a surface. One of these is electrical breakdown of atmospheric air. Fractal fibers which are formed as a result of this are interwoven, and form a knot of fractal fibers. Such a knot may be the framework of ball lightning. An example of such a plasma is a laser plasma which is formed under the action of the radiation having intensity in the range 106_107 W/cm2 and pulse duration i04—103 s. Then the laser energy is spent on evaporation of the surface material and is not absorbed by the forming plasma. Some parameters of such a plasma are given in table 6.5 for the case of irradiation of a copper surface. Here T 0, N0, p0 are the vapor temperature, number density and pressure near the evaporating copper surface, respectively. Evaporating atoms, after some collisions form a vapor beam. The theory of this process has been developed [152—154]. In particular, if the vapor pressure is large compared with the pressure of the surrounding gas, the beam propagates with the velocity of sound. In this case we have, from the laws of the conservation of energy and particles, —

T1=0.69T0,

N1=0.25N0,

(6.30)

where T1 is the temperature of atoms in a beam near the surface and N1 is their number density. The vapor beam is expanded in the adiabatic regime until the vapor pressure is less than the saturated vapor pressure at a given temperature. The temperature and number density of atoms on the saturation line are designated by T~and N~,and their numerical values are given in table 6.5. After this moment the condensation of atoms on ions takes place. The rate of the condensation process is proportional to the ion number density and it is important that the ionization equilibrium is violated [155].Then the ion number density is higher than in the equilibrium case, and the condensation process proceeds rapidly. Table 6.5 presents the numerical values of the temperature T1 at which the ionization equilibrium is violated, and the electron number densities Ne on the saturation surface as well as the ratio of N~to the Table 6.5 Parameters of copper laser plasma. (The calculations assume that 30% of the energy of the laser radiation is spent on material evaporation and the accommodation coefficient the sticking probability of an atom in collision with a surface is equal to 0.2.)

—

Specific2)power (W/cm

—

(K) T0

N0 (cm~)

106

7340 5440 4410

1.1 x 1020 4.1><10’~ 1.6 x 10~

Specific power

T~

N

(W/ cml)

(K)

1 (cm 3)

T (K)

1.5 x 1019 6.4 x 1018 2.9 x 1019

5300 5000 4800

3x106

3270 3 x 106 106

2720 2360

p0 (atm) 110 30 9.4

(K) T1 4910 3640 2950 N1 (cm~) 2.1 x i~’~ 1.4 x iO’~ 1.0 x iO’~

N, (cm~) 2.8 x 1019 1.OxlO’9 4.0 x 109

(atm) 19 5.0 1.6

N11N3

r0 (nm)

r (s)

62 1200 1.7 x i0~

17 12 7.3

0.27 0.18 0.07

199

BM. Smirnov, Physics of ball lightning

equilibrium number density on the saturation line N~.This characterizes the degree of violation of the ionization equilibrium on the saturation line. Along with the condensation process, the coagulation process takes place, i.e. the joining of liquid drops. These processes terminate when the temperature falls to the melting point. After this the particles became solid and join in fractal structures. Table 6.5 gives the values of the average radii of the particles under the assumption that particles are liquid during a time t = i0~s. Note that according to experiments [112,1131, the radii of metallic particles of Pt, Fe, Ti, Ag are found in the range 10—30 nm. The next stage of the process comes when hard particles join in fractal clusters. The value r in table 6.5 is the time of fractal cluster formation under the assumption that the average radius of the fractal cluster is equal to 100 To. Then using the fractal dimension of the aggregate, D = 1.8 [113],we have that the mean fractal cluster consists of 4000 single particles. The last stage of the process is the joining of fractal clusters in fractal fibers. This process can take effect only in an external electric field and proceeds due to the electrostatic interaction of cluster dipoles which are induced by the electric field. To give some estimate for the conditions of the experiment [1511 when the average density of the metal in the chamber equals p i0~g/cm3 and the average density of the metal in clusters is j~ i0~g/cm3 (this gives Nr~—2 x 10~),we obtain a fiber radius r 30 pm and a time of the growth of the structures r 10 mm. This is found in a qualitative agreement with the experiment [1511. Thus the above processes lead to the production of fractal fibers. They are formed as a result of the plasma relaxation, and a plasma is formed from evaporating material as a result of laser irradiation of a solid surface. Other ways of creating this plasma are possible, for example, as a result of breakdown. In my opinion, at present the best model of ball lightning is a knot of interwoven fractal fibers. Firstly, such a system can be produced as a result of electric breakdown of air near a surface. Relaxation of a plasma that is formed as a result of surface evaporation leads to production of fractal fibers. Thus there is a connection between this model and the conditions of the origin of ball lightning. Secondly, the structure of this system satisfies some of the requirements that follow from observations (see section 6.1), such as the conservation of the form and size of ball lightning during its evolution. Furthermore, this structure, just like aerogel, explains the nature of the internal energy of ball lightning due to the large area of the internal surface. Thus the fractal fiber model of ball lightning deserves attention just as the aerogel model does. Now let us compare the model of fractal fibers and the aerogel model. Let us analyze the process of energy release that takes place in the form of propagation of a thermal wave as a result of the densification process. Estimates show that the wavefront that propagates in aerogel is large compared with the correlation radius R~of the aerogel. Thus aerogel is a uniform medium for such a wave. Therefore, only one spherical thermal wave can exist inside the aerogel. Inside the considered system, that is a knot of fractal fibers, many thermal waves can propagate simultaneously, and each of these propagates along a separate fiber. The process of propagation of a thermal wave in fractal fiber is more complicated than in the case of aerogel because the heat extraction process is important. The typical time of heat extraction must exceed the time of the energy release. In practice this condition implies that the maximum temperature of the wavefront exceeds the melting point of the structure. The heat extraction process changes the temperature distribution in a wave (see fig. 21). Moreover, the fractal fiber model as a nonuniform system can explain the observational cases of transformation of the form of the ball lightning during its evolution (the ball goes from the spherical to the band form and back). The uniform aerogel model cannot explain this. The conditions of the -~

—

—~

200

B.M. Smirnov, Physics of ball lightning

Fig. 21. The temperature distribution for the thermal wave propagating (a) in aeroge! and (b) along a fractal fiber.

formation of ball lightning resemble the conditions of the production of fractal fibers and differ from those of aerogel production. Thus, ball lightning is, probably, a knot of interwoven fractal fibers. In spite of this, aerogel is a suitable object for modeling some properties of ball lightning.

7. Mechanics and gas dynamics of ball lightning 7.1. Elastic properties of ball lightning Let us analyze the elastic properties of ball lightning using aerogel as a model of ball lightning. The Young modulus of aerogel decreases as its density decreases because it depletes the number of “chains” which counteract external pressure. The Young modulus of an aerogel, E, can be approximated by the functional form E

=

E0(pIp0)~.

(7.1)

Measurements of the Young modulus for silica aerogel, xerogel and glass were made in ref. [1561.The 3 [120], treatment of these data give for the parameters of formula (7.1) and for densities p —0.1 g/cm = 10658±018N/rn2, f~ = 2.8 ±0.2, p 2. 0 =be0.13 The values of Young’s modulus may usedg/cm to determine the speed of sound in the substance. The longitudinal c,~and transverse c sound velocities are defined by the relations [861

/ E(1-u) \i/2 c =1 I [ \p(l+u)(I—2u)/

/ E \1/2 c=~ j ~ \2p(l+a)/

,

(7.2)

B.M. Smirnov, Physics of ball lightning

201

where a- is the Poisson coefficient. Let us take the scaling law for the sound velocities by analogy with (7.1), c = c0(p/p0)~

(7.3)

.

Then an analysis of the experimental data [120] yields the same coefficient a forc the longitudinal and 3 we obtain transverse sound waves a = 0.9 ±0.2.*) Further, for p0 = 0.13 g/cm 0 170 ±30 m/s for the longitudinal sound wave and c0 = 110 ±20 mis for the transverse one. The Poisson coefficient of a silica aerogel is a- =0.12

i~:~-

(7.4)

Formula (7.1) allows us to estimate the range of densities for which an aerogel can exist in the atmosphere. Suppose that the aerogel is destroyed if its relative change in length is of order unity and the pressure gradient a human (7.1) shout we (80 db), sound pressure amplitude is 2. Then on corresponds the basis ofto formula obtaini.e.,forthethe limited aerogel density 0.2 N/m p = 0.4 x i0~°3 g/l. Together with other estimates, this demonstrates that in principle one may prepare an aerogel with specific gravity comparable to that of air. The elastic properties of ball lightning are observed in some events when ball lightning falls under the influence of gravity and recoils from a horizontal surface just as a ball [8, 166]. In such an observational event [8], three ball lightnings were formed at the end of a soldering iron as a result of a short circuit. The largest one which had a diameter 4 mm made 10—12 recoils from a horizontal surface during 3—4 s (see fig. 24). Stakhanov estimated the acceleration in this case, a = 2h1r2 1 mis2 (h 1.5 cm is the jump altitude; r is the time of the jump upwards). Because it is smaller than the free fall acceleration g = 9.8 mis2, he concluded that the mean specific gravity of ball lightning is of the order of that of air. Let us repeat this estimate taking into account the fact that the skeleton of the ball lightning is small in comparison with the weight of the air inside the ball lightning. Then the inertial mass of ball lightning is determined by the air inside it, and the weight force is the weight of the skeleton only. Designate the specific weight of air as Pair’ and the mean specific weight of the skeleton as Assume ~ = P~Pair~ 1 Then the acceleration of ball lightning under the influence of gravity is a = ~g, and we have ~ 1/7. Compare the weight of ball lightning with the resistance force of air. Because the Reynold number is Re 20, the resistance force of air is given by formula [159]: ~.

F=

(7.5)

CPair~V2TTT2,

where C is the resistance coefficient; v is the ball lightning velocity; r is the ball radius. The condition that the resistance force should be small compared with the gravity of the skeleton yields: ~ >

(3i2ir)(Cirg)h2ir2.

(7.6)

After using numerical values we have ~> 1/3. It can be seen that there is a contradiction between the ~ *) Note that the relation f3 = 2a + 1 follows from formulae (7.1)—(7.3). Then on the basis of Young modulus data we have a complete agreement with these results.

=

0.9

±0.1

in

202

B.M. Smirnov. Physics of ball lightning

values obtained by different means. This can be explained by the limited accuracy with which the used parameters are determined. Taking this fact into account we obtain for the mean specific weight of ball lightning for the considered event (~ = 1003±02) —

p=lO

—3.7±0.3

g/cm

3

7.2. Interaction between ball lightning and the surrounding air Let us analyze the behavior of the air near the surface of ball lightning. Thermal processes take place inside the ball lightning, resulting in the transfer of the released energy to the surrounding air. We analyze the gas-dynamical and thermal processes which result in such a heat transfer. Assume that the ball lightning is some material with the form of a ball, and that the shape and size of ball lightning are conserved during its life time. Let us study the motion of the air near the ball lightning in these conditions. A general picture of this air motion is represented in fig. 22. A mass of air is heated near the ball lightning, rises allowing cold air to move to its place. Far from the ball lightning the air motion is similar to the smoke motion flowing from a tube. Using this analogy we can arrive at some estimate on the gas dynamics of air based on the theory of Zeldovitch [1601related to the motion of smoke from a tube. The Navier—Stokes equation of moving air has the following form [1591: (v-V)v=~z~v+giXT/T,

(7.7)

where v is the air velocity; g is the free fall acceleration; i.’ is the kinematic viscosity coefficient, which in the considered range of temperatures is approximated well by the dependence [142] v = 2/ 5; T is the air temperature; ~T is the difference in air temperature between a given point = 0.159 cm and far from ball lightning. Note that the considered regime corresponds to a small amount air heating ~ T ~ T and the Reynolds number Re = vR/~ is large Re

~-

(7.8)

\~~7

1 (R is the cross-sectional radius of the air flow).

nrt Fig. 22. The character of the interaction of a heated ball with the surrounding air.

B.M. Smirnov, Physics of ball lightning

203

One can see that for this regime the right-hand side of eq. (7.7) is small compared with the left. Indeed, p

—

pv/R2

v2i(R Re) ~ v2iR.

Taking this into consideration, we obtain the following estimate for the typical velocity of a moving air mass at a point on the radius R of the flow cross section: v

[gR iXT(R)IT]~2.

(7.9)

On the basis of this formula we have for the power which is transferred by heated air [9, 161], P— c~pLIT oR2

—

c~pg”2R512AT3/2/Tu2

,

(7.10)

where cA,, is the specific heat of the air and p its density. Because the total transferred power does not depend on the cross section, we have R512 LIT312 = const., i.e., LIT(R)-=R513.

(7.11)

The interaction between moving air and ball lightning creates the carrying power (lift) which acts on the ball lightning. The force due to the air flow acting on the ball lightning is estimated by F-=pv2R2---pgR3LIT/T.

(7.12)

Introducing a numerical coefficient we can express this formula in the form F= agpR 0irR~LIT(R0)iT,

(7.13)

where R0 is the radius of the ball lightning. The numerical coefficient a was obtained from a modeling experiment [161] and equals a = 11 ±5. Let us make some estimate valid for an average ball lightning. that the the ball Reynolds number 5[LIT(RWe assume 112 near lightning, and is large. From and (7.11) we have Re—10 0)iTI Re R213. Thuseqs. the(7.8), above(7.9) assumption is fulfilled. From formula (7.10), taking account of the numerical coefficient we have for air heated close to an average ball lightning, —

LIT—40x 10szooK.

(7.14)

Formula (7.10) gives the power of a heated ball having the radius of an average ball lightning found in the atmosphere, P = A LIT312

,

(7.15)

where the numerical coefficient is A = 1010~04Vs’/K312. Assume that the lift is of the order of the ball lightning weight, i.e., that the ball lightning can fly. Then the ratio Z of the ball lightning weight to the

204

B.M. Smirnov, Physics of ball lightning

air weight inside the ball lightning equals [9, 11, 1611 Z=

10±08,

(7.16)

i.e., the specific weight of the average ball lightning material is of the order of the air specific weight (see section 11.1). Thus the above analysis of the interaction of the ball lightning with the surrounding air leads to the following conclusions. At first, the air temperature near the ball lightning is of the order of 100 K. *) Subsequently, the mean specific gravity of the ball lightning is of the same order as that of air. The weight of an average ball lightning is 1008~z11 (cf. table 12.1). The observational parameters of ball lightning we used have led us to the conclusion that its properties do not change in the course of its existence. 7.3. Interaction of ball lightning with air flow As stated above, ball lightning is a composite phenomenon, hence a productive way to study it is to analyze separately its individual properties. The synthesis of such a study would allow one to arrive at a general picture of this phenomenon. A number of papers by Gaidukov [125—1281 have paved the way in this direction, providing us with an understanding of the character of the motion of ball lightning in air, and an explanation of its behavior as it flows around obstacles and of its capture by an air stream. For the analysis of the gas dynamics of ball lightning one assumes that it is a self-contained object and that, during its interaction with air flow, no air molecules attach to its surface. For the analysis of the motion of ball lightning as it flows in atmospheric air and as it passes through wide gaps, one may model ball lightning as an undeformed ball [162—165].Such a model allows one to explain some observational phenomena, such as the capture of ball lightning by a vortex-type flow (for example, during an airplane motion), its capture by heated smoke flowing from a pipe, etc. [162,163]. A more complicated motion of ball lightning takes place during its passage through holes and slots with dimensions remarkably smaller than its diameter (see fig. 23). A suitable model of ball lightning in such cases is an incompressible ideal liquid [1651.Moving through a small hole together with the air flow, a ball lightning produces a cylindrical spray and in this way flows from one side of the obstacle to the other. Subsequently, due to the action of internal forces, which provide the surface tension of the ball lightning, the substance of the ball lightning resumes its ball form.**) The Gaidukov papers [162—165]on the gas dynamics of ball lightning in air flow are of interest not only from the viewpoint of explaining some observational facts. They formulate the model of the internal structure of ball lightning. According to these papers, the ball lightning is composed of mutually interacting elements. This interaction among the elements ensures the surface tension which causes the spherical form of ball lightning. However, this interaction is weak, and under the influence of air flow the ball lightning can change its form. On the basis of these conclusions one can model the skeleton of ball lightning as a system of interacting fractal clusters. The size of an individual fractal cluster is of the order of 1—10 tim. That *) Note that a suitable model of ball lightning with respect to its thermal interaction with the surrounding air is an electric iron, which has size, power and temperature of the same order of magnitude as an average ball lightning. **) The transport of ball lightning through slots was studied in ref. [166]using erosion plasma to model the ball lightning.

B.M. Smirnov, Physics of ball lightning

205

if Fig. 23. Transport of ball lightning through a hole under the influence of an air flow. The ball lightning (a) emits a cylindrical stream, (b) flows with it from one side of the hole to the other and (c) is reassembled in a ball.

follows from the size of a particle in a fractal cluster (—1—10 nm) and conditions of cluster stability. Problems concerning the interaction of these clusters inside the skeleton of ball lightning, and the character of the skeleton constructed in this way demand a special analysis. 7.4. Ball lightning as an acoustic source The appearance of ball lightning is accompanied usually by a whistle, cracking, sizzling, sputtering and similar sounds. Further we estimate the parameters of ball lightning as an acoustic source by using of pyrotechnical model. We assume that there are hot zones inside the ball lightning which move as a result of the propagation of waves due to chemical reactions. Gaseous products formed as a result of chemical reactions lead to an increase in the pressure of the surrounding air and cause the generation of sound waves. The interference of sound waves from separate hot zones creates the net sound effect. Note that a typical time for cooling the hot zones (and hence of their motion) is of the order of 10 3—i0~s. Therefore the produced sound lies in the range of frequencies detectable by our ears. Let us determine the amplitude of sound waves at a distance R from the ball lightning by assuming that there are n sound sources (hot zones) producing sound waves with randomly distributed phases. Designating by dVldt the rate of air expansion from an individual source we have for the amplitude of the sound wave

206

B.M. Smirnov, Physics of ball lightning

Lip

=

cp(dV/dt)V~iI4irR2,

(7.17)

where c = 3.3 x i04 rn/s is the sound velocity in atmospheric air; p = 1.2 x i0~g/cm3 is the air density. In the conditions under consideration the above rate corresponds to the volume of the gaseous product formed from one hot zone per unit time: dV/dt = ~rR~v,where R 0 is a radius of a hot zone assumed to be cylindrical; v is the velocity of propagation of a thermal wave. By introducing 00 the velocity of propagation of the wave in the massive sample we have —

—

v

=

00

X

(R0/r0),

(7.18)

where r0 is the initial radius of an active substance. From this and eq. (7.11) follows 2r Lip = cpR~v0v7i/4R 0.

(7.19)

On the basis of this formula we can estimate properties of the fractal fiberfrom model of ball(7.19) lightning. 4 we have formula that Using for an average ball lightning R0 1 mm, V() 100 cm/s, n i0 the sound power of an average ball lightning is about 60 dB at a distance of R = 3 m. From this it can be seen that a ball lightning is an acoustic source having average power in accordance with observational data. The conclusion obtained that ball lightning is an acoustic source of medium intensity may follow from other suggested mechanisms giving rise to sound (electrical, discharge, combustion, etc.) using the power of the average ball lightning. j—

—

—

8. Energy and transfer processes in ball lightning 8.1. Types of energy in ball lightning Let us study the sources of energy active inside a ball lightning which causes the destruction of objects. Most cases where objects are destroyed by the action of ball lightning can be explained by release of an electric energy. Indeed, the destruction of objects may be attributed to a quick release of that energy during an electric current flow through the objects. The splintering of a log, melting of a hole, destruction of metallic objects including melting of the wires of electric, radio and telephone circuits, and other forms of rapid destruction can be explained by this mechanism (table 3.9 shows some examples). The source of electric energy could reside inside the ball lighting or it could be external. Our aim is to understand which of these possibilities is realized. Let us estimate the maximum electric energy which could be carried by the average ball lightning having a charged skeleton. In this case the electric field strength near the surface of the ball lightning equals electric field strength at breakdown in dry air, i.e., E = 30kV/cm. The electric energy of a charged ball equals Emax=E2R~/3~

(8.1)

where R 0 is the ball radius. From this we obtain the maximum electric energy of the average ball lightning as approximately Emax = 4 J. This value is remarkably lower than the typical energy released in observational ball lightning. Such a contradiction leads us to the conclusion that in the observed above

B.M. Smirnov, Physics of ball lightning

207

cases, the electric energy is released by an external source. This source is atmospheric electricity and the ball lightning causes an electrical breakdown of air when the air contains large electric fields. *) The electric charge that goes through wires in these cases, is estimated to be of the order of 1 C [10, 32], which is several orders of magnitude larger than the quantity which a ball with the radius of mean ball lightning can carry. Indeed, on the basis of formula (8.1) one can obtain an estimate of the maximum charge of this ball, which would correspond to electric breakdown of the air, equal ~~106 C. Thus, an external source of electric energy is involved in a number of observational events of ball lightning. But this situation is only possible in cases when the ball lightning occurs out of doors, or in rural wooden buildings where an external electric field can permeate. However, half of the observations of ball lightnings have taken place inside spaces where an external source of energy cannot act. In the following we analyze the source types which could be responsible for the internal energy of ball lightning. To solve of this problem we consider the relatively high probability of the occurrence of ball lightning (4 x 10~1.2 ball lightnings per flash of usual lightning) and its correlation with usual lightning. If we accept the association of ball lightning with usual lightning, we must reject various mechanisms of energy storage in ball lightning such as antimatter [79],cosmic rays [80], nuclear processes [81,82], etc. (see also the detailed analysis of such hypotheses by Singer [3,4], Garfield [83], and Charman [5]). Then the internal energy of ball lightning could only be plasma energy or chemical energy. Further, we show that in the case of a low temperature plasma the processes of energy dissipation proceed very rapidly, and that the plasma energy cannot be confined within the ball lightning in the course of its life time. Table 8.1 relates to the case when the energy of ball lightning resides in the charged particles of a plasma, and gets released as a result of recombination processes. Table 8.1 uses recent information [168—170]on the rates of some processes leading to recombination of the charged particles of a plasma. This table provides convincing evidence that the plasma models of ball lightning cannot be valid. The plasma models assume that the energy of ball lightning is stored in charged particles and is released during their recombination. The amount of released energy due to the Table 8.1 Plasma models of ball lightning Hypothetical model plasma of electrons and ions

plasma of positive and negative ions

plasma of clusters of ions

Examples for recombination in excited air

Rate constants at room temperature

Released energy JIa (J s/cm’)

e + N —* 2N e+O—~2O e+NO*~_,N+O e+N 4~—’2N2 0 + O~+ 0, —* 302 NO* + NO~+N 2—~NO+NO2 +N, e+ H 3O~H,O—* recombination e + H,O~(H,O),—~recombination e + NH~(NH,)4 —+ recombination

2 x i0’ cm’/s 2x10’cm’/s 4x 10’cm’/s 6cm’/s

1 x 10” 1x10~ 42 xx 1O~’~ 10_12

1.6x 1O 1.6 x 10_25 cm6/s 6/s lx 10~25cm

6 X 10_I) lx 10~2

2.4 x 10_6 cm’/s 5 x 10’ cm’/s 3 x 10’ cm’/s

1 x 10_12 5 x ill” 8 x 10~°

~ For example, we give the description of one case from the Stakhanov collection [167]which took place in the settlement Krivaja Balkaof the Odessa province (USSR). A ball lightning with a diameter of 60—70cm fell on the iron roof of not so large house. when it touched the roof, an explosion took place and ball lightning broke up into many small balls. As a result of the explosion (besides the broken window panes) the television set, the refrigerator and the electric meter had fused; the covered electric wiring was torn from the wall; melting traces were found on a radiator, pipes and other metallic items. Soot was found near a socket in which the refrigerator was plugged.

208

B.M. Smirnov, Physics of ball lightning

recombination of each pair of charged particles is of the order of the ionization potential J of the atomic particles. The balance equation for the number density of the charged particles has the form dN/dt

—aN2,

(8.2)

where N is the number density of the charged particles in the quasineutral plasma, and a is the recombination coefficient. It follows from this that the total energy of ball lightning stored per unit volume Q throughout its life time r amounts to Q 7—J/a.

(8.3)

One can compose a set of possible concrete processes responsible for the recombination of particles in excited air. Using the well known values of the recombination coefficients, one can findfollows the energy 3 that from parameter J/a from formula (8.3) and compare it with the value 30 X 10±08 J s/cm observational data. The discrepancy of several orders of magnitude between these values shows the invalidity of the ball lightning plasma models. Similarly one can analyze other ways to store energy in ball lightning when this energy is carried by electronically excited atoms or molecules, vibrationally excited molecules, charged aerosols or dust, or chemically active components. Table 8.2 shows, on the basis of recent information on the rate constants, that the life time of excited atoms and molecules in atmospheric air is remarkably smaller than the life time of ball lightning. Thus one can conclude that chemical processes constitute the only practical way of storing the relatively high specific energy in ball lightning for a sufficiently long period [1, 172]. This result can be understood from general considerations. The characteristic time for molecules to collide in excited atmospheric air (which is on the order of i0~s) is many orders of magnitude smaller than the observed life time of ball lightning. Then only strongly prohibited processes can proceed for such a long time. Such is the case of chemical processes in which sub-barrier transitions of nuclei lead to the reconstruction of their structure and so the processes last sufficiently long. It is of interest that the chemical mechanism of energy conservation in ball lightning was proposed in the middle of last century by Arago [1]. Because during thunderstorms nitrogen compounds are usually —

Table 8.2 The quenching of metastable atoms and molecules in atmospheric air Excited particle

Scheme of the decay process in air

O2(’~g)

2O2(IIIg)•_9O2 + O2(I~g)+

O,(~) 0,—~20,

Rate constant 11711 (cm’/s)

Lifetime in air (s)

2 x 10_I) 2 x l0~’~

0.1 0.1

O,(’~~)

O2(~) + N 2—* 02 + N,

2 x l0_17

0.01

N2(A’~)

4 x l0_12 5 x 10_lI

5 x l0~ 4 x i0~’

O(’D) 0(’S)

N2(A’~+ O2—~N2+ 02 1D) + O,—~O+ 02 O( O(1S) + O~~O + 02

3x

5 x l0~’

N~N,(v>0)

N~+N

0~ O,(v >0)

10~13

2—+2N, N~+CO2—~N2+CO2

lx iø~’~ 6xl0~’

O~+ 02_* 20,

1

x

10_I?

0.02 0.02 0.02

209

B.M. Smirnov, Physics of ball lightning

formed, he believed that some unstable chemical compositions could cause the above processes in ball lightnings. Thus chemical processes inside a ball lightning can lead to heat release and glowing. But there are additional requirements related to the process of heat release. To demonstrate this let us consider a concrete process ozone decay in atmospheric air. Ozone is chosen as a composition which is formed with large efficiency in electrical processes in atmospheric air. Ozone decays in atmospheric air according to the scheme —

0+02+ ~2~O3+,

O+O~~2O,.

(8.4)

This decay has the form of a thermal wave. The temperature behind the wave Tm is related to the initial temperature T0 and the initial concentration of ozone in air c by Tm~To=50c,

(8.5)

where the temperatures are expressed in kelvin and the ozone concentration in per cent. Using the known values of the rate constants of the above processes, one can find [9] that the thermal wave velocity depends on the final temperature through u exp(—5800i T) (T is expressed in kelvin). From this it follows [9] that an increase of the final temperature from 400 K to 700 K (the initial temperature is taken to be 300 K) corresponds to an increase of the wave velocity approximately from 0.01 cm/s to 10cm/s. This leads to a sensitive dependence of the time rate of the process on the final temperature, in contradiction to the observational data, according to which a sharp change of the average glowing of the ball lightning is absent during its evolution. Thus the amount of energy released by the chemical process must not depend strongly on the parameters of the process itself. 8.2. Energy release of ball lightning Let us give some estimates on the energy released by ball lightning. Accepting the chemical nature of its internal energy source, let us estimate the weight of the chemically active substance in it. There are two storage mechanisms of the active substance in the skeleton of the ball lightning. In the first case the active substance is found in the pores of the skeleton. In the second case the active substance constitutes the skeleton. Table 8.3 lists types and amounts of active substances for mean ball lightning. Note that the last case of table 8.3 supposes that the energy gets released as a result of the joining together of silica aerogel macroparticles in compact glass. It is assumed that SiO., molecules have a Table 8.3 The weight of active substance providing the energy of a mean ball lightning Active substance

Specific energy content (kJ/g)

Weight (g)

pyrotechnical composition coal stearin ozone silica aerogel

6 30 40 3 3

3 0.7 0.5 7 3

210

B.M. Smirnov, Physics of ball lightning

spherical form and that the interaction takes place only between neighboring molecules. Then the binding energy of the surface molecules is half of the binding energy of the molecules found inside the macroparticles. The typical specific area of the aerogel internal surface is taken as 900 m2/g. It yields for the specific energy that is released as a result of the aerogel transformation to glass the value 3 kJ/ g. The average weight of the active substance of ball lightning equals iOO3404 g according to the data of table 8.3. The weight of the air inside the mean ball lightning equals 1009~z02g, i.e., the ratio of these weights is 1006±06. Note that the ratio of the weight of the ball lightning skeleton to the weight of the air inside the ball lightning equals 10~08 according to formula (7.10). Thus the active substance occupies the smaller portion of ball lightning. —3.6±1.2 We found earlier in subsection 3.3 that the internal energy of ball llghtnlng is the fraction 10 of the energy of atmospheric electricity. But the energy which is spent on the formation of ball lightning exceeds its internal energy. Let us estimate these values on the basis of man-made ball lightning. Silberg [63, 64] reported that accidental short circuits in electrical systems of submarines resulted in the appearance of green fire balls near the circuit breaker electrodes. These balls flowed into the engine room, and had a diameter of approximately 10—15 cm and a life time of about 1 s. According to Silberg’s estimate [63,64], the electric energy released during such short circuits was 0.4—4 MJ. Using these parameters, estimate the energy spent on a single atom of the substance when the ball lightning originates. Assume that the input energy gets transferred to the skeleton and air inside it. In addition, assume that the material of the skeleton is copper. Take into consideration that the ratio of the skeleton weight to the weight of the air inside it equals 10±08.Then we have in the given conditions that the weight of the skeleton equals 1010?~11 g and the weight of the air inside the skeleton is 1009±02 g. Assuming that the amount of energy spent per copper atom and per air molecule is the same, we obtain for the energy which is spent per copper atom the value Lie = 1024±10 eV. Note that the energy of metallic copper atomization (i.e., the energy which is spent per atom as a result of the decomposition of metallic copper into atoms) equals 3.5 eV (10°~ eV). This comparison means that the energy released during an electric short circuit is sufficient to provide the energy copper evaporation which is required for the skeleton formation. One can see that these data do not contradict the concept under study that the skeleton of ball lightning is formed as a result of the relaxation of the evaporating material. Let us estimate which part of the enclosed energy gets transferred to the internal energy of the fire balls. Assume that the internal energy e is proportional to the volume of a ball, and that for a ball of radius R it is -

C

=

s

3,

(8.6)

0(R/R0) where e 0 is the internal energy of the average ball lightning and R0 is its radius. For the above case we have s = 1005±07 kJ and a fraction of 1027~~ of the electric energy released during the electric short circuit is spent on the formation of the fire ball. 8.3. Transfer of the energy and gas inside porous ball lightning Let us study the processes of gas transfer inside a ball lightning. Assume that the skeleton of ball lightning is modeled by a sparse porous structure, such as aerogel. The skeleton is a chemically active substance or some amount of a chemically active substance is found in its pores. As a result of the chemical reactions, in which the chemically active substance participates, which take place in separate

B.M. Smirnov, Physics of ball lightning

211

zones inside the ball lightning, these zones are heated. Gaseous products of the reactions leave the skeleton through pores. Further, we analyze the character of this process, and estimate the pressure which is created inside the ball lightning by gaseous products of the reactions. Assume the following relation between the mean size of pores R~,the free path length of the molecules inside the framework A, and a mean size r0 of the particles composing the skeleton

(8.7)

R~>>A>>r0.

Then the gas flow velocity at each point in space is the same regardless of the distance between the particles and the skeleton. Gas molecules which collide with the particles of the skeleton return to their average velocity as a result of collisions with other molecules. Therefore the gas flow velocity changes slowly in space. The force exerted by the skeleton on a unit volume of the gas is estimated by F—Sj~Adp/dx,

(8.8)

where S is the specific surface of the porous skeleton; j~iis the average density of the skeleton; dpidx is the gradient of the gas pressure. From the equation F = d(vp)/dt (with v the average velocity of the gas flow and p the gas density) one can estimate the difference in the gas pressure Lip between the inside and outside of the skeleton,

(8.9)

Lip— (p/j~)v/SA.

Some numerical estimates can be obtained using the parameters of the aerogel and supposing that the released energy goes to the formation of gaseous products. Choosing the mean power of ball lightning to be P = ~ W, the specific energy content of a chemically active substance to be q = 6 kJ/g, and the average atomic weight of the reaction products to be A = 36, we have for the total flow of molecules through the skeleton (the number of atoms or molecules per unit time) i’ = P1 qA —5 x 1021 ~I• Because i.’ = 47rR~vp(with R0 the radius of ball lightning and v the flow velocity on its boundary) we have v 0.1 cm/s. Then formula (8.9) gives Lip —0.1 Pa— 106 atm. Thus the change in the gas pressure inside the skeleton of ball lightning due to a chemical reaction is relatively small. —

8.4. Time variation of heat transfer processes in ball lightning In the considered models, the ball lightning is implicitly assumed to be a stationary phenomenon whose parameters vary over times of the order of its life time. But the following study of heat transfer from the hot zones inside the ball lightning to its boundary shows a different character of these processes. Let us assume that these processes are stationary and draw inside the ball lightning isotherms, i.e., surfaces of equal temperature, and designate by S(T) the total area of the isotherm with temperature T. Then the heat flow through this surface is J=

—

K(T)S(T) dT/dr,

(8.10)

where K(T) is the thermal conductivity coefficient, and r is the radius of the surface curvature. We assume that the heat transfer is determined by thermal conductivity. The boundary conditions are

212

B.M. Smirnov, Physics of ball lightning

S = 4i~R~,T = T0 on the boundary of the ball lightning (R0 is the ball lightning radius, T0 is its boundary temperature which is close to the nearby air temperature and S = 4irr~n,T = Tmax~with r1 a typical radius of a hot zone. n is the number of hot zones and Tmax is their typical temperature. Let us assume that the thermal conductivity coefficient coincides with the air thermal conductivity coefficient, i.e., the skeleton of the ball lightning is sufficiently sparse. Then with a 10% accuracy in the temperature range 300—2000 K we can use the approximation [141] 8, T K(T)

=

K(T0)(T/T0)°

0

=

300K, K(T0) = 2.7 mW/cm K.

(8.11)

Suppose that heat arises only in hot zones, i.e., J=const.

(8.12)

inside the ball lightning but outside the hot zones. Assuming that S(T) and S(r) are monotonic, take them the form S(T)— T~ ,

S(r)~ra

(8.13)

.

On the basis of eqs. (8.10)—(8.13) we obtain the following relations between the exponents: y

=

1.8a/(a

—

1),

a = y/(y

—

1.8).

(8.14)

For the monotonic functions S(T) and 5(r) these exponents vary within the bounds 1
y<3.f,.

(8.15)

Now let us determine the energy parameters of ball lightning on the basis of the above relations. The heat power transferred inside the ball lightning is S(T)K(T)T

r(T)

=

(a—i) 1.8

=

J0(ct

—

1),

(8.16)

i.e., ~P~ ‘I~),and for the mean ball lightning tP~= i0’°~°’~ W. Assuming that hot zones radiate as black bodies, we have for the radiation power ~rad

where 6

= =

8 = Pmax(Tmax/ T 8 , (8.17) a-T~axS(Tmax)= 41TR~UT~(Tmax/ T0) 0) 4— y = (2.2a 4)/(a 1) and for a medium ball lightning ~max = 250 x 10~°6W.The —

—

energy which must enter a ball lightning to approach the stationary state is R 0

E=

J

T(r)

S dr

c~p(T’)dT’,

(8.18)

where c~is the specific heat capacity of air, p(T) = p(T0)T0/T is its density. The main contribution to

B.M. Smirnov. Physics of ball lightning

213

the integral in (8.18) comes from the intervals where the temperature is closest to the temperature of the ball lightning surface. We have

i.8(a + 1)

=E 9(a—1) ~E 0 (a + 1)

(8.19) 0

and for a mean ball lightning E0 = 400 X i02504 J. Let us analyze these results. One can see that the energy required to approach the stationary state is small in comparison with the observational energy of the mean ball lightning. We obtain heat losses inside the ball lightning due to the radiation of hot zones. But the maximum value of this power is at least one order of magnitude smaller than the power of the mean ball lightning. This discrepancy can be attributed to an error in the determination of the energy of the mean ball lightning, because in most cases of observed destruction that energy is determined under the influence of an external source of electric energy. But in the cases when ball lightning is governed by the internal source of energy, the maximum light flow estimated for the stationary model of ball lightning is according to this analysis, at least, one or two orders of magnitude less than the observed one. Thus we have a discrepancy between the observational data and the stationary model of ball lightning. This leads to the conclusion that the thermal processes inside the ball lightning are not stationary (see also fig. 9). This conclusion is confirmed by some observational data.

9. Radiative processes in ball lightning 9.1. Spotted structure of glowing ball lightning According to observational data the mean ball lightning has the following parameters when considered as a light source. The brightness is 1500 ±200 Im, the light output equals 0.6 X 102205 lm/W. Let us compare the ball lightning as a light source with a radiating black body having the radius of the ball lightning. Then the temperature of such a black body whose brightness is the same as that of the average ball lightning equals 1360 ±30 K. The temperature of the above black body whose light output is that of the mean ball lightning equals 1800 ±200 K. To bring these values into agreement one can assume that only a small part of the surface of the black body ball radiates. This fraction equals 10_17~08. From this it follows that ball lightning is a transparent optical radiator with an optical thickness ~= 10_17~08. Let us analyze the result related to the temperature of a glowing zone. We saw earlier (section 7) that the mean difference LIT between the mean temperature of the ball lightning surface and the temperature of the surrounding air is LIT = 40 x 10±06K, based on the power and radius of the mean ball lightning. The power transferred to the surrounding air is proportional to LIT3t2 (LIT~T, Tis the air temperature). Thus we must conclude that the temperature of a glowing zone which is close to 2000 K cannot be maintained throughout the space inside the ball lightning. From this it follows that the ball lightning has a spotted structure. There are many hot zones with a temperature near 2000 K inside the ball lightning. These hot zones are formed as a result of heat release processes inside the ball lightning and have a limited life time. These hot zones cause the ball lightning to glow. The mean increase of the temperature inside the ball lightning is of the order of tens of kelvin. Thus we have a spotted, glowing ball lightning, with many hot zones existing simultaneously inside it.

214

B.M. Smirnov. Physics of ball lightning

The above scheme provides a valid description if the skeleton of the ball lightning does not absorb the radiation of the hot zones. Let us estimate the skeleton transparency by using the aerogel model for it. Consider an aerogel as a sum of small balls, of which the radii are small in comparison with the wavelength of the radiation. The aerogel transparency does not depend on these radii, but depends only on the aerogel density. The optical density of SiO2 aerogel is [125] ii ±3 cm2/g and the optical thickness at the center of the ball lightning is ~ = 2T~R 0= iO_06±09. It coincides with the above value = ~o1.7 0.8 that follows from observational data in the limits of their accuracy. This result would be different if it corresponded to a skeleton consisting of opaque material. For example, the optical thickness of a skeleton consisting of soot particles equals ~ = 1026±09. Thus we conclude that the ball lightning skeleton consists of transparent material. ±

9.2.

Air equilibrium inside the ball lightning

We conclude from the above that the glowing of a ball lightning is caused by separate zones at high temperatures. The radiation from these zones can arise due to radiative transitions between states of atoms or molecules, and due to radiating hot macroparticles or heated surfaces. Let us consider the first case, when radiation is created by excited atoms or molecules, and study the equilibrium between excited and ground states of atomic particles in such systems. The main processes involving an excited atom or molecule M* taking place in heated atmospheric air are as follows: M*+X2

kq

M + x2>

k~

M*>

hr

>M+X2,

(9.1)

M*

(9.2)

+

X2,

M + 11w.

(9.3)

Here X2 designates an air molecule; M is the considered atom in the ground state; kq is the rate constant of quenching of an excited atom M* by collisions with air molecules; kex is the rate constant of the inverse process; r is the radiative life time of the excited state. The rate constants of excitation ~ and quenching kq of atomic particles in air are related by kex = kq

exp(—LiE/T)g~Ig0,

(9.4)

where LIE is the excitation energy, g0, g~are the statistical weights of the ground and excited atomic states, respectively. Processes (9.1)—(9.3) lead to the following expression for the probability that an excited atomic particle decays to the ground state by photon radiation, but not quenching: =

(1+ {kq(N2)[N2]

+

kq(O2)[O2]}Ty’,

(9.5)

where [X2] is the number density of X2 molecules; kq is the rate constant of excited-state quenching by molecules X2. Table 9.1 contains the values of the probability f3 for the resonance excited states of alkali metal atoms. As it is seen these atoms are found in thermodynamic equilibrium with air (/3 ~ 1). Resonance

B.M. Smirnov, Physics of ball lightning

215

Table 9.1 The parameters of the resonance excited states of alkali metal atoms and the quenching rate constants for these states in air at atmospheric pressure and temperature 1800 K

atom, excited state

excitation energy (eV)

radiative life time (109s)

quenching rate constants~ 3/s) (10_b cm

Probability of

N

2 02 radiation (%) 2P) 1.85 27 5.6 — 1.1i~ Na(32P) Li(2 2.10 16 6.6 U 2.0 K(42P) 1.61 27 5.0 14 1.3 Rb(52P 1.56 28 5.0 19 1.1 2P122) Cs(6 12) 1.39 31 U — 05b ~I The average accuracy of these rate constants [171,1861 is about 20%. bi The ratios of the quenching rate constants for molecules of oxygen and nitrogen are taken equal, 3 ±1.

excited atoms have a short radiative life time. Therefore some excited atoms or molecules are found in thermodynamic equilibrium with atmospheric air. For this reason the radiation power of atoms or molecules does not depend on the way in which they were excited, but is completely determined by the air temperature. Thus the specific power of radiation of hot zones depends on their temperature and the compound, not on the way this excitation came about. The condition /3 1 means that excited atoms or molecules in air are found in thermodynamic equilibrium with atmospheric air. From this it follows that the radiation of individual atoms or molecules found in a hot region of space, is determined by the temperature of this zone, and is independent of the excitation mechanism. -~

9.3. Pyrotechnical composition as an analog of ball lightning As a radiative source, ball lightning is similar to illuminating pyrotechnical compositions [187—189], which produce radiation of a definite colour. In both cases the radiation is a result of the chemical energy transformation and the luminosity of a given colour is created by admixtures which are present in relatively small amounts. Therefore pyrotechnical compositions can be used for modeling ball lightning processes. Note that pyrotechnical compositions are more effective sources of light than ball lightning, whose light efficiency exceeds that of ball lightning by an order of magnitude. Taking into account the above analogy let us model the chemically active substance of ball lightning by pyrotechnical compositions. Taking into account that the specific energy content of pyrotechnical compositions 6 kJ/gthis[187] we inobtain the that density of an active substance in ball as 22°7g/l.is From follows particular combustion of an active substance can lightning heat the air 0.3 x i0a ball lightning up to temperatures of some thousands kelvin. inside Further we introduce the phenomenological model of ball lightning [9, 11, 92, 120, 190, 191] in our analogy between ball lightning and pyrotechnical compositions. Assume that ball lightning has a rigid skeleton similar to an aerogel and has an average specific weight of the order of that of air. Only a small portion of the pores of this skeleton may be occupied by an active substance. The heat released as a result of chemical reactions involving this active substance, causes the heating of the surrounding space and creates radiation. Our task is to analyze the character of this process. The pyrotechnical model assumes that the active substance is found inside the skeleton of the ball

2

216

B.M. Smirnov, Physics of ball lightning

lightning in the form of separate fibers rather than grains. To justify such an assumption we refer to the results of ref. [192], which studied the structure of dyes (rodamin B and malachite green) absorbed by porous glass. It appears that the dye absorbed inside a porous glass forms a fractal structure with fractal dimension D = 1.74 ±0.12. Taking into consideration the fiber-like structure of the active substance inside the ball lightning, the process now involves the propagation of the wave of a chemical reaction along separate fibers, together with the transfer of that wave from one fiber to another. Note that in the considered cases the skeleton may itself be the active substance. Let us make some estimates for the propagation of the wave of the chemical reaction and luminosity along an individual fiber of the active substance. Denote the initial radius of the fiber of an active substance element by r0. After the chemical reaction has taken place, the expanding products occupy a cylindrical volume of radius R1~, 112, (9.6) = [(AIA0)p0/p4(T)] where A 0 is the average atomic mass of the air molecules, A is the average atomic mass of the reaction products, p0 is the initial density of the active substance and pa(T) is the air density at the final temperature. Using the corresponding the same 3) weparameters obtain fromofformula (9.6)active pyrotechnical composition of yellow colour (A = 36, p0 = 2 g/cm R 0/r0 = 80(T/ 1000)1/2, (9.7) where the temperature of the reaction products is expressed in K. The velocity of the wave propagating along a fiber of the active substance is v = v0R01r0

,

(9.8)

where 00 is the wave velocity in the massive sample. For pyrotechnical compositions we have 00 = 0.1—1 cm/s [187—189].We will use these values below. To estimate the initial radius of an active substance element we use the fact that the luminosity (light efficiency) of a hot gaseous volume depends on its size. Let us require that the observational light efficiency of the mean ball lightning coincides with that of the heated volume under consideration. We can estimate some parameters for the ball lightning, having yellow colour, whose luminosity is created by radiation from excited sodium atoms. We use the general calculational scheme described elsewhere [9, 11, 190, 191]. Assume that the sodium concentration in the active substance is the same as its typical concentration in the surface layers of the Earth, i.e., 2.8% [193]. In these calculations the luminous volume is a cylinder of radius R0 and of length VT, where ~ is the time interval of heat transfer. The results of the calculations are given in table 9.2 and are obtained assuming a light efficiency of the luminous volume of 0.7 lm/W and a wave velocity in the massive sample V() = 1 cm/s. Let us restate the phenomenological model of ball lightning on the basis of the above data and add some further numerical estimates. An active substance inside the skeleton of ball lightning is found in the form of separate fibers or, more exactly, in the form of fractal clusters. The wave of a chemical reaction propagates along the individual fibers and is accompanied by luminosity due to formation of heated zones. This wave can be transferred to neighbouring fibers. Let us estimate the average number of fibers h~that participate simultaneously in the process under consideration. The power imparted by the process to an individual fiber is

B.M. Smirnor, Physics of ball lightning

217

Table 9.2 Parameters resulting from the model according to which the wave of a chemical reaction propagates inside the ball lightning [190,191] (symbols defined in the text)

p

=

T (10’ K)

r

1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6

14 ii 8.5 7.0 5.8 4.8 4.2 3.5 3.1 3.0

0 (gm)

r

(l0’ s)

73 39 23 14 9.3 5.8 3.9 2.7 2.0 1.6

R0 (mm)

si (10’)

1.1 0.88 0.70 0.59 0.50 0.43 0.38 0.32 0.29 0.28

0.38 0.61 1.0 1.4 2.0 2.8 3.7 5.3 6.6 7.0

irr~vQp0,

(9.9)

where Q is the specific energy of the chemical of the active substance and p0 is the density active 3) of wethe have for substance. Using the parameters of a pyrotechnical composition (Q = 6 kJ/g, p0 = 2 g/cm the average number of burning fibers, ii

Pip,

(9.10)

where P = 2 kW is the power of typical ball lightning. Table 9.2 contains values of ñ for a mean ball lightning, obtained from (9.10), and their accuracy is estimated to be reliable within a factor 10±12. Note that the process accelerates according to the law T exp(—t/T), where r is typical time of increase of the number of fibers which participate in the process, depending on the length of an individual fiber and on its inflammability. For a typical ball lightning we obtain T 1 s from the observational data. Thus on the basis of observational data and the above analysis we can describe the ball lightning as a light source. It has medium brightness and low efficiency for transforming chemical energy to glowing, similar to that of matches or candles. The ball lightning luminosity is created in small regions of space heated up to temperatures close to 2000 K. This heating takes place due to the chemical reactions of an active substance which is absorbed by the ball lightning skeleton and exists in the form of separate fibers inside the ball lightning. The waves of the chemical reaction propagate simultaneously along a large number of fibers (—10~).They produce the impression of volume luminosity. —

9.4. Candle flame as a model of ball lightning Flames that are formed as a result of combustion of organic fuels have optical parameters close to unity. Among the various types of flame, the properties of the candle flame have been studied well [194, 195]. In what follows we describe the candle model of ball lightning; thus the candle flame parameters are used for modeling the energy and optical properties of ball lightning. Table 9.3 contains some candle flame parameters [197, 198]. In this model we assume that a ball lightning consists of many candle flames, and list in table 9.4 the corresponding parameters based on that assumption. In arriving

218

B.M. Smirnov, Physics of ball lightning Table 9.3 Parameters of a candle flame [197,1981 Table 9.4

temperature power light flow light output optical thickness (in an optical spectrum range) rate of fuel (stearin) combustion rate ofoxygen usage rate of nitrogen supply

1810

±50K

.

±

25 ±6 Im 0.4 ±0.llm/w i0_06203

1.65 ±0.01 mg/s 4.8 mg/s 23mg/s

.

The parameters of the mean ball lightning consisting of candle flames number of separate sources 1016r03 optical thickness l0_17~3 initial amount of fuel (stearin) 0.5 x lOro2 g amount of oxygen used 1.5 x l0~02g amount of air containing this oxygen 6 x io~°~ g amount of air inside the ball lightning 7 x i0~02g

at these results we assumed that the light flow of the ball lightning coincides with the total light flow set of candle flames. One can describe the candle model of ball lightning in the following way. At first fuel is absorbed by the skeleton of the ball lightning in its pores. Evaporation of the fuel by the action of the flame heat leads to its combustion in the flame. The number of burning zones inside the ball lightning is large. There are interactions between neighboring zones. The development of one of these is accompanied by a decrease in the oxygen concentration in the surrounding region and causes the quenching of the neighboring hot zones. Such an interaction determines the number of simultaneously existing hot zones inside the ball lightning. Note that the candle model differs from the pyrotechnical model of ball lightning. First, in the pyrotechnical model the active substance includes simultaneously the fuel and the oxidizer. Second, the active substance in the pyrotechnical model is at rest. In both models there are many hot zones of ball lightning to glow. The difficulty in both models is the estimation of the number of these zones. 9.5. Coal model of chemical source with prolonged glowing Let us consider some model leading to prolonged glowing resulting from the release of chemical energy. This model allows us to analyze a character of glowing and transfer processes inside the ball lightning. A general scheme of the considered model is the following. Coal particles of radius R = 0.1—1 mm are placed between two aerogel plates and heated by laser radiation with low power. After heating the particles, the laser is switched off and coal particles are burned inside an aerogel according to the diffusion regime. The analysis of heating process allows us to determine the parameters of this model the temperature of the particles, the combustion time and the stability of combustion. Let us analyze the behavior of a heated particle inside the aerogel. The heat balance equation for a heated particle with radius R in the stationary regime has the form —

(9.11) where I~measures the heat extracted due to thermal conductivity, radiation, and i~Iis the power of heat release. We have =

4ITR[TK(T)

—

2aa-(T4

T0K(T0)]I(a + 1),

~‘~2 =

—

~2

T~).

the heat extracted due to (9.12)

4irR

Here T is the particle temperature; T 0 the aerogel temperature far from the particle and T ~ T0 K the thermal conductivity coefficient and a = d ln K / d ln T; a is the greyness coefficient (for a black body a = 1); a- is the Stefan—Boltzmann coefficient. —

B.M. Smirnov, Physics of ball lightning

219

We have two regimes of burning. The heat release in the kinetic regime is determined by the rate of chemical reactions and is (9.13) where p~is the density of the material of the particles; Ea is the activation energy of the combustion process; q0 exp(— Eai T) is the specific power of heat release. These parameters have been measured for activated coal and are given by [199—202] 1°W/g, q0=(3±1)x10 valid in the temperature range 800—1800 K. Heat release in the diffusion regime is limited by the oxygen transfer and is given by

(9.14)

Ea=34± kcal/mol,

~dif

=

4irD [0

2]R LIe/(1

+

/3),

(9.15)

where D is the diffusion coefficient of oxygen molecules inside the aerogel; [02] is the number density of oxygen molecules; Lie is the energy of heat release per oxygen molecule; /3 = —d ln(D [02])/d In T. Let us designate by T1 the solution of eq. (9.11) for the kinetic regime and by T2 this solution for the diffusion regime in the stationary case. The analysis shows that self-maintaining combustion is possible for T2> T1 further, T2 is the stable state of this process, T1 relates to the threshold of the thermal explosion instability. Therefore it is enough to heat a particle up to temperature T1, beyond which the thermal instability develops heating the particle up to temperature T2. Note that the diffusion and kinetic regimes can exist separately if the condition ~kjn(T2)~ tP~(T2)is met. In the following we give the results of calculations for activated coal, because in this case we know 3. For these the rate constants of combustion for an aerogel density (6.13). j~= 0.05 g/cm we assume parameters the thermal conductivity[199—202], coefficientand is determined by formula Further I( 3, a = 0.8, 2(T) T (i.e., a = 1). In addition we use the parameter values /3 = —0.5, Pc = 0.8 gicm LIe = 4.02 eV (corresponding to the total combustion of activated coal). Then we have the following expressions for the powers: —

39X2R,

~2

=

570X4R2,

~dif

=

630cpR/Vi~,

~kin

=

10’1R3 exp(—17iX),

(9.16)

where X= T/1000K. The powers are expressed in mW, R in mm. c is the oxygen concentration in the gas; p is the gas pressure in atm. The results of the calculations for pure oxygen at atmospheric pressure are given in table 9.5, the Table 9.5 The results of the ball lightning model obtained from total combustion of activated coal assuming pure oxygen at atmospheric pressure R(mm)

T

0.2 0.4 0.8

1410 1220 1060

2(K)

T,(K)

~

r(s)

~(%)

e(J)

990 940 890

16 9

8 28

14 10

0.01 0.09

4

105

17

0.6

220

B.M. Smirnov, Physics of ball lightning

data correspond to the total combustion of coal. The parameter ~ = ~khfl(T2)/~I’dIf(T2) is the stability reserve, the quantity r is the combustion time; e is the energy required for the sample to burn; i~is the portion of the released heat which goes to heat (the rest goes to radiation), =

~I~/(cP~ + ~~2)

(9.17)

1)dIf 1I~).Therefore, even Note the thatpower in formula (9.16)conductivity only Pdjf depends on the aerogel (~ sensitive to this value, this though of thermal ~1-~ is uncertain and the density results are effect is not reflected on the general picture because one can govern the parameters of the processes by changing fi. In addition it should be noted that the output parameters of this model experiment may be extended by using other fuels. The above model is of interest for two reasons. First, because we used a real physical object, the model proves the reality of the ball lightning model as a chemical source of light [1,9, 172]. Indeed, inside porous systems, processes of heat transfer due to thermal conductivity are weaker than in open systems. Therefore a source of chemical energy placed in transparent porous media leads to an excessive light output in comparison with the case when it is found in open air. Second, this model offers the possibility to study chemical, thermal and radiative processes in ball lightning simultaneously, thus allowing us to understand these processes in detail. —

9.6. Glowing according to fractal fiber model Let us consider the glowing of ball lightning on the basis of a model when the ball lightning skeleton is a knot of fractal fibers. The glowing processes inside the ball lightning in the framework of this model are the following. Glowing waves propagate along separate fibers. The heat release at the wave front takes place as a result of an increase in the size of the fiber particles or a decrease in the specific area of the internal surface of the fractal fibers. The number of waves may be increased when fibers cross, or decreased when fibers include regions with lower densities. A high temperature at the front of a wave decreases later due to the thermal conductivity of air. To make some estimates on the basis of this model, take parameters of a fractal fiber from experiment [151]: radius r = 20 ~m; average material density p —0.01 g/cm3 wave velocity v 100 cm/s; the specific energy storage is chosen on the basis of silica aerogel parameters, e 2 kJ/g. The power released in a separate wave is p ~irr2ve 0.03 W. Then —i05 waves must exist simultaneously in an average ball lightning. We considered different models that can describe the glowing of ball lightning, showing that there are different systems in nature whose glowing resembles that of ball lightning, and allowing us to analyze that glowing in terms of real physical systems. Of all these models, the last one, describing ball lightning as a knot of fractal fibers, is preferable at present. —

—

‘—

—

10. Electrical properties of ball lightning 10.1. Electrical parameters According to observational data ball lightning is attracted to metallic objects and wires. Let us estimate the electric charge and other electrical parameters of ball lightning on the basis of an

B.M. Smirnov, Physics of ball lightning

221

assumption that an electric force exerted on it by metallic objects is equal to its weight. The electrical parameters of the average ball lightning obtained from this assumption are given in table 10.1 [9, 190, 191]. The electric charge q of the average ball lightning results from the relation q2/4R~=P,

(10.1)

where R 0 is the radius of ball lightning (2R0 is the distance between its center and the center of the charge induced inside the metallic object) and P is the weight of the skeleton of the ball lightning. Let us analyze the data of table 10.1. (1) The electric energy of the average ball lightning is smaller than its total energy by approximately five orders of magnitude. (2) The surface tension of water at room temperatures lies within the range of values for the surface tension of the average ball lightning. (3) The average charge density of ball lightning (i.e., the ratio of its charge to its volume) is larger than the density of atmospheric ions by some orders of magnitude. This means that ball lightning is formed in the atmosphere under nonequilibrium conditions. Some conclusions can be drawn from values of C. the Dividing ball lightning According to the analysis 720S it by charge. the average volume of the ball the average value of that charge is 8 x iO_ lightning we obtain the charge density 1089~08 e/cm3. This value is some orders of magnitude higher than the number density of ions in a real atmosphere, which is of the order of i02_103 e/cm3. The discrepancy between these values means that the skeleton formation must take place in a nonequilibrium plasma whose density is higher than the atmospheric one. This conclusion follows also from an analysis of the processes of particle charging in a plasma. The typical times of charging a hard particle and a cluster in the plasma under consideration are smaller than the formation time of the ball lightning skeleton. The equilibrium charge of a particle or a cluster in a quasineutral plasma is proportional to the radius, provided it exceeds the free path length A of the molecules in the plasma (for atmospheric air A 0.06 ~m).The equilibrium charge of a particle or a cluster of radius R in a quasineutral plasma is determined by the Fucks formula [203], q

=

(RT/e) ln(D÷/Dj,

(10.2)

where D 22, D are, respectively, the diffusion coefficients of positive and negative ions in the plasma, and T is the gas temperature. This formula is valid for q e and R ~ A, and can be used for the analysis of a real situation. ~‘

Table 10.1 The electrical parameters of the mean ball lightning. The accuracy is shown in parentheses

charge charge density surface ratio of tension charge to mass electric field strength near a surface electric potential electric energy electric pressure on a surface

8 x l0~(l0~° ‘)C 4 x l08(10~03)2e/cm’ 5O.2(10r08) x l0~8(i0?05) J/m C/g 4 x l0_3(l0~06) V/cm 5 x 104(i0~06)V 0.04(l0~°°) J i.2(lOro6) Pa

222

B.M. Smirnov, Physics of ball lightning

The ion types found in real air and their diffusion coefficients depend on the humidity and other parameters of the air. Using the data of ref. [204], D22 = 0.029 cm2/s and D = 0.043 cm2/s, and the data of ref. [205], D÷= 0.028 cm2/s and D = 0.036 cm2/s, we obtain for atmospheric air on the basis of formula (10.2), q/R

=

(—6 ±1) x i04 e/cm.

(10.3)

From this it follows that the equilibrium charge of the ball lightning skeleton is approximately 106 e, as compared to the value of table 10.1, 10127~z05 e. Therefore we conclude that a large part of the charge of the ball lightning skeleton can be conserved during the aggregation only if this process proceeds in a nonequilibrium plasma. 10.2. Electrical processes of ball lightning Let us analyze the electrical processes which proceed during the formation and evolution of ball lightning. The aggregation of hard particles leads to the formation of the skeleton of ball lightning. These processes take place in a unipolar plasma, and for this reason the skeleton of ball lightning gains electric charge. Table 10.2 contains the times for the various processes involved in unipolar plasma evolution. The chosen parameters of the processes can correspond to a real situation. According to table 10.2, the first stage in these processes is a charge transfer from the plasma to the hard particles. The aggregation of hard particles leads to the formation of a charged cluster. It is important that the charge on the cluster does not influence the rate of the aggregation process. (The corresponding estimates are given in ref. [9].) 10.3. Initial conditions of ball lightning formation From the analysis of the electric properties of ball lightning, one can propose the following scheme for the formation of the ball lightning skeleton. An electric current which is caused by an external electric field flows through a certain region of the atmosphere. It causes air ionization and leads to charge separation. A vapour of skeleton material is ejected in such a plasma. This vapour becomes condensed into particles which join into clusters. Because the process proceeds in a unipolar plasma, particles and clusters gain an electric charge. Let us analyze the processes of charged cluster formation on the basis of a simple model, assuming

Table 10.2 Times in s of electrical and aggregation processes in a unipolar atmospheric plasma. The charge density is 10°e/cm’, the particle radius is 3 nm and the concentration of the material of the particles in air is 1 g/g 19, 190, 191] charging of particles plasma decay as a result of the attachment of ions to particles recombination of positive and negative ions separation of charges in a plasma under the action of an external electric field I kV/cm aggregation of particles to a cluster discharging of the cluster framework in

atmospheric plasma

2.5 x 10~’ 3 x 10_lb 5 x iO~~ 0.01 0.2 10’

B.M. Smirnov, Physics of ball lightning

223

that an electric current flows from the air through the surface of a solid material. This current causes the surface material to evaporate and creates a plasma which contains hard particles. The condition of the electric current continuity on the interface gives E1cr1=E7o-2,

(10.4)

where E1 and E., are the respective electric field strengths in the air and in the solid, and a-1 and a-2 are their respective conductivities. From this relation it follows that, due to the difference in conductivity between an atmospheric plasma and a solid material, an electric field discontinuity arises on the interface. It creates the electric charge on the interface. According to the model under consideration, the surface charge is found in a thin layer, and can be ejected into the atmospheric plasma during surface evaporation under the action of an electric current. Thus a unipolar plasma is created. The plasma charge transfers rapidly to the hard solid particles. Thus during the first stage of cluster formation we have air with charged, solid particles and the total charge of the particles differs from zero. Let us make some estimates on the basis of the above model. Assume for simplicity that the air conductivity is small compared with the material conductivity, i.e., that the electric field jump at the interface coincides with the electric field strength E1 in air. Assume that the area of the interface exceeds substantially the cross section of ball lightning, i.e., we are limited to the one-dimensional case. Then according to the Poisson equation we have Ei=4~eJNdx,

(10.5)

where N is the charge number density on the interface and x is the direction perpendicular to the interface. Introducing R the unipolar plasma size after the material evaporation we obtain for the number density of the excess charge in the plasma, —

N = E1/4ireR.

—

(10.6) 3Ne =

~ELet us take the plasma as a ball of a radius R. The charge inside this ball is equal to q = ~ irR 1R~.Let us assume that solid particles join in a cluster of the radius R. Then this cluster has a charge q and creates near its surface an electric field of strength E, 2=E E = q/R

1i3.

(10.7)

Thus the electric field strength of a charged cluster is of the order of the electric field strength in the initial air through which an electric current flows. The totality of processes, which cause the skeleton to assemble, is such that the electric field in the plasma is frozen on the particles and remains on the ball lightning skeleton. To obtain parameters which correspond to those observed for a typical ball lightning, it is necessary for the initial electric field strength in air to be about E 10 kV/cm. 10.4. Aggregation of charged solid particles The aggregation of solid particles into the skeleton of the ball lightning studied in section 6 was analyzed subject to the assumption that the aggregating particles are neutral. But the skeleton of the

224

B.M. Smirnov, Physics of ball lightning

ball lightning has a charge. It means that some of the particles joining the cluster are charged. Let us analyze the influence of this charge on the aggregation process. During the first stage of the process the charge of the aggregating particle is inessential if the Coulomb interaction energy of the clusters is smaller than their thermal energy T, q2/R

-~

T,

(10.8)

where R is the size of the cluster. Let us make some numerical estimates. The cluster mass is equal to [206]

m= m 0(R/r0)’~,

(10.9)

where m0 is the mass of an individual particle, r0 its radius, D the fractal dimension of the cluster, equaling 1.77 for the cluster—cluster regime of3,aggregation (see refs. If the to density of the and the ratio of the[206,207]). cluster charge its mass is material composed of particles is p~ = 2 g/cm 5 x 10~~ C/g, according to the analysis we obtain from eq. (10.8) R ~ 400 p~mfor r 0 = 3 nm. In this case clusters of size of the order of R 1 p~moccupy the total amount of space, i.e., the skeleton formation stops before the cluster charge can become important for the cluster aggregation. Thus one can conclude from the above that the electric charge of the clusters joining to form the skeleton of the ball lightning, does not influence the rates of the aggregation processes. 10.5. Creation of the electrostatic potential of ball lightning Let us consider the mechanisms which could create a strong electrostatic potential in air and cause the above phenomena. Evidently they have the same nature as the processes in a thunderstorm cloud [49, 211—214] which lead to the origin of its electrostatic potential. After the formation of charged drops and pieces of ice in the cloud, the charge separation takes place because charged aerosol particles of size 20—50 urn fall down. As a result of this process the cloud of average thickness L 4 km acquires an electrostatic potential U = 30—50 MV; the average electric field strength near the Earth’s surface is 130 VIm [49, 2151 and the breakdown electric field strength for dry air is 33.MV/m. It corresponds an Note that the typicaltoion excess indensity the charge density of either U/2~L e/cm number in dry number air at these altitudes is of sign the order of =~ 300e/cm3. The presence of many small particles in the atmosphere with a radius smaller than 60 nm is important for the creation of a cloud potential. The attachment of air ions to these particles causes a decrease in the atmospheric conductivity. For this reason, an electric current inside a cloud is small and does not discharge it. Evidently, a similar mechanism creates the electrostatic potential of ball lightning. Let us consider two possible versions. In the first version, we initially have an atmospheric plasma with a charge number density more than i09 e/cm3. In this plasma the charge separation occurs because large particles with size of the order of 10—100 urn fall down. After this process small particles with size 1—10 nm aggregate in a plasma and create the charge skeleton of ball lightning. In the second version, the charge separation in a dusty atmosphere proceeds in the same way as in a thunderstorm cloud, but the size of the dust region and the electrostatic potential are lower than in the usual cloud. After this the charge separation has occurred in a dust cloud, and near the interfaces a weak discharge arises in which the ball lightning skeleton is formed.

B.M. Smirnov, Physics of ball lightning

225

Let us obtain estimates for the first version. If a particle of a radius 20 um is introduced into a plasma of dry atmosphere, it obtains a charge approximately —100 e due to the difference in the mobilities of positive and negative ions. If the number density of the particles is large, the particle charge is less than the above one. A particle with a radius of 20 um falls down in air with a velocity 5 cm/s, i.e., the formation of unipolar plasma takes some seconds or minutes. Assume that the density of the material of the particles is equal to 2 g/cm3. Then the ratio of the charge of the particles to their weight is equal to 2 X 10_to g/C. Using the average charge number density for ball lightning, 10_80~05 C/cm3, we find that the initial number density of the dust in the atmosphere must be equal to i0’”°5g/cm3. This lowest estimate for the dust density is two orders of magnitude larger than the atmospheric density. The physical impossibility of such a situation rejects this version. The second version gives the following physical picture of the phenomenon. At the beginning we have a dust cloud with an electric potential of the order of 100 kV. If in this cloud the same processes take place as In a thunderstorm cloud, its size must be of the order of 10 rn. In a gap between this cloud and a surface arises a weak gas discharge, in which the ball lightning skeleton can be formed due to the presence in the gap of the dust material. Note that this picture refers to the formation of ball lightning in open air. Inside rooms other ways may exist to bring about the creation of the electric potential of ball lightning. These ways are connected with interactions involving surface discharge. 10.6. Ball lightning as a source of plasma In addition to the glowing of ball lightning, it is of interest to investigate hot zones as sources of plasma. The plasma of the hot zones is found in thermodynamic equilibrium, and the degree of ionization of this plasma is determined by adrnixtures with a small ionization potential. The main contribution to the ionization of air inside the ball lightning may come from sodium. The content of sodium on the surface of the Earth is 2.7% [1931. The ionization of sodium takes place in hot zones. Assume that the sodium content in hot zones is 2.7%, and that the temperature of these zones is 1800 ±200 K. Then we obtain for the electron number density Ne in these zones, Ne

=

10125±07cm3

-

This value is large enough. (For comparison the number density of ions in the atmosphere found near the Earth is in the range 102_103 cm3.) The same order of magnitude applies to the number densities of electrons in hydrocarbon flames (Ne 1012 cm3 [216]). The free electrons in hot zones affect some properties of ball lightning. In particular, due to these free electrons, hot zones are a source of electromagnetic radiation in the radio-frequency range. Free electrons may cause electric breakdown in the presence of strong electric fields in the atmosphere. This fact has probably become apparent in many observations of ball lightning, but its physics is incomprehensible. To analyze the electrical properties of ball lightning we can make use of the concept of a charged skeleton [89]. Thus the charge of the ball lightning ensures the skeleton stability and determines its electrical properties. Another concept [217] is based on the suggestion that ball lightning is found in a strong electric field and that there is corona discharge near it. To confirm this, the authors of ref. [217] performed an elegant experiment. A ball of a radius 2 cm was constructed from wire of diameter 0.15 mm, with a weight about 100 mg. (For comparison, the weight of air inside the ball is about 40 mg.) This ball was placed in atmospheric air between two electrodes with a vertically directed electric

226

B.M. Smirnov, Physics of ball lightning

field. The distance between the electrodes was 30 cm, the electrostatic potential between them lay in the range 50—160 kV. Under these conditions corona discharge was maintained near the ball, and the electrostatic forces compensated for the ball weight, i.e., the ball was suspended between the electrodes. This model experiment [217]showed that the ball position was stable and could change by the action of metallic and conducting objects in the surrounding space. Unfortunately, this experiment does not model ball lightning. The ball used consisted of a few coils only, and therefore the electric field strength near the coils exceeded remarkably its average value near the ball surface. Estimates for a ball that is constructed from fibers [9, 11] and fractal clusters [218] show no evidence of corona discharge near it, if the ball parameters are those of the average ball lightning, in electric fields strengths of the order 1 kV/cm (such fields cause breakdown in clouds). One may expect that some electrical phenomena in ball lightning are the result of the transformation of chemical energy inside it. The ball lightning is then similar to the usual accumulator with a special distribution of charges and currents that depend on the interaction with external conductors. The behavior of such a system was analyzed by Kadomtsev [219]. In conclusion we note the principal aspects concerning the electrical properties of ball lightning. (1) The analysis shows that ball lightning is formed as a result of electric breakdown of air or an electric discharge near a surface. The subsequent evolution of the evaporating material leads to the formation of a rigid skeleton for the ball lightning. The same processes lead to the production of fractal fibers (section 6.6). In both cases the presence of a strong electric field is essential, as it provides the growth and the charge of the structure. (2) Because hot zones develop inside a ball lightning, which are sources of a plasma, the ball lightning causes along its trajectory an enhanced ionization of air. The presence of a strong electric field in the atmosphere may cause electric breakdown along this trajectory and lead to destruction of objects. (3) The electric charge of the ball lightning is responsible for the stability of its skeleton [891. The production of plasma in the hot zones leads to the neutralization of this charge. If the skeleton is similar to a chemical source of electricity, this charge may be restored. Note, that the loss of the electric charge of the ball lightning does not lead to the destruction of its skeleton.

11. Properties of the skeleton 11.1. Specific weight The concept of mean ball lightning (i.e., ball lightning with mean values of parameters) is useful for the analysis of its properties. One can use different information and models of ball lightning to determine the definite parameters of ball lightning. Because these models, which refer to different aspects of ball lightning, must be connected, one can expect to obtain from different treatments close values of the parameters. Further we continue this analysis to estimate two parameters of ball lightning: the specific weight and the surface tension. The values of the specific weight obtained from different considerations are collected in table 11.1. Some comments are in order. The first estimate of the ball lightning is obtained from the energy released in it under the assumption that the temperature of the glowing region is close to 2000 K. The second estimate, given in section 7.4, is based on an observation of the recoil of a ball lightning from a surface. The third estimate takes into account the lift resulting from the convective motion of the air near the ball lightning, and is obtained by supposing that the lift equals the weight of the ball lightning.

B.M. SmiTnov, Physics of ball lightning

227

Table 11.1 Specific weight of mean ball lightning

Specific weight Means of estimation high temperature of glowing regions

(g/cm >10

io’’~°~

recoil from a surface the lift equals the weight

10_50r08

the observational specific internal energy conservation of the spherical form observational optical thickness

iO_40r09

Table 11.2 . . . Surface tension of mean ball lightning 2) Means of estimation Surface tension (J/m ________________________________________________________ conservation of the spherical form 10~7*05 transition through holes 10_10~ 5 electric character of surface tension <10 ‘~‘

<10_45~10 i

40n1 0

average value

0_

The fourth estimate uses the specific energy content of different chemical materials and the above specific internal energy of ball lightning (section 9). The next estimate is based on the action of the surface tension of the skeleton which overcomes the weight of the ball lightning and creates its spherical form. The last estimate uses the observational optical thickness of the mean ball lightning as the ratio of its brightness to the brightness of a black body, the temperature of which coincides with the temperature of the glowing regions of ball lightning. Parameters of silica aerogel were used in the above estimates for the parameters of the skeleton of ball lightning. The data of table 11.1 allow to estimate the specific weight of mean ball lightning. On the basis of these data we have for the average value of the specific weight of the skeleton, ~

=

g/cm~

(11.1)

-

Only the statistical accuracy for the mean value is shown. Note that because of the model-dependent assumptions (for example, use of silica aerogel parameters) each estimate can have a lower accuracy. These independently established quantities of the ball lightning density corroborate to increase the reliability of the final result. 11.2. Surface tension According to observational data ball lightning has a spherical form in 90% of the cases (see section 3.1). This and other facts (e.g., the formation of balls after the decay of ball lightning into fragments, the restoration of the spherical form after passing through small holes and slits, the rolling up of a stick-shaped ball lightning into a ball, etc.) testify for the presence of surface tension. In what follows we estimate the coefficient of the surface tension of ball lightning by different methods. The estimated values of the surface tension of the mean ball lightning are collected in table 11.2. Let us give some comments on these estimates. The first estimate is based on the conservation of the spherical form of ball lightning during its motion. Then the pressure exerted by its weight P must be small compared with the pressure from the surface tension. We have aiR 0> PI4irR~, a

>

~R~/3,

(11.2)

where a is the coefficient of the surface tension of ball lightning; R0 is its radius; ~óis the mean density of the skeleton. The second estimate used the character of the passage of ball lightning through small holes which was analyzed by Gaidukov [160—163].When ball lightning approaches a small hole with different

228

B.M. Smirnov, Physics of ball lightning

amounts of pressure on either side of the hole, a force arises which acts on the ball lightning. This force equals F= 12~py2/R~,

(11.3)

where p is the air density, y is the ball lightning volume passing through the hole per unit time. The pressure acting on the ball lightning surface is a- = F/1rb2, where b is the hole radius. Because this pressure must deform the ball lightning, the coefficient of its surface tension must be bounded by the value a
—

P=q2/4R2,

(11.5)

and the coefficient of the surface tension of ball lightning is a

=

q2/4R~= P/irR 0.

(11.6)

Summarizing data of table 11.2 we have 2 a = 10_15~0.5 J/m As can be seen, the coefficient of the surface tension of ball lightning is comparable with the coefficient of the surface tension of water (0.07 J/ m2). It is essential that the different estimates coincide in the limits of their accuracy. This fact and the use of different methods for obtaining the result increase it~ reliability. -

11.3. Physics of the surface tension and restructuring processes The above values of the coefficient of the surface tension allow us to understand the character of the processes near the surface of ball lightning. We proceed to estimate the coefficient of the surface tension for the two mechanisms. The first of these is similar to the case of a liquid. Let us cut the skeleton of ball lightning into two parts and assume that the formed surface will not change later. Then the surface tension is created by surface molecules that have half as many neighboring molecules compared with the internal ones. The coefficient of surface tension is comparable with the energy per unit area e that is spent to cut the skeleton. Let us estimate this energy by using a simple model of the skeleton consisting of bonded balls of radius r 0. Designating by N the number density of balls we obtain for the cross section of the balls cut

B.M. Smirnov, Physics of ball lightning

229

per unit area by the plane, =

(2~r~/3)2r0N = 4~r~NI3,

where 2irr~/3is the average cross section of a ball, and 2r0N is the number of cut balls per unit area. The centers of these balls are placed at a distance less than r0 from the plane, and these balls are cut by the plane. Use the relation ~ = ~i~r~ p0N, where 1E1 is the average density of the skeleton; p0 is the density of the condensed material. Thus we have: =

(~ip0)e0,

where e0 the energy per unit area that must be spent on cutting the condensed material. Using 3 e~ = 7.3 J/m2) and mean ball lightning (j~= i0~~°4 parameters of silica aerogel (p0 = 2.2 g/cm g/cm3) we have a

<

e = iO

3’~05

J/m2

-

Comparing it with the above quantities of the coefficient of surface tension one can conclude that this mechanism is not responsible for the surface tension of the mean ball lightning. Let us consider the opposite case when the ends of the skeleton have a high mobility and can stick together. As a result of this, the bonds between the ends may be broken and be restored. Then the density of the skeleton near the surface is higher than inside it. This mechanism admits processes of restructuring of the surface of the skeleton as in the case of fractal aggregates [220,221]. The coefficient of the surface tension in this case is estimated by a i~NR~e 0, where ~ is the portion of the particle surface that forms bonds with neighboring particles; R~is the correlation radius which determines the depth of restructuring.2Accepting ~ 0.1 as a basis for the parameters of obtain silica aerogel and mean r 15cm3 R~—10um)we a—0.2J/m2. As ball can lightning (e0=7.3J/m be seen, this mechanism is0=1.5nm; suitable forN—4x10 creating surface tension. From this it follows that the substance of the ball lightning has a sparse and mobile structure. Bonds between elements of this structure can be formed and can decay during the evolution of ball lightning. Therefore the skeleton of ball lightning has properties of a liquid during some processes, for example, during its transport through holes and slits. Besides, restructuring processes lead to densification of the structure during the evolution of ball lightning. The above results show the special structure of the substance of ball lightning. Indeed, ball lightning has a rigid and sparse skeleton. Because of the low density of the skeleton, its elements have a high mobility, resulting in properties both liquid and solid. One more peculiarity of the skeleton that follows from the above analysis is an irreversible process of skeleton densification. This process is intensified by thermal waves propagating inside the skeleton. The process of densification which leads to the breakup of some bonds between the particles of the structure and to the formation of new bonds is called “restructuring”. The restructuring process has been studied well for fractal clusters. There are models (e.g., refs. [220, 221]) to describe the restructuring of fractal clusters. The breakup of some bonds and the formation of other bonds at new locations make the structure more solid and denser. As a result of this process the number of bonds per particle is increased, and densification of the structure takes place. This process was observed in the experiment of ref. [222]where the aggregation of gold particles of

230

B.M. Smirnov, Physics of ball lightning

radius 7.5 nm in a solution was studied. One can control the rate of the aggregation by changing the solution acidity. For low rates of aggregation (up to one per day) the regime of cluster—cluster aggregation limited by reaction (RLCA-regime [209, 210]) takes place, and clusters with fractal dimension 2.0—2.1 were formed. For high rates of aggregation, when a typical duration of the process was of the order of 1 mm or less, a cluster with the fractal dimension 1.7—1.8 was formed. It corresponds to the cluster—cluster regime of aggregation limited by diffusion [207,208] (CCA-regime). In the considered experiment [222]the transition from one type of cluster to another was observed. In some events at the first stage a CCA-cluster with a fractal dimension of about 1.8 was formed, and later it transformed to a RLCA-cluster with a fractal dimension of about 2.1 as a result of the restructuring process. Thus, one can conclude that the skeleton of ball lightning is a special state of substance that has the density of a gas, and can have properties of a solid or liquid. This state is not stable, and “is altered” with time due to the restructuring process that leads to densification of the structure. The life time of such a state of the substance, according to recent experience, is estimated at several days.

12. Conclusions The above analysis shows that the concept of mean ball lightning is useful for the purpose of comparison. Using this concept and basing our analysis on observational data, we obtained additional parameters of this phenomenon. These parameters are collected in table 12.1, allowing us to estimate various properties of ball lightning. Ball lightning is a many-sided phenomenon. Therefore different models are useful for its analysis. Table 12.2 contains a list of some models which were used in this review. Certainly, this list does not include many models which could help in the analysis of this phenomenon. This paper reflects the contemporary state of the understanding of ball lightning. Let us formulate briefly the main results. Ball lightning has a rigid skeleton. This skeleton is formed as a result of nonequilibrium electric phenomena in atmospheric air near a surface, in particular, as a result of the electric breakdown of air. The skeleton is an unusual sort of substance that has the specific gravity of a Table 12.1 Parameters of mean ball lightning Parameter

value

means of estimation

diameter life time brightness internal energy specific gravity weight of the skeleton typical air heating near ball lightning size of particles in the ball lightning skeleton surface tension temperature of glowing zones light output optical thickness number of glowing zones size of an individual glowing zone Young’s modulus of the skeleton

23 ±4 cm 8 x 10~°s 1500 ±200 Im 7 x 10~02 Im 10~’°~° g/cm’ 10 ‘~° g 40 x 10~06K 3 x l0~°° nm 2 0.03 x 10~0 J/m 1800 ±200 K 0.6 X i0~05lm/W

observation observation observation observation various methods (section 11.1) various methods (section 11.1) heat transfer from the skeleton to a convective flow of air duration of the skeleton formation various methods (section 11.2) light stream observation observation various methods (section 9) various methods (section 9) aerogel model (section 7.1)

10~7~08 102 5407 10_08~0‘cm

5 x 10~’Pa

B.M. Smirnov, Physics of ball lightning

231

Table 12.2 Models of ball lightning using real systems Model

Modeled property

Meaning of the model

aerogel fractal fibers external electric source

structure structure energy

chemical energy incompressible liquid electric iron pyrotechnical model

energy gas dynamics gas dynamics glowing

ball lightning has a rigid skeleton similar to an aerogel rigid skeleton of ball lightning is a knot of interwoven fractal fibers processes inside ball lightning are maintained by the electric energy of an external source (atmospheric electricity) ball lightning is governed by an internal source of chemical energy material of ball lightning is an ideal incompressible liquid interaction of ball lightning with surrounding air is the same as near an electric iron the processes of chemical energy transformation to glowing are similar to burning

candle coal charged skeleton

glowing glowing electric property

corona

electric property

pyrotechnical compositions glowing processes are similar to soot glowing in candle flame long glowing is modeled by combustion of coal inside aerogel charge of the skeleton provides its stability and causes attraction to metallic objects corona discharge under action of an external electric field creates the stability of the skeleton

gas, and can have properties of a solid or liquid. The best model of the ball lightning structure is a knot of fractal fibers. The glowing of ball lightning is a result of the propagation of thermal waves along separate fibers. These waves use the surface energy of fractal fibers, and create hot zones with a temperature of about 2000 K that cause the glowing of ball lightning. Thus the glowing of ball lightning has a spotted structure. The state of ball lightning substance deserves most of our attention. Fractal clusters (fractal aggregates) are elements of the ball lightning structure. They take part in the restructuring process which is accompanied by the breaking of some bonds between clusters and by the formation of new bonds. Because of the irreversible character of this process, the structure changes, and its density increases with time. The life time of such a structure is estimated at several days. As can be seen, ball lightning opens new prospects in physics that must be the object of detailed investigations.

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Note added in proof Let us consider briefly the development of the problem during the last two years. During that time a study of ball lightning observations was in progress. New information became available for separate events of ball lightning observations using video [1,2] and camera [3]. These results of photographing ball lightnings were analyzed in detail. The analysis of the observations in their totality includes some outstanding old events [4], as well as new methods of treating data containing thousands of observations [5—7]. New experiments for the complete modeling of ball lightning on the basis of gas discharge were carried out [8—11]. In what follows we elaborate on the concept of the structure of ball lightning, which plays a central role in the understanding of the nature of ball lightning. As we discussed in the paper, ball lightning has a solid skeleton whose small fragments are fractal aggregates. This is essential not only for the ball lightning problem. The ball lightning skeleton is a newly discovered physical object [12]. It is a system of interwoven fractal fibers called “a fractal tangle”. A fractal tangle has simultaneously properties of solids, liquids and gases. Belonging to the group of rarefied porous substances, the fractal tangle has properties of such systems [13]. For this reason a study of rarefied porous systems is of interest for this object.

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In particular, within the framework of the problem considered, this study is important for the new technology of aerogel production [14]. It produces samples of silica aerogel which have minimum density 3 g /1 and preserve their transparency. Such an aerogel has a chain-like micro structure with an average chain width of 2 nm and a distance between links about 20 nm [14]. Along with aerogels which are described in this paper and consist of metallic oxides, organic aerogels have also been produced (see, e.g. ref. [15]). Because of the specific character of the considered concept, additional corroborations are essential. One of the new approaches is Gaidukov’s analysis [16, 17] of the motion of ball lightning in air flows. He has stated, on the basis of a comparison of a gas dynamic theory with observational data, that the boundary layer between ball lightning and air flows is absent. This is only possible in the case where the ball lightning skeleton consists of small fragments which possess a small part of the ball lightning volume. Within the context of the present concept, a ball lightning skeleton is a rarefied porous substance with fragments of small size, i.e., the above condition is fulfilled. Note that the considered model does not give a total description of the phenomenon. In particular, it does not involve a number of separate glowing zones. Therefore, one can expect a modification of the model in the future, just as formerly the aerogel model was changed by the fractal tangle model. Within the framework of this paper, we considered the ball lightning structure as a sum of small bounded balls. Probably, it will be changed in the future, and elements of the polymer structure (polymer net) will be introduced in the ball lightning structure. The polymer structure of ball lightning was discussed in ref. [18].

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