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> r,r, this this is is sufficient sufficientto to excite excite aa weak weak counterpropagating counterpropagating leader leader process. process. With II » Of will have have no 10cm cm long, long, it it will no effect effect on on the the Of course, course, if if this this leader leader is is only only about about 10 lightning lightning trajectory. trajectory. Its Its energy energy is, is, however, however, large large enough enough to to ignite ignite an an inflaminflammable gas mixture, if there is mable gas mixture, if there is any any in in the the vicinity, vicinity, since since the the channel channel temperature temperature is is close close to to 5000 5000 K K and and its its lifetime lifetime is is as as long long as as that that of of aa lightning lightning leader. leader.
1.6.4 1.6.4 High potential infection This term has used in This unsuitable unsuitable term has long long been been used in Russian Russian literature literature on on lightning lightning protection. It protection. It means means that that the the surface surface and and underground underground service service lines, lines, which which get get into into aa construction construction to to be be protected, protected, may may introduce introduce in in it it aa potential potential different from from the the zero zero potential of the construction metalwork metalwork connected connected the construction different potential of to possible if not to earth earth connection. connection. This This may may become become possible if aa service service line line is is not linked the grounding but connected linked to to the grounding of of the the construction construction but connected or or passes passes close close to the the earth to earth connection connection of of another another construction construction loaded loaded by by lightning lightning current current during during aa stroke stroke (figure (figure 1.9). 1.9). This This may may also also be be aa natural natural earth earth connection connection formed the moment with the formed at at the moment of of lightning lightning contact contact with the earth earth due due to to an an intense intense breakionization ionization in in it. it. If If the the introduced introduced potential potential is is high, high, it it causes causes aa spark spark breakdown nearby metallic down between between the the service service line line and and aa nearby metallic structure structure of of the the object, object, whose whose potential potential is is zero zero owing owing to to the the earth earth connection. connection. The The scenario scenario of of the the emergency emergency that that follows follows has has been been described described above. above. To potential infection, To avoid avoid sparking sparking induced induced by by high high potential infection, all all metallic metallic service service lines lines subject subject to to explosion explosion zooms zooms are are linked linked to to the the earth earth connection connection of potentials are of the the construction. construction. All All metalwork metalwork potentials are equalized. equalized. The The connection, connection,
Copyright © 2000 IOP Publishing Ltd.
20
Introduction: lightning, its destructive effects and protection
I -/ Figure 1.9. high potential 1.9. Schematic Schematic input input of of high potential from from remote remote lightning lightning strokes. strokes.
however, becomes by additional way there however, becomes loaded loaded by additional current, current, which which finds finds its its way there from aa remote remote lightning lightning stroke, stroke, using using the the service service line line as as aa conductor. conductor. from When the earth connection connection resistance is low low and and the the service service line line goes goes through through When the earth resistance is so that that the the current current leakage leakage through through the side the ground ground with with aa high the high resistivity resistivity so the side surface is is not not large, large, nearly all of of the the lightning lightning current current arrives arrives at at the the conconsurface nearly all nection from from the the stroke stroke site. site. This This situation situation appears appears to to be somewhat similar similar nection be somewhat to measures must to aa direct direct lightning lightning stroke. stroke. Sometimes, Sometimes, special special measures must be be taken taken to to restrict restrict the the infection infection current. current. A detailed detailed treatment treatment of of the the problem problem of of current current and potential infections will offered in in chapter chapter 6. 6. and potential infections will be be offered
creeping along the earth's earth’s surface surface 1.6.5 Current inrush from a spark creeping This phenomenon is familiar familiar to to all all communications communications men men who have to repair This phenomenon is who have to repair communications cables cables damaged damaged by lightning. The The damaged damaged site site can can be be communications by lightning. detected easily, easily, because it is is indicated indicated by furrow in in the the ground ground extending extending detected because it by aa furrow furrow may may be be as as long long as as several several dozens dozens far away away from from the the stroke stroke site. site. A furrow far of metres, or IOO-200m 100-200m in in aa high ground. Such Such aa long long gap gap can can of metres, or high resistivity resistivity ground. be bridged bridged by spark because because of of the the electric electric field field created created by by the spark current current be by aa spark the spark spreading out out through through the the ground. ground. The The mechanism of spark spark formation formation along along spreading mechanism of conducting surface surface differs differs from from that of aa 'classical' ‘classical’ leader leader propagating propagating aa conducting that of through air. air. A creeping creeping spark spark can can develop develop in in low low fields fields and and have have aa very very through high high velocity. velocity. Underground cables cables are are not not the the only only objects objects suffering suffering from from creeping creeping Underground spark current. current. Similarly, Similarly, it it can can find find its its way way to to underground underground service service lines lines spark and by lightning and to to the the earth earth connections connections of of constructions constructions well well equipped equipped by lightning protectors. But aa protector protector palisade cannot stop stop lightning. lightning. When convenprotectors. But palisade cannot When the the conventional way from from the the earth earth surface surface is is blocked, it breaks through from from beneath, tional way blocked, it breaks through beneath, making in the ground. Lightning thus behaves much like like aa making aa bypass bypass in the ground. Lightning thus behaves very very much clever general general in in ancient ancient times, times, who ordered his soldiers to make aa secret secret clever who ordered his soldiers to make underground instead of of attacking attacking openly openly the the impregnable impregnable underground passageway passageway instead castle walls. It is is reasonable suggest that the contact contact of of aa creeping creeping spark spark castle walls. It reasonable to to suggest that the with with combustible combustible materials materials is is as as frequent frequent aa cause cause of of aa fire fire as as aa direct direct lightning lightning stroke. stroke. The details details of of the the creeping creeping discharge discharge mechanism mechanism have have been unknown The been unknown until quite recently. recently. They They are are analysed analysed in in chapter chapter 6. 6. until quite
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Lightning hazards
21 21
1.10. This This lightning lightning has missed the teletower tip tip by by over over 200m. 200m. Figure 1.10. has missed the te1etower
1.6.6 1.6.6 Are lightning protectors protectors reliable? reliable?
Lightning believed to reliable, since Lightning protectors protectors are are believed to be be reliable, since their their design design has has changed but little Nevertheless, the photochanged but little over over two two and and aa half half centuries. centuries. Nevertheless, the photograph judgement: the graph in in figure figure 1.10 1.10 makes makes one one question question this this judgement: the lightning lightning struck 200 m m below below its its top, top, i.e., i.e., the the Tower Tower struck the the Ostankino Ostankino Television Television Tower Tower 200 could not protect rule. Most could not protect itself. itself. This This is is not not an an exception exception to to the the rule. Most descending descending discharges to what discharges missed missed the the Tower Tower top top more more or or less less closely, closely, contrary contrary to what had had been expected. been expected. This This is is aa serious serious argument argument against against the the vulgar vulgar explanation explanation of of the major major principle principle of the of protector protector operation operation that that lightning lightning takes takes aa shortcut shortcut at at the the final final stage stage of of its its travel travel to to the the earth. earth. There There are are also also other other arguments, arguments, perhaps not not as perhaps as obvious obvious but but still still convincing. convincing. Breakdown under strictly Breakdown voltage voltage spread spread is is registered registered in in long long gaps gaps even even under strictly identical probability 9 \l1 varies varies with with the the pulse pulse identical conditions. conditions. The The breakdown breakdown probability (figure 1.11). 1.11). Deviations Deviations from from the 50% probaprobaamplitude of of test test voltage voltage U (figure amplitude the 50% bility voltage, U bility voltage, Use%,, are appreciable appreciable and and may may be be 10-15% 10-15% either either way. way. SO %, are Curve the probability Curve 22 in in figure figure 1.11 1.11 shows shows the probability function function \l1( !F( U) U ) for for aa shorter shorter gap. ranges, both both curves breakdown probabilities probabilities gap. In In certain certain voltage voltage ranges, curves promise promise breakdown remarkably means that remarkably different different from from zero. zero. This This means that if if two two different different gaps gaps are are tested that any tested simultaneously, simultaneously, there there is is aa chance chance that any of of them them (the (the smaller smaller and and the the larger larger gap) gap) will will be be bridged. bridged. In In general, general, this this situation situation is is similar similar to to that that arisarising point to to strike ing when when aa lightning lightning discharge discharge is is choosing choosing aa point strike at. at. It It does does not not always take the shortest shortest way way to to aa protector but, instead, instead, may may follow follow aa always take the protector but, longer path in order order to to attack attack the the protected protected object. object. longer path in For path problem, problem, one has to For solving solving the the lightning lightning path one has to treat treat aa multielectrode multielectrode system system consisting consisting of of several several elementary elementary gaps. gaps. For For lightning, lightning, all all elementary elementary Copyright © 2000 IOP Publishing Ltd.
22
destructive effects and and protection protection Introduction: lightning, its destructive
,,
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UIU SO % Distributions of breakdown voltages voltages in in air air gaps gaps of of various various lengths lengths with with aa Figure 1.11. Distributions of breakdown sharply non-uniform electric electric field. sharply non-uniform field.
gaps have have aa common common high high voltage voltage electrode (the leader that has has descended to gaps electrode (the leader that descended to aa certain certain altitude), altitude), while while the the zero zero potential potential electrodes electrodes are are formed by the the formed by earth's with grounded grounded objects objects and protectors distributed distributed on on it. it. The The earth’s surface surface with and protectors problem calculation of of problem of of protector protector effectiveness effectiveness thus thus reduces reduces to to the the calculation breakdown probabilities probabilities for for the the elementary gaps in in aa multielectrode multielectrode breakdown elementary gaps system. The The general general formulation formulation of of this this problem problem is is very very complex, complex, since since the the system. spark spark development development in in the the elementary elementary gap gap in in real real conditions conditions cannot cannot be be taken taken to to be be independent. independent. The The discharge discharge processes processes affect affect one one another another by by redistributing redistributing their their electric electric fields, fields, which which eliminates eliminates straightforward straightforward use use of of statistical statistical relations relations describing describing independent independent processes. processes. We We cannot cannot say say that that the the spark spark discharge discharge theory theory for for aa multielectrode multielectrode system system has has been been brought brought to to any any stage stage of of completion. completion. But But what what has has been been done, done, theoretically theoretically and and experimentally, experimentally, allows allows the the formulation formulation of of certain certain concepts concepts of of the the lightning lightning orientation orientation mechanism mechanism and and the the development development of of engineering engineering approaches approaches to to estimate estimate the the effectiveness effectiveness of of protectors protectors of of various various heights heights (see (see chapter chapter 5). 5). Investigation Investigation of of multielectrode multielectrode systems systems is is also also important important from from another another point point of of view: view: we we must must find find ways ways of of affecting affecting lightning lightning actively. actively. It It would would be be reasonable reasonable to to leave leave the the discussion discussion of of this this issue issue for for specialized specialized chapters chapters of of this this book, book, but but they they will, will, however, however, attract attract the the attention attention of of professionals professionals only, only, or or of of those those intending intending to to become become professionals. professionals. It It is is not not professionals professionals but amateurs amateurs who, who, most most often, often, try try to to invent invent lightning lightning protectors. protectors. They They but have have at at their their disposal disposal aa complete complete set set of of up-to-date up-to-date means: means: lasers, lasers, plasma plasma jets, jets, corona-forming corona-forming electrodes electrodes for for cloud cloud charge charge exchange, exchange, radioactive radioactive
Copyright © 2000 IOP Publishing Ltd.
Lightning as a power supply
23 23
sources, high voltage generators generators stimulating stimulating counterpropagating counterpropagating leaders, leaders, etc. etc. sources, high voltage That lightning lightning management management has has aa future future has confirmed by by laboratory laboratory That has been been confirmed studies on on sparks sparks of of multimetre multimetre length. length. These These experiments experiments and and their their implicaimplicastudies tions be analysed analysed below, so there there is is no no point in discussing discussing them them here. here. Still, Still, tions will will be below, so point in it is is hard to resist resist the the temptation make some some preliminary comments hard to temptation to to make preliminary comments it addressed to those who like to to invent invent lightning lightning protection addressed to those who like protection measures. measures. When explaining explaining the the leader leader mechanism mechanism at at the the beginning of this this chapter, chapter, When beginning of we noted that that the the leader leader tip tip carries carries aa strong strong electric electric field field sufficient sufficient for for an an we noted intense air air ionization. ionization. It It is is very difficult to to act act on on this this field field directly, directly, because because intense very difficult it would would be necessary to to create create charged charged regions regions close close by, by, whose whose charge charge dendenit be necessary sity and and amount amount would would be comparable with with those those in in the the immediate immediate vicinity vicinity of of sity be comparable the tip. tip. Pre-ionization Pre-ionization of of the the air air by by radioactive radioactive sources sources is is oflittle of little use use because because the of the the low low air air conductivity conductivity after after radiative radiative treatment. treatment. The The initial initial electron electron of density behind the ionization wave front in the leader process is higher density behind the ionization wave front in the leader process is higher 12 cm than than 10 10l2 ~ m-3,- and and ~ , in in aa 'mature' ‘mature’ leader leader it it is is at at least least an an order order of of magnitude magnitude higher. These and and even even much much lower lower densities densities are are inaccessible inaccessible to to radiation radiation at at aa higher. These distance of dozens of metres from the radiation source which must present no distance of dozens of metres from the radiation source which must present no danger to life. The same is true of a gradual charge accumulation due to danger to life. The same is true of a gradual charge accumulation due to aa slow corona corona formation formation between special electrodes. electrodes. Besides, Besides, one one cannot cannot preslow between special predict the polarity of a particular spark to decide which charge is to to be be dict the polarity of a particular spark to decide which charge is pumped into pumped into the the atmosphere. atmosphere. Quite another another thing thing is is plasma generation. In In principle, we could could create create aa Quite plasma generation. principle, we plasma channel comparable comparable with with the the lightning lightning rod height, thus thus increasing increasing its its plasma channel rod height, length. A high laser could, could, in in principle, used as as aa plasma plasma source. source. It It length. high power power laser principle, be be used is clear clear that that it it should should be be aa pulse pulse source source and and the the plasma produced should should have have is plasma produced short lifetime. lifetime. It It must must be be generated generated exactly exactly at at the the right right moment, when aa aa short moment, when lightning leader leader is is approaching approaching the dangerous region region near the object object to to be be lightning the dangerous near the protected. This is is aa new associated with with synchronization synchronization of of the the protected. This new problem problem associated laser operation operation and and lightning lightning development, development, giving giving aa new to the the task of laser new turn turn to task of lightning protection, which does does not not at at all all become easier. lightning protection, which become easier. Finally, we should always always bear in mind mind that that most lightning discharges discharges are are Finally, we should bear in most lightning multicomponent. In about about half of them, subsequent components components do do not multicomponent. In half of them, the the subsequent not follow the path of of the the first first component. component. In In fact, fact, these these are are new discharges follow the path new discharges which individual handling. To prepare laser light light source source for for which would would require require individual handling. To prepare aa laser operation cycle cycle for for aa fraction fraction of of aa second second is is possible difficult techniaa new new operation possible but but difficult technically. The The cost cost of of such such protection is anticipated anticipated to close to of gold. gold. cally. protection is to be be close to that that of It is is not our intention intention to to intimidate intimidate lightning lightning protection inventors. We It not our protection inventors. We just want want to to warn warn them them against just against excessive excessive enthusiasm. enthusiasm.
1.7
Lightning Lightning as a power supply
The question question of of whether whether lightning lightning could could serve serve as as aa power supply cannot cannot be be The power supply answered positively, how much much we wish it it to to be be one. one. Some Some authors authors answered positively, no no matter matter how we wish
Copyright © 2000 IOP Publishing Ltd.
24 24
Introduction: lightning, Introduction: lightning, its destructive effects and protection
of to harof science science fiction fiction books books force, force, quite quite inconsiderately, inconsiderately, their their characters characters to harness lightning lightning in in order order to to use its electric electric power. Even without without this this service, service, ness use its power. Even lightning people by thought. The lightning has has done done much much for for people by stimulating stimulating their their thought. The energy of of aa lightning lightning flash flash is is not not very very high. high. The The voltage voltage between cloud energy between aa cloud and MV even and the the earth earth can can hardly hardly exceed exceed 100 l00MV even in in aa very very powerful powerful storm, storm, the transported transported charge charge is is less less than than 100 lOOC, and maximum maximum energy energy release release is is the C, and 10 10 J. This 10'OJ. This is is equivalent equivalent to to one one ton ton of of trinitrotoluene trinitrotoluene or or 2-4 2-4 ordinary ordinary airborne bombs. A family family cottage cottage consumes consumes more more power power for for heating, heating, illumiillumiairborne bombs. A nation, nation, and and other other needs needs over over aa year. year. Actually, Actually, only only aa small small portion portion of of the the lightning power is accessible accessible to to utilization, utilization, while while most most of of it it is is dissipated dissipated in in lightning power is the the atmosphere. atmosphere. Normally, aa person person lives Normally, lives through through 40-50 40-50 thunder thunder storm storm hours hours during during aa year. per year. All All storms storms send send to to the the earth earth an an average average of of 4-5 4-5 lightning lightning sparks sparks per 2 square kilometre kilometre of of its its surface surface providing power of of less less than than 11 kWjkm kW/km2 square providing aa power 2 per km , this MW, which 500 x 400 400km2, this is is about about 200 200MW, which is is per year. year. In In aa country country of of 500 aa very very small by an small value value compared compared with with the the electrical electrical power power produced produced by an indusindustrial trial country. country. Just Just imagine imagine the the immense immense net net which which would would be be necessary necessary for for trapping lightning lightning discharges discharges in in order order to to collect collect such such aa meagre meagre power! trapping power! Other power sources, Other natural natural power sources, such such as as wind, wind, geothermal geothermal waters, waters, and and tides, tides, are infinitely infinitely more more powerful than lightning, lightning, but but they they are are still still not not utilized utilized are powerful than much. much. Clearly, Clearly, we we should should not not even even raise raise the the problem problem of of lightning lightning power power resources. resources.
1.8
intending to read read on on To those intending
There will will be more popularized stories about about lightning lightning in in this this book. There be no no more popularized stories book. Nor Nor shall we we mention mention ball lightning here. here. The The next chapter will contain aa shall ball lightning next chapter will contain thorough analysis of of available available data data and and theoretical treatments of of the the long long thorough analysis theoretical treatments spark, because because we we believe that without without these these preliminaries preliminaries the the lightning lightning spark, believe that mechanism may not not become become clear clear to to the the reader. reader. Nature Nature has has eagerly eagerly employed employed mechanism may so lightning lightning is is quite quite likely likely to to represent represent the standard solutions solutions to to its its problems, standard problems, so the limiting case case of of the the long long spark. spark. It It would be useful useful for for readers readers to to familiarize familiarize limiting would be p a r k Discharge, because it is is totally themselves with our previous themselves with our previous book book SSpark because it totally concerned with with this this phenomenon. But even even without without it, it, they they will will be be able able to to concerned phenomenon. But find here here basic basic information information on on long long sparks. sparks. We We have tried to to describe describe their their find have tried general mechanisms mechanisms and and to give predictions predictions as as to to their their extension extension to to air air general to give gaps of of extremal extrema1 length. length. Even Even for for this reason alone, alone, the the next next chapter chapter is is not not gaps this reason summary of of the the previous previous book. Lightning is is as as complicated complicated aa phenomenon aa summary book. Lightning phenomenon as the the long long spark spark and and is is definitely definitely more more diverse. diverse. It It is is aa multicomponent multicomponent as process. Since its its subsequent subsequent components components sometimes sometimes take take the the path path of of an an process. Since earlier component, component, we must consider consider the the effects effects of of temperature temperature and and residual earlier we must residual conductivity in in the the spark spark channel channel on on the the behaviour of new new ionization ionization conductivity behaviour of waves. waves.
Copyright © 2000 IOP Publishing Ltd.
To those intending to read read on
25
Even Even aa simple simple model model should should not not treat treat aa kilometre kilometre spark spark in in terms terms of of electrical circuits circuits with with lump lump parameters. parameters. A lightning lightning spark spark is is aa distributed distributed electrical system. system. The The time time for for which which the the electric electric field field perturbation perturbation spreads spreads along along the the sparks sparks is is comparable comparable with with the the duration duration of of some some of of its its fast fast stages. stages. The The allowance for for the the delay delay can, can, in in some some cases, cases, change change the the whole whole picture picture allowance radically. This This requires requires new new approaches approaches to to lightning lightning treatments. treatments. ExperimenExperimenradically. tal data data and and theoretical theoretical ideas ideas concerning concerning the the lightning lightning leader leader and and return return tal stroke are are discussed discussed together. together. First, First, there there are are not not many many of of them. them. On On the the stroke other we have have tried tried to to point point out out ideological ideological relationships relationships between between other hand, hand, we experiment experiment and and theory theory and and to to offer offer aa more more or or less less consistent consistent physical physical description. description. Spark discharges discharges in in aa multi-electrode multi-electrode system system are are the the subject subject of of aa special special Spark chapter. We We present present available available data data and and analyse analyse possible possible mechanisms mechanisms of oflightchapter. lightning orientation. orientation. This This is, is, probably, probably, the the most most debatable debatable part part of of the the book. book. ning Field of lightning lightning orientation orientation are are very very difficult difficult to to carry carry out out primarily primarily Field studies studies of because constructions of even 100-200 m high are affected by descending because constructions of even 100-200 m high are affected by descending discharges only once or observer must discharges only once or twice twice aa year. year. The The observer must have have exceptional exceptional patience substantial support support to patience and and substantial to be be able able to to reveal reveal statistical statistical regularities regularities in lightning trajectories. From field observations, one usually borrows the in lightning trajectories. From field observations, one usually borrows the statistics of lightning strokes at objects of various height and, sometimes, statistics of lightning strokes at objects of various height and, sometimes, protected objects, objects, such as power power transmission transmission the statistics the statistics of of strokes strokes at at protected such as lines with with overhead overhead grounding grounding wire wire connections. This material, material, however, however, is lines connections. This is too scarce to build build aa theory. theory. For For this this reason, reason, one has to to refer refer to to laboratory too scarce to one has laboratory experiments on long sparks generated in No one one has has ever ever experiments on long sparks generated in 10-15m 10-15m gaps. gaps. No proved (or will ever so) the the geometrical geometrical similarity of sparks; therefore, proved (or will ever do do so) similarity of sparks; therefore, experimental experimental data data can can be be extended extended to to lightning lightning only only qualitatively. qualitatively. NevertheNevertheless less theoretical theoretical treatments treatments must must be be brought brought to to conclusion conclusion when when one one develops develops recommendations recommendations on on particular particular protector protector designs. designs. We We analyse analyse the the reliability reliability of of engineering engineering designs, designs, wherever wherever possible. possible. The The last last chapter chapter of of the the book book discusses discusses lightning lightning hazards hazards and and protection protection not not only only in in terms terms of of applications. applications. Even Even the the classical classical theory theory of of atmospheric atmospheric overvoltages overvoltages in in power power transmission transmission lines lines required required the the solution solution of of comcomplicated plicated electrophysical electrophysical problems. problems. Thorough Thorough theoretical theoretical treatments treatments are are necessary necessary for for the the analysis analysis of of lightning lightning current current effects effects on on internal internal circuits circuits of of engineering engineering constructions constructions with with metallic metallic casings, casings, on on underground underground cables, cables, aircraft, aircraft, etc. etc. The The range range of of problems problems to to be be considered considered is is not not limited limited to to electromagnetic electromagnetic field field theory. theory. We We shall shall also also discuss discuss gas gas discharge discharge mechanisms mechanisms of of aa spark spark creeping creeping along along aa conducting conducting surface, surface, the the excitation excitation of of leader leader channels channels in in air air with with the the composition composition and and thermodynamic thermodynamic characteristics characteristics locally locally changed changed by by hot hot gas gas outbursts, outbursts, and and the the lightning lightning orientation orientation under under the the action action of of the the superhigh superhigh operating operating voltage voltage of of an an object. object. These These theoretical theoretical considerations considerations will will not not screen screen our our practical practical recommendations recommendations concerning concerning effective effective lightning lightning protection protection and and the the application application of of particular particular types types of of protectors. protectors.
Copyright © 2000 IOP Publishing Ltd.
26 26
Introduction: lightning, lightning, its destructive effects and protection
References References [l] Bazelyan Bazelyan E E M, M, Gorin Gorin B B Nand N and Levitov Levitov V V 11978 I 1978 Physical and Engineering Funda[1] of Lightning Protection (Leningrad: (Leningrad: Gidrometeoizdat) Gidrometeoizdat) p 223 223 (in (in Russian) Russian) mentals of 57 493 [2] McEachron McEachron K K 1938 1938 Electr. Engin. 57493 [2] [3] Berger K K and and Vogrlsanger Vogrlsanger E E 1966 SEV 57 No No 13 1966 Bull. SEV 13 1I [3] Berger [4] Newman Stahmann JJ R, 1967 JJ.. Geophys. Geophys. Res. [4] Newman M M, M, Stahmann R, Robb Robb JJ D, D, Lewis Lewis E EA A et a1 al1967 72 4761 4761 [5] Uman Uman M M A 1987 1987 The The Lightning Discharge (New (New York: York: Academic Academic Press) Press) p377 p 377 [5] [6] R Band B and Kroninger Kroninger H H 1975 1975 Electra 4123 41 23 [6] Berger Berger K, K, Anderson Anderson R [7] Malan D D and and Collens Collens H H 1935 1935 Froc. Proc. Roy. Roy. Soc. Soc. London Ser A 152 152 595 595 [7] Schonland Schonland B, Malan EM M and and Raizer Raizer Yu Yu P P 1997 1997 Spark Discharge (Boca (Boca Raton: Raton: CRC CRC Press) Press) [8] Bazelyan Bazelyan E [8] p 294 p294 191 Schonland Schonland B B 1956 1956 The The Lightning Discharge. Handbuch der Physik 22 (Berlin: (Berlin: [9] Springer) p 576 576 Springer)
Copyright © 2000 IOP Publishing Ltd.
Chapter 2
The streamer-leader streamer-leader process process in in a long long spark This will deal with the This chapter chapter will deal with the spark spark discharge discharge in in aa long long air air gap. gap. We We have have already material should be ignored by already mentioned mentioned in in chapter chapter 11 that that this this material should not not be ignored by the know much the reader. reader. But But for for the the long long spark, spark, specialists specialists would would know much less less about about lightning. voltage laboratories lightning. Today, Today, high high voltage laboratories are are able able to to produce produce and and study study long tens and long sparks sparks of of several several tens and even even hundreds hundreds of of metres metres long long [1-3]. [l-31. Many Many of of the the long long spark spark parameters parameters and and properties properties lie lie close close to to the the lower lower boundary boundary of of respective respective lightning lightning values. values. Most Most effects effects observable observable in in aa lightning lightning discharge the laboratory. discharge were, were, sooner sooner or or later, later, reproduced reproduced in in the laboratory. One One exception exception is but the technology rather is aa multicomponent multicomponent discharge, discharge, but the obstacles obstacles lie lie in in the the technology rather than phenomenon. It would be be very very costly than in in the the nature nature of of the the phenomenon. It would costly to to instal instal and and synchronize high voltage power generators, synchronize several several high voltage power generators, making making them them discharge discharge consecutively consecutively into into the the same same air air gap. gap. researchAs for for the the fine fine structure structure of of gas-discharge gas-discharge elements, elements, long long spark spark researchbe otherwise, ers of lightning lightning observers. observers. This This could could not not be otherwise, since since aa ers are are far far ahead ahead of laboratory be reproduced by starting laboratory discharge discharge can can be reproduced as as often often as as necessary, necessary, by starting the the generator generator at at the the right right moment, moment, within within aa microsecond microsecond fraction fraction accuracy, accuracy, and and strictly timing the strictly timing the switching switching of of all all fast fast response response detectors. detectors. But But with with lightning, lightning, the the situation situation is is different. different. It It strikes strikes every every square square kilometre kilometre of of the the earth's earth’s sursurface year. So, face in in Europe Europe approximately approximately 22 to to 44 times times aa year. So, even even such such aa high high conconstruction m) is by lightning struction as as the the Ostankino Ostankino Television Television Tower Tower (540 (540m) is struck struck by lightning only times aa year. these, only only 25-30 25-30 times year. Of Of these, only 2-3 2-3 discharges discharges are are descending, descending, while to aa cloud. Normally, lightning while the the others others go go up up to cloud. Normally, lightning observations observations have have to be be made to made from from afar, afar, so so that that many many details details of of the the process process are are lost. lost. The The be filled gaps gaps in in the the study study of of its its fine fine structure structure must, must, of of necessity, necessity, be filled in in laboratory laboratory conditions. conditions. The The long long spark spark theory theory is is far far from from being being completed, completed, and and there there is is no no the process. process. Still, has lately been some adequate adequate computer computer model model of of the Still, there there has lately been some unwise progress, primarily primarily owing progress, owing to to laboratory laboratory investigations. investigations. It It would would be be unwise to these data try to to use the description to discard discard these data and and not not to to try use them them for for the description of of 21 27 Copyright © 2000 IOP Publishing Ltd.
28 28
The streamer-leader process in a long spark The streamer-leader
lightning. basic lightning. In In this this chapter, chapter, we we shall shall outline outline our our conception conception of of the the basic phenomena in a long spark. We shall present some newer data and ideas phenomena in a long spark. We shall present some newer data and ideas which book [4] which emerged emerged after after the the book [4] on on the the long long spark spark had had been been published. published. We should like to emphasize again that many details of the We should like to emphasize again that many details of the spark spark physics physics are are still still far far from from being being clear. clear.
2.1 2.1
What a lightning lightning researcher should should know know about a long long spark
The how aa spark The key key point point is is how spark channel channel develops develops in in aa weak weak electric electric field, field, by by 1-2 1-2 orders necessary to the electron orders of of magnitude magnitude lower lower than than what what is is necessary to increase increase the electron density under normal Naturally, we density in in air air (E (Eii :::::; x 30 30 kV/cm kV/cm under normal conditions). conditions). Naturally, we speak Near an speak of of aa discharge discharge in in aa sharply sharply non-uniform non-uniform field. field. Near an electrode electrode with this is radius with aa small small curvature curvature radius radius (suppose (suppose this is aa spherical spherical anode anode of of radius x l-lOcm), 1-lOcm), the the field field is is Ea(ra) at the the voltage U x 50-500kV. 50-500kV. rr,a :::::; Ea(r a) == Ea E a > E, Ei at voltage V:::::; = lOr lor,a This is is the the site site of of initiation initiation of of aa discharge discharge channel. channel. At At aa distance distance r = This from the the electrode electrode centre, centre, the channel tip enters the the outer outer gap gap region, region, from the channel tip enters where where the the initial initial value value of of E := =E Ea(r,/r)* is one one hundredth hundredth of of that that on on the the a (r a /r)2 is electrode. This This weak weak field field is is incapable incapable of of supporting supporting ionization. ionization. Nevertheless, Nevertheless, electrode. the moves on, plasma. the channel channel moves on, changing changing the the neutral neutral gas gas to to aa well-ionized well-ionized plasma. There is is no no other other reasonable reasonable explanation explanation of of this this fact fact except except for for aa local local There enhancement the developing enhancement of of the the electric electric field field at at the the tip tip of of the developing channel. channel. The The enhancement enhancement is is due due to to the the action action of of the the channel's channel’s own own charge. charge. Indeed, Indeed, aa conconductive with the the anode be charged much ductive channel channel having having aa contact contact with anode tends tends to to be charged as as much as its its potential U , relative relative to to the the grounded grounded cathode. cathode. Current Current arises arises in in the the as potential Va channel, the positive positive electric channel, which which transports transports the electric charge charge from from the the anode anode (more (more exactly, voltage source, which the the anode exactly, from from the the high high voltage source, to to which anode is is connected). connected). (In (In reality, the channel reality, electrons electrons moving moving through through the channel toward toward the the anode anode expose expose low low mobility positive positive ions.) ions.) Such Such would exactly the the mechanism mechanism of of charging charging aa mobility would be be exactly metallic be pulled pulled out metallic rod rod if if it it could could be out of of the the anode anode like like aa telescopic telescopic antenna. antenna. Then the the strongest strongest field field region region would would move move through through the the gap gap together together with with Then the rod rod tip. tip. We We can can say say that that aa strong strong electric electric field field wave wave is is propagating through the propagating through gap, in in which ionization occurs occurs and and produces new portion of the the plasma plasma aa gap, which ionization produces aa new portion of it as as an an ionization ionization wave, wave, and and tthis h s term term is is commonly commonly channel. We We can can also also name name it channel. used. used. The wave wave mechanism of spark spark formation formation was was suggested suggested as as far far back back as as The mechanism of the by L Raether. The thus formed the 1930s 1930s by L Loeb, Loeb, JJ Meek, Meek, and and H H Raether. The channel channel thus formed was termed termed aa streamer streamer (figure (figure 2.1). 2.1). Experiments Experiments showed showed that that the the streamer streamer was 7 velocity could could be be as as high high as as 10 107m/s. In lightning, lightning, this this velocity is demondemonvelocity m/s. In velocity is strated by by the the dart dart leader leader of of aa subsequent subsequent component. component. Even Even the the mere mere fact fact strated that these these velocities velocities are are comparable comparable justifies our interest interest in in the the streamer streamer that justifies our what determines velocity mechanism. mechanism. It It is is important important to to know know what determines the the streamer streamer velocity and how it with the potential. For have to and how it changes changes with the tip tip potential. For this, this, we we have to analyse analyse
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What a lightning lightning researcher should should know about a long spark spark
29 29
grounded cathode
:ni"',>,~ fUlode _ _cathode t ~
t x
Figure 2.1. A schematic (x), external schematic cathode-directed cathode-directed streamer: streamer: U, Uo(x), external field field potential; potential; U(x), potential along along the the conductive conductive streamer streamer axis. axis. U ( x ) ,potential
processes taking taking place place in in the the streamer streamer tip tip region region where where ionization ionization occurs. occurs. It It is is processes necessary to to find out how how the the processes processes of of charged charged particle particle production production are are necessary find out related to to electron electron motion motion in in the the electric due to to which which the the charged related electric field, field, due charged region travels travels through through the the gap gap like like the the crest crest of of aa sea sea wave. wave. region The specific specific nature nature of of spark breakdown is is not not restricted restricted to to the the ionization ionization The spark breakdown wave front, front, because because its its crucial crucial parameter parameter is is the the channel tip potential potential U Its wave channel tip U,. I . Its value may may be be much much smaller than the the potential potential Ua, of of the the electrode, electrode, from which value smaller than from which the streamer has started, started, since the channel channel conductivity is always always finite finite and the streamer has since the conductivity is and the voltage voltage drops drops across across it. it. Therefore, Therefore, the the analysis analysis of of streamer propagation the streamer propagation for large distance will require require aa knowledge knowledge of of the the electron electron density behind for aa large distance will density behind the wave front and the current along the channel in order to eventually the wave front and the current along the channel in order to eventually calculate calculate the the electric electric field field in in the the travelling travelling streamer streamer and and to to derive derive from from it it the voltage voltage drop drop on on the the channel. channel. Incidentally, Incidentally, the the field field and and the the current current the preset preset the the power power losses losses in in the the channel. channel. It It will will become become clear clear below below how how important important this this parameter parameter is is for for spark spark theory. theory. The The streamer streamer creates creates aa fairly fairly dense dense plasma. plasma. Without Without this, this, it it would would be be unable unable to to transport transport an an appreciable appreciable charge charge into into the the gap. gap. A quantitative quantitative description description of of the the ionization ionization wave wave propagation propagation provides provides the the initial initial electron electron density in the channel and defines its initial radius. Behind the density in the channel and defines its initial radius. Behind the wave wave front, front, the streamer channel channel may may the streamer streamer continues continues to to live live its its own own life. life. A streamer expand, expand, through through ionization, ionization, in in the the radial radial electric electric field field of of its its intrinsic intrinsic charge, charge, provided provided that that the the latter latter grows. grows. The The cross cross section section of of the the current current flow then then becomes becomes larger. larger. The The channel channel continuously continuously loses loses the the majority majority curcurflow rent rent carriers carriers -- electrons. electrons. The The rates rates of of electron electron attachment attachment to to electronegative electronegative particles particles and and electron-ion electron-ion recombination recombination strongly strongly affect affect the the fate fate of of the the discharge If the the air air through through which which aa streamer streamer propagates propagates is is discharge as as aa whole. whole. If cold cold and and the the power power input input into into the the channel channel is is unable unable to to increase increase its its temperatemperature ture considerably considerably (by (by several several thousands thousands of of degrees), degrees), the the process process of of electron electron loss loss is is very very fast, fast, since since the the attachment attachment alone alone limits limits the the electron electron average average lifetime lifetime 7 s.s. This to to 10lop7 This is is aa very very small small value value not not only only at at the the scale scale of of lightning lightning but but
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30 30
The streamer-leader The streamer-leader process in a long spark
4 also also of of aa laboratory laboratory spark, spark, whose whose development development in in aa long long gap gap takes takes 10lop43 10behind lop3s. s. One One must must be be able able to to analyse analyse kinetic kinetic processes processes in in the the channel channel behind the the ionization ionization wave wave front. front. Without Without the the knowledge knowledge of of their their parameters, parameters, one one will be unable unable to will be to define define the the conditions, conditions, in in which which aa streamer streamer breakdown breakdown in in air will will be air be possible. possible. Here and and below, below, we we shall shall mean mean by of aa gap gap by by Here by aa breakdown breakdown the the bridging bridging of aa channel which, like by aa falling channel which, like an an electric electric arc, arc, is is described described by falling current-voltage current-voltage characteristic. then limited by the the resistance resistance of characteristic. The The channel channel current current is is then limited mostly mostly by of the high voltage source. Such Such aa situation situation in in technology technology is is usually called short short the high voltage source. usually called circuiting. circuiting. Current rise increase of of the gap voltage inevitably suggests suggests aa Current rise without without an an increase the gap voltage inevitably considerable heating of the gas thermal expansion, considerable heating of the gas in in the the channel. channel. Due Due to to thermal expansion, the the molecular density N decreases, decreases, thereby increasing the the reduced reduced electric electric field field thereby increasing molecular density E / N and and the the ionization ionization rate constant (see (see [4]). Another Another consequence consequence of of the the rate constant heating is is aa change change in in the the channel channel gas gas composition composition because because of of aa partial heating partial dissociation of of 02, 0 2 ,N2 and H H 2200 molecules molecules and and the the formation formation of of easily easily dissociation N 2 and ionizable NO The significance significance of of many of charged charged ionizable NO molecules. molecules. The many reactions reactions of particle and loss loss changes. changes. The The importance importance of of electron electron attachment attachment particle production production and decreases, because negative ions ions produced in aa hot hot gas gas rapidly disintegrate to decreases, because negative produced in rapidly disintegrate to set free free the the captured captured electrons. electrons. The The electron-ion electron-ion recombination set recombination rate rate becomes becomes lower. But But of of greater greater importance importance is is associative associative ionization ionization involving involving 0 and and N N lower. atoms. The The reaction reaction is is accelerated accelerated with with temperature temperature rise rise but but it it does does not atoms. not depend directly directly on on the the electric electric field. field. This This creates creates prerequisites for aa falling falling depend prerequisites for current-voltage characteristic. characteristic. current-voltage Clearly, aa researcher dealing with with long long sparks sparks and and lightning lightning cannot cannot Clearly, researcher dealing avoid considering considering the the energy energy balance in the the discharge discharge channel, channel, which which deterdeteravoid balance in mines the the gas gas temperature. It is is here here that that the the final final result result is is most most likely likely to to mines temperature. It depend on on the the scale scale of of the the phenomenon phenomenon and and the the initial initial conditions. conditions. In In the the depend laboratory, aa streamer streamer crossing crossing aa long long gap gap seldom seldom causes causes aa breakdown breakdown laboratory, directly. A streamer streamer propagating propagating through through cold cold air air remains cold. It It will directly. remains cold. will be be shown below below that that the the specific specific energy energy input input into into the the gas gas is is too too small small to to heat heat shown it. Even Even during during its its flight, flight, the the old, old, long-living long-living portions portions of of aa streamer streamer lose lose it. most of of their their free free electrons. electrons. In In actual actual fact, fact, it it is is not not aa plasma plasma channel channel but most but rather its its nonconductive nonconductive trace trace which which crosses crosses aa gap. gap. The The researcher must posrather researcher must possess special special skills skills to to be able to to produce an actual actual streamer streamer breakdown breakdown of of aa sess be able produce an cold air air gap gap in in laboratory laboratory conditions. conditions. cold The situation situation with with lightning lightning may may be different. Most Most lightnings lightnings are are multimultiThe be different. component structures. structures. With With the the next next voltage voltage pulse, pulse, the the ionization ionization wave wave often often component propagates through through the the still still hot hot channel channel of of the component. It It is is not not propagates the previous previous component. cold air air but but quite quite aa different different gas gas with more favourable favourable chemical chemical composicomposicold with aa more tion and and kinetic kinetic properties. Surrounded by by cold cold air, air, the the hot hot tract tract shows shows some some tion properties. Surrounded features of of aa discharge discharge in in aa tube with aa fixed fixed radius radius and, and, hence, hence, with with aa more more features tube with concentrated energy energy release. It seems seems that that the the mechanism mechanism of of the the phenomenon concentrated release. It phenomenon known as as aa dart dart leader leader is is directly directly related streamer breakdown. One should should known related to to streamer breakdown. One
°
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What a liKhtning lightning researcher should know lorig spark What knol\' ahout a lonK
31 31
streamer zone
and aa scheme scheme of of aa positive positive leader. leader. Figure 2.2. A photograph photograph and be to give be ready ready to give aa quantitative quantitative description description or or make make aa computer computer simulation simulation of of this process. this process. Long gaps gaps of of cold cold air air are are broken broken down down by by the the leader leader mechanism. During Long mechanism. During the plasma channel (5000- 10 000 000 K) K) is is travelling travelling the leader leader process, process, aa hot hot plasma channel (5000-10 through the the gap. gap. Numerous streamers start start at at high high frequency frequency from from the the through Numerous streamers leader leader tip, tip, as as from from aa high high voltage voltage electrode, electrode, and and form form aa kind kind of of fan. fan. They They fill up up aa volume of several several cubic cubic metres metres in in front front of of the the tip tip (figure (figure 2.2). 2.2). fill volume of This This region region is is known known as as the the streamer streamer zone zone of of aa leader, leader, or or leader leader corona, corona, by analogy analogy with with aa streamer streamer corona corona that that may may arise arise from from aa high high voltage voltage elecelecby trode in in laboratory laboratory conditions. conditions. The The total total current current of of the the streamers streamers supplies supplies trode with energy energy the the leader leader channel channel common common to to the the streamers, streamers, heating heating it it up. up. with with charges charges of of streamers streamers that that are are being being The streamer streamer zone zone is is filled filled up The up with formed formed and and those those that that have have died. died. As As the the leader leader propagates, propagates, the the zone zone travels travels so that the leader leader channel channel enters enters aa space space through the the gap gap together together with with its its tip, tip. so through that the filled with aa space filled with space charge, charge, 'pulling' 'pulling' it it over over like like aa stocking. stocking. A A charged charged leader leader cover thus formed this 2.2). It It is is this cover is is thus formed which which holds holds most most of of the the charge charge (figure (figure 2.2). charge that changes the space charge that changes the the electric electric field field in in the space around around aa developing developing spark and and lightning. lightning. It It is is neutralized neutralized on on contact contact of of the the leader leader channel channel with spark with the earth, creating creating aa powerful powerful current current impulse impulse characteristic characteristic of of the the earth, the return return of interrelated interrelated events, events, stroke of of aa spark. spark. Thus, Thus, we can follow follow aa chain chain of stroke we can which unites the simplest simplest element element of of aa spark spark (streamer) (streamer) with the leader leader which unites the with the process possessing possessing aa complex complex structure structure and and behaviour. behaviour. process All the properties properties All details details of of the the leader leader development development directly directly follow follow from from the of aa streamer streamer zone. zone. In In lightning, lightning, it it is is entirely entirely inaccessible inaccessible to observation of to observation because of the the relatively relatively small small size size and and low low luminosity. luminosity. Today, Today, there is no no because of there is other way way but to study study long long sparks sparks in in laboratory laboratory conditions conditions and and to extrapoother but to to extrapolate the obtained to to extremely extremely long long gaps. gaps. This This primarily concerns aa late the results results obtained primarily concerns stepwise negative leader, whose whose streamer streamer zone zone has has an an exclusively exclusively complex complex stepwise negative leader, Copyright © 2000 IOP Publishing Ltd.
32
The streamer-leader process in a long spark The streamer-leader
structure. polarities, starting structure. It It contains contains streamers streamers of of different different polarities, starting not not only only from from the leader but also them. the leader tip tip but also from from the the space space in in front front of of them. The The leader leader channel channel of of aa very very long long spark, spark, let let alone alone oflightning, of lightning, is is its its longlongAn appreciable part of to the est est element. element. An appreciable part of the the voltage voltage applied applied to the gap gap may may drop drop on on this this element. element. This This is is why why one one should should know know the the time time variation variation of of the the channel properties mostly channel conductivity. conductivity. The The channel channel properties mostly depend depend on on the the current current flowing flowing through through aa given given channel channel cross cross section. section. If If the the current current is is known, known, it it is is not belongs to not particularly particularly important important whether whether it it belongs to aa long long spark spark or or lightning. lightning. The parameter parameter that that changes changes is is the during which which one one observes observes this this The the time time during process: two orders than for process: for for lightning, lightning, it it is is one one or or two orders of of magnitude magnitude longer longer than for aa spark. process ofleader production spark. By By analysing analysing the the self-consistent self-consistent process of leader current current production in the the streamer streamer zone zone and and its its effects effects on on plasma plasma heating heating and and conversion conversion in in the the in channel, one one can can derive derive the the conditions conditions for for an an optimal optimal leader leader development development in in channel, gap of of aa given given length. length. There There are are reasons to believe believe that that these these conditions conditions are are aa gap reasons to realized in in lightning lightning when when it it develops develops in in an an extremely extremely weak weak field. field. Nature realized Nature always strives strives for for perfection, it is is animated animated but because optimal optimal always perfection, not not because because it but because conditions most often lead lead to to the the highest highest probability of an an event. event. conditions most often probability of To to To conclude conclude this this section, section, long long spark spark theory theory is is of of value value in in its its own own right right to specialists in in lightning lightning protection. protection. Lightning Lightning current current is is the the cause cause of of the the most most specialists common type type of of overvoltage overvoltage in in electric electric circuits. circuits. The The amplitude amplitude of of lightning lightning common overvoltages reaches reaches the the megavolt megavolt level. level. In In order order to to design design aa lightninglightningovervoltages resistant circuit, circuit, one one must must be be able able to to estimate estimate breakdown voltages in in air air resistant breakdown voltages gaps of of various various lengths lengths and and configurations. configurations. This This can can be done only only with with aa gaps be done clear understanding of the long spark spark mechanism. mechanism. clear understanding of the long
2.2 2.2.1
A long long streamer The streamer tip as an ionization wave
Let us us consider consider aa well well developed developed 'classical' ‘classical’ streamer, streamer, which has started started from from Let which has high voltage voltage anode anode and and is is travelling travelling towards grounded cathode. cathode. The The aa high towards aa grounded main ionization process occurs in in the the strong strong field field region near the streamer main ionization process occurs region near the streamer tip. We We shall shall focus focus on on this this region. region. The The front front portion of aa streamer streamer is is tip. portion of 2.3 together together with with aa qualitative qualitative axial axial distribudistribushown schematically schematically in in figure figure 2.3 shown E , electron electron density density ne, ne, and and aa difference difference between tion of of the the longitudinal longitudinal field field E, tion between the densities densities of of positive ions and and electrons, electrons, or or the the density density of of the the space space charge charge the positive ions p= = e(n+ e(n+ -- ne) ne) (the (the time time is is too too short short for for negative ions to to be formed). negative ions be formed). The The strong strong field field near near the the tip tip is is created created mostly mostly by by its its own own charge. charge. In In front front of the the tip tip where where the the space space charge charge is is small, small, the field decreases decreases approximately approximately as as of the field E == E ~/r)2, / Y ) ’ ,which is characteristic characteristic of of aa charged charged sphere sphere of of radius Here, Em, ((rTm which is radius .,rYm. Here, E, is the the maximum streamer field field at at the the tip tip front front point. point. In In fact, fact, the at Em is maximum streamer the radius radius at It approxiapproxiwhich the field is is maximum should be termed the the tip tip radius radius .,rYm' It which the field maximum should be termed mately coincides coincides with with the initial radius radius of of the the cylindrical cylindrical channel channel extending extending mately the initial behind the tip. tip. The The front front portion portion of of aa conventionally conventionally hemispherical hemispherical tip tip behind the
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A long streamer streamer
33 33
x, Figure 2.3. Schematic Schematic representation representation of of the the front front portion portion of of aa cathode-directed cathode-directed streamer qualitative distributions distributions of of the the electron electron density density ne, ne, the the density density difference difference streamer and and qualitative n, (space charge), along the n+ - ne ne (space charge), and and longitudinal longitudinal field field E along the axis. axis.
should be called the ionization wave front. The streamer tip charge charge is is primarily primarily should be called the ionization wave front. The streamer tip concentrated in in the the region region behind behind the the wave wave front. front. The The field field there there becomes becomes low, low, concentrated dropping to to aa value value E, Ec in the channel, channel, small as compared compared with with E,. Em. The lines of in the small as The lines of dropping force going radially away from the the tip tip in of it become straight lines inside force going radially away from in front front of it become straight lines inside the tip tip and along the the streamer streamer channel. channel. the and align align axially axially along Let us us mentally mentally subdivide the continuous process of subdivide the continuous process of streamer streamer developdevelopLet ment into The strong strong field region in front of of the the tip tip is is the the site ment into stages. stages. The field region in front site of of ionization ionization of of air air molecules molecules by by electron electron impact. impact. The The initial initial seed seed electrons electrons necessary necessary for for this this process process are are generated generated by by the the streamer streamer in in advance. advance. Their Their production of quanta, quanta, accompanying accompanying the the ionization ionization production is is due due to to the the emission emission of process process because because of of electronic electronic excitation excitation of of molecules. molecules. In In our our case, case, highly highly excited molecules are active so that the quanta emitted by them ionize N2 molecules are active so that the quanta emitted by them ionize excited N 2 molecules, whose ionization potential is lower than that of N the O2 molecules, whose ionization potential is lower than that of N2. The the O 2 2 . The radiation is is actively actively absorbed, absorbed, but but its its intensity intensity is is high high enough enough to to provide provide radiation 5 _10 6 cm -3 at an M~ of of about about 10 105-106~m-3 at aa distance distance of of 0.10.1an initial initial electron electron density density no 0.2 cm from 0.2cm from the the tip. tip. Each Each of of these these electrons electrons gains gains energy energy from from the the strong strong field, field, giving giving rise rise to to an an electron electron avalanche. avalanche. Since Since the the number number of of avalanches avalanches developing of developing simultaneously simultaneously is is very very large, large, they they fill fill up up the the space space in in front front of the the streamer streamer tip tip to to form form aa new new plasma plasma region. region. Owing Owing to to the the electron electron outflow outflow towards towards the the channel channel body, body, the the positive positive space space charge charge of of the the plasma plasma becomes becomes exposed. exposed. Simultaneously, Simultaneously, electrons electrons that that have have advanced advanced from from the the ahead ahead region region neutralize neutralize the the positive positive charge charge of of the the 'old' ‘old’ tip tip which which turns turns to to aa new new channel channel portion, portion, thereby thereby elongating elongating the the streamer. streamer. The The gas gas in in the the wave wave front front region region must must be be highly highly ionized ionized for for the the electronelectronion ion separation separation to to produce produce an an appreciable appreciable charge charge capable capable of of creating creating aa strong strong ionizing ionizing field field in in front front of of the the newly newly formed formed tip. tip. For For this this reason, reason, the the region region of of
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34 34
The streamer-leader streamer-leader process in a long spark
concentrated concentrated tip tip charge charge is is somewhat somewhat shifted shifted towards towards the the channel channel body body relative relative to to the the intensive intensive ionization ionization site site (figure (figure 2.3). 2.3). Normally, Normally, the the electric electric field field is is pushed pushed out out of of aa good good plasma plasma conductor, conductor, and and the the space space charge charge (if (if the the conductor conductor is is charged) charged) quickly quickly concentrates concentrates near near its its surface surface as as aa 'surface' ‘surface’ charge. The The plasma plasma of of aa fast ('fast' in in the the sense that will will be charge. fast streamer streamer (‘fast’ sense that be specified specified below) possesses possesses aa fairly high conductivity, conductivity, and and these these properties properties apply apply to to below) fairly high such streamer. Therefore, strong field field and space charge such aa streamer. Therefore, the the region region of of strong and space charge in in the looks like like aa thin layer, as as is is shown shown in in figure figure 2.3. 2.3. the tip tip looks thin layer, >> rr,,m, its its velocity velocity and and the the tip tip parameters parameters change change If the the streamer streamer length length is is I » If little during during the the time time the the tip tip travels travels aa distance distance of of its its several several radii. radii. This T h s means means little and the the axial axial coordinate coordinate x, all all parameters parameters that, depending depending on on the the time time tt and that, of the the type type E(x, E ( x . t) t) = = E(x E ( x -- Vst), V,t), and and what what is is shown shown in in are the the functions functions of are figure 2.3 2.3 moves moves to to the the right right as as aa whole, whole, without without noticeable noticeable distortions. distortions. The The figure picture changes changes only only as as the the streamer streamer velocity velocity changes changes relatively relatively slowly. slowly. This This picture kind of of process process represents represents aa wave, wave, in in this t h s case case aa wave wave of of strong strong field field and and kind ionization. ionization. The The external external parameter parameter determining determining the the wave wave characteristics characteristics (its (its , velocity maximum field Em, tip radius r , electron density behind the V,, maximum field E,, tip radius r,, electron density behind the velocity V m s wave is the the tip tip potential potential V U,. It is is indeed indeed an an external external characteristic characteristic of of the the wave nn,) t . It c ) is tip, although although it it partly partly depends depends on on the the properties properties of of the the wave wave itself. itself. The The potenpotentip, Ut, is is equal equal to to the the anode anode potential potential Va U , minus minus the the voltage voltage drop drop on on the the tial V tial streamer channel. channel. The The channel channel properties, properties, however, however, are are initially initially determined determined streamer so that that the the problem problem of of streamer streamer developdevelopby the the ionization ionization wave wave parameters, parameters, so by ment is, is, strictly strictly speaking, speaking, just just one one problem. problem. Still, Still, itit can can be be approximately approximately ment of subdivided into into two two parts: parts: the the ionization ionization wave wave problem problem and and the the problems problems of subdivided voltage drop drop and and current current in in the the channel. channel. Both Both parts parts will will be be related related by by the the voltage dependencies of of V V,( Utt) ) and and current current iilj at at the the channel channel front front on on velocity velocity V V,. dependencies s( V s'
2.2.2 2.2.2 Evaluation Evaluation of streamer streamer parameters parameters
The formulas formulas to to be be derived derived in in this this and and subsequent subsequent sections sections of of this this chapter chapter do do The not claim claim high high accuracy. accuracy. The The streamer streamer and and leader leader problems problems are are very very complex, complex, not and and aa rigorous rigorous solution solution can can be be obtained obtained only only by by numerical numerical computation. computation. But But simplified analytical analytical treatment treatment may may also also be be useful useful because because it it provides provides an an aa simplified understanding understanding of of basic basic laws laws and and relations relations among among the the process process parameters. parameters. of the the phenomphenomIn other other words, words, one one is is able able to to get get aa general general idea idea of of the the physics physics of In enon under under study study and and to to estimate estimate the the order order of of values values of of its its characteristics. characteristics. enon Let us us consider consider aa fast fast streamer, streamer, whose whose velocity velocity is is much much higher higher than than the the Let of electron drift drift velocity velocity in in the the wave. wave. Streamers Streamers are are fast fast in in many many situations situations of electron practical practical interest. interest. The The calculation calculation of of electron electron production production can can ignore ignore the the slight drift drift of of electrons electrons from from aa given given site site in in space space for for the the short short time time the the slight ionization wave wave passes passes through through it. it. In In this this case, case, the the ionization ionization kinetics kinetics ionization along the the streamer streamer axis axis is is described described by by the the following following simple simple equations: equations: along e 3 yn,. -an at == vn e,
at
1
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s
f
nc C = exp exp vvidt =exp -n dt = exp no = 1
a0
f
dx
1/'1
V
s
(2.1 )
A long streamer 11:
8
"
35
1011 S-l
6 4
2 E, kVlcm
0
200
400
600
Figure 2.4. Ionization Ionization frequency frequency of of air air molecules molecules by by electron electron impact impact under Figure under normal normal and electron electron drift drift velocity velocity V. V, conditions (from (from the the data data on on ionization ionization coefficient coefficient acy and conditions in [II]). [l 11). in
where vi = Vi vi(E) is the the frequency frequency of of electron electron ionization ionization of of molecules. molecules. Its Its time where Vi = (E) is time integral been transformed x along integral has has been transformed to to the the integral integral over over the the coordinate coordinate x along dt corresponding corresponding to to the the the wave wave axis, axis, according according to to the the equality equality dx dx == V s, dt the coordinate the sharp coordinate system system moving moving together together with with the the wave. wave. Due Due to to the sharp increase increase of the the ionization ionization frequency frequency with field (figure (figure 2.4), 2.4), the the region region where where the field is is of with field the field not not much much less less than than its its maximum maximum makes makes the the largest largest contribution contribution to to the the electron of the the same same order order of of electron production. production. This This region region in in the the wave wave is is of magnitude the tip tip radius radius (figure we can write the 2.3). So So we can write the approximate approximate magnitude as as the (figure 2.3). expressions (2.1) and and streamer streamer velocity: velocity: expressions for for the the integral integral (2.1) (2.2)
This was first by Loeb has been been used used since This type type of of formula formula was first suggested suggested by Loeb [5] [5] and and has since that [6-lo]. The The that time, time, in in this this or or modified modified form, form, in in all all streamer streamer theories theories [6-10]. velocity no and and final final nn,c velocity of of aa fast fast streamer streamer is is weakly weakly related related to to the the initial initial no electron maximum field Em and the electron densities densities and and is is determined determined only only by by the the maximum field E, and the tip radius radius rr,.m' tip The Em and which determine not independent. The quantities quantities E, and rm, which determine V s, are are not independent. They They . For an isolated conductive sphere with aa are by the tip potential potential V are interrelated interrelated by the tip U,. For an isolated conductive sphere with t uniformly distributed Q', we we have have V U == rmE r,E, m == Q' Q 'j47fcorm, /~TTTE~Y,, uniformly distributed surface surface charge charge Qt, 11 12 )-1 ~ F/m is permittivity. A where where co E~ = = (367f ( 3 6 Xx~ 10 lo")-' FZ 8.85 8.85 xX 1010-12F/m is vacuum vacuum permittivity. streamer hemispherical rounded streamer looks looks more more like like aa cylinder cylinder with with aa hemispherical rounded end end (see (see figure perfect conductor 2.3). We We can can show show [4] [4] that that in in aa long long perfect conductor of of this this shape, shape, figure 2.3). the potential potential at one one half half of of the at the the hemisphere hemisphere centre centre is is created created by by charges charges concentrated hemisphere surface half by by those concentrated on on the the hemisphere surface and and the the other other half those on on the the so that that the the tip tip charge charge is is Q Q= = 27fcormVt. ~ T T T E ~The The Y , Ufield field ~ . at at the the tip cylinder surface, surface, so cylinder tip front of that that in in an an isolated isolated front point point is, is, to to good good approximation, approximation, only only one one half half of
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The streamer-leader streamer-leader process in a long spark The
sphere sphere with with the the same same potential, potential, or or
0;t == 2E 2E,r,. U m rm ·
(2.3)
The The tip tip charge charge moves moves because because of of the the electron electron drift drift under under the the field field action. action. The The electron electron density density in in the the wave wave plasma plasma and and the the respective respective plasma plasma conducconductivity tivity must must provide provide the the charge charge transport transport with with the the same same velocity velocity as as that that of of the the wave. wave. This This permits permits estimation estimation of of the the plasma plasma density density in in the the streamer streamer just (2.2),the the electron electron just behind behind its its tip. tip. With With the the same same assumptions assumptions as as those those in in (2.2), density density in in the the strong strong field field region region on on the the streamer streamer axis axis increases increases as as M no noexp exp (Vimt) (vimt)for for the the time time 6.t:::::: At M rm/V rm/Vs. During this this period period of of time, time, the the nnee :::::: s. During electron M no no exp exp (Vim6.t) (vi,&) ,, and and the the electron electron electron density density rises rises to to its its final final value value nncc :::::: Ve, == peE, (where f-te pe is is electron electron drift towards towards the the channel channel with with velocity velocity V drift f-teEm (where mobility taken, taken, for for simplicity, simplicity, to to be be constant) constant) exposes exposes the the charge charge which which mobility E, in in the the region region of of the the new new streamer streamer tip. tip. creates the the field field Em creates The electron electron charge charge that that flows flows through through aa unit unit cross cross section section normal normal to to the the The A t is is axis in in the the wave wave front front region region over over time time 6.t axis At !:>.t
f-teEmnc PeEmnc (2.4) . Jo Vim vim It leaves leaves behind behind it it aa positive positive charge charge of of the the same same surface surface density density q. q. It It is is this this It E,. We We shall shall see see soon soon that that the the effective effective thickness thickness charge that that creates creates the the field field Em. charge Ax « << rm' r,. In In electrostatics, electrostatics, the the field field of of such such aa of aa positively positively charged charged layer layer is is 6.x of E, M q/ q /fOq (this ( t h s equality equality is is layer at at the the conductor conductor surface surface is is equal equal to to Em:::::: layer absolutely exact exact for for the the surface surface charge charge of of aa perfect perfect conductor). conductor). By By substituting substituting absolutely from (2.4), (2.4),we we get get an an estimate estimate for for the the density density behind behind the the ionization ionization wave: wave: q from
ep,E,noSo q == ef-teEmno
exp(6,t) dt dt == exp(vimt)
~
M Eovim/epCL,. nn,c :::::: fOVim/ef-te'
The plasma plasma density density nn,c is is not not related related directly directly to to the the streamer streamer velocity velocity and and is is The essentially determined determined by by the the maximum maximum field field value value which which defines defines the the ionizaionizaessentially tion frequency. frequency. tion Let us us make make sure sure that that the the tip tip charge charge is is indeed indeed concentrated concentrated in in the the thin thin layer. layer. Let The effective effective time time for for the the charge charge to to be be formed formed in in the the layer layer approximately approximately is is The 6.t e ::::::
(At - t)t ) exp(vimt) exp(vi,r) dt/ dt J:l exp(vimt) J:l (6.t exp(q,t) dt dt -
Vi~l.1
= = vi;.
The space space charge charge layer layer of of thickness thickness 6.x Ax moves moves to to aa new new site site at at velocity velocity The Ax/& e which which is is equal equal to to the the streamer streamer velocity velocity V V,, since the the ionization ionization 6.x/6.t s ' since (2.2),we we obtain obtain wave moves moves as as aa whole. whole. Hence, Hence, using using (2.2), wave
Ax = =V VsAt, M Vs/Vim V , / q , :::::: M rr,/ln(n,/no). 6.x (2.5) s6.t e :::::: m/ In(nc/no). (2.5) The final final plasma plasma density density is is many many orders orders of of magnitude magnitude larger larger than than the the initial initial The density no, no,so so that that the the logarithm logarithm in in (2.5) (2.5)is is aa large large value. value. Therefore, Therefore, we we have have density 6.x Ax « << rr,.m. The formulas formulas derived derived here here claim claim for for nothing nothing more more than than an an illustration illustration of of The functional relations relations among among streamer streamer parameters. parameters. Numerical Numerical factors factors allowing allowing functional
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A long streamer
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the of magnitude specific value deliberthe transition transition from from an an order order of magnitude to to aa specific value have have been been deliberately ignored. is justified simplified all all initial conditions ately ignored. This This is justified because because we we have have simplified initial conditions and the of the the phenomenon phenomenon and the derivation derivation procedure procedure in in order order to to reveal reveal the the physics physics of in significance of it is ‘equipped’ with in question. question. The The significance of aa formula formula will will increase increase if if it is 'equipped' with numerical numerical factors, factors, even even approximate approximate ones. ones. Since Since we we know know the the origin origin of of these these factors, judge about factors, we we can can partly partly judge about the the theory theory validity validity and and meaningfully meaningfully compare the the analytical analytical results results with with computations computations and and measurements. measurements. compare more rigorous rigorous treatment treatment of of aa fast fast ionization ionization wave, wave, using using oneoneA more dimensional equations equations consistent consistent with with the the streamer streamer physical physical model model [4,9], [4,9], dimensional yields the yields the expressions expressions
v= _ v, s - (2k -
qVimrm mrm
In(nm/no)'’ (2k - 1) 1) ln(n,/no)
n, =
kePe
’
nm nC = ln-
nm
no
(2.6) (2.6)
where where k is is the the power power index index from from the the approximate approximate formula formula vi(E) Vi (E) "" N E Ekk and and n, nm is is the electron electron density density in in the the wave wave front at the the point of maximum maximum field field (the the front at point of (the density is is an an order order of of magnitude magnitude smaller smaller than than the the maximum maximum achievable achievable density density )' In the field range characteristic of an air streamer, k ;::::: 2.5. = 2.5. density nn,). In the field range characteristic of an air streamer, c The issue issue of of the the streamer streamer tip tip radius radius or or maximum maximum field field represents represents the the The most most complicated complicated and and least least convincing convincing point point in in streamer streamer theory. theory. It It is is likely likely that values are under the the action that their their values are established established under action of of aa self-regulation self-regulation mechanmechanism related related to to the the proportionality proportionality Vs, = ;::::: q(E,) Vi (Em) and to the the rapidly rapidly increasing increasing ism and to (at first) then slowly growing dependence dependence of of q Vi on (figure 2.4). 2.4). If, If, at at (at first) and and then slowly growing on E (figure constant tip potential, potential, the the tip tip radius radius turns turns out out to to be be too too small small and and the the field field constant tip Em respectively respectively too too high, high, corresponding corresponding to to the the slow growth of of vi(E), Vi (E), the the E, slow growth channel front end end will forward quickly channel front will not not only only move move forward quickly but but it it will will expand expand as as fast fast under under the the action action of of aa strong strong transverse transverse field. field. The The value value of of rm, will will rise rise while while that that of of Em' E,, according according to to (2.3), (2.3), will will fall. fall. Suppose, on the contrary, that the radius is too too large large and and the the field field Em E, Suppose, on the contrary, that the radius ,rYm is is too low, corresponding to a rapid growth of Vi (E). Any slight plasma is too low, corresponding to a rapid growth of vi(E).Any slight plasma proprotrusion in the tip tip front will locally the field. trusion in the front will locally enhance enhance the field. The The ionization ionization rate rate will will greatly increase there, there, and the protrusion protrusion will will run run forward forward as of aa greatly increase and the as aa channel channel of smaller radius. Some qualitative considerations considerations of this kind kind were were suggested smaller radius. Some qualitative of this suggested in in an old work work of ofCravath Loeb [12], but these these authors authors discussed discussed aa lightning lightning an old Cravath and and Loeb [12], but channel other mechanisms, mechanisms, because because lightning lightning develops develops via via the the leader leader channel obeying obeying other process. These These ideas ideas were were used used in in [6, 7] to to formulate formulate an an approximate streamer process. [6,7] approximate streamer theory. theory. A A semi-qualitative semi-qualitative criterion criterion was was suggested suggested for for choosing choosing the the maximaximum field feasible in the streamer tip. According to [6,7], Em corresponds mum field feasible in the streamer tip. According to [6,7], E, corresponds to vi(E). This This criterion criterion was was to the the saturation saturation point point or or bending bending in in the the function function vJE). refined refined in in [13] [13] by by establishing establishing aa quantitative quantitative relation relation of of Em E, to to the the slope slope of of the normal field the Vi q ((E) E ) function function and and to to the the charge charge and and normal field distributions distributions over over the streamer tip surface. was shown that the the field Em at the tip tip front front the streamer tip surface. It It was shown that field E, at the point is is established established such that the the normal normal field field on its lateral point such that on its lateral surface surface corresponds to to the the point point of transition from from the the rapid rapid to to the the slow of corresponds of transition slow growth growth of the ?vJE) function (figure (figure 2.4). 2.4). the ( E ) function
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The streamer-leader streamer-leader process in a long spark
The The streamer's streamer’s choice choice of of maximum maximum field field was was manifested manifested during during aa numerical numerical simulation simulation of of short short fast fast streamers streamers within within the the framework framework of of aa complete complete two-dimensional two-dimensional formulation formulation [14-16]. [ 14- 161. The The mechanism mechanism of of autoautothe calculation matic E, was was demonstrated demonstrated in in [13] [13] in in the calculation of of matic establishment establishment of of Em long long streamer streamer development development with with arbitary arbitary initial initial conditions conditions and and considerably considerably simplified been able No one one has has been able yet yet to to simplified equations equations (see (see also also [4 [4 Suppl.]). Suppl.]). No calculate calculate aa long long streamer streamer within within aa complete complete two-dimensional two-dimensional model. model. We We can can conclude conclude from from these these results results that that the the field field at at the the front front point point of of an an air air streamer streamer propagating propagating at at atmospheric atmospheric pressure pressure and and room room temperature temperature seems kV!cm. The tip radius E, ~ M 150-170 150-170kV/cm. The tip radius varies varies with with the the tip tip potenpotenseems to to be be Em tial, tial, approximately approximately satisfying satisfying (2.3). (2.3). E, == 170 170 kV!cm kV/m We shall shall give give aa numerical numerical example example as as an an illustration. illustration. At At Em We 2 in ~ 1.1 /-le ~ IV s)s) and s-1 ,pe M 270 270 cm cm2/V and frrn, == 0.1 0.1 cm cm (corresponding (corresponding (vim M 1.1 X x lOll 1011 S-I, in air air (Vim 6 to kV), the cm -3 from to U U,t == 34 34 kV), the streamer streamer velocity velocity for for no no ~ M 10 lo6 cmP3 from formula formula (2.6) (2.6) is is 6 1.7 x lo6 mjs and the electron density in the newly born channel close to close to 1.7 10 m!s and the electron density in the newly born channel 13 1013 ~ m 3 -. Within ~ . 20% accuracy, accuracy, these these values values coincide coincide portion is is nn,c M~ 99 x 10 portion cmWithin 20% with the results of integration of unreduced equations in the one-dimensional with the results of integration of unreduced equations in the one-dimensional model illustrated illustrated in in figure figure 2.3 2.3 [10]. [lo]. They They show show not 2- or or 3-fold 3-fold model not more more than than aa 2disagreement with numerical simulations of streamers in a two-dimensional disagreement with numerical simulations of streamers in a two-dimensional model developed developed by different workers. workers. This This type type of of computation computation is is extremely extremely model by different complicated not particularly So the the simple simple formulas formulas (2.6) (2.6) are are complicated and and not particularly advanced. advanced. So useful since provide rough rough working useful since they they can can provide working estimations. estimations. They They are are also also applicable which similarity applicable to to aa gas gas of of lower lower density, density, in in whch similarity laws laws are are operative. operative. v, M N f ( E / N ) , where N is the number of molecules per 1 cm3, is aa Since is the number of molecules per 1cm3 , f f is Since Vi ~ Nf(E/ N), where I pe M N ‘ , we have function of the type given in figure 2.4, and function of the type given in figure 2.4, and /-le ~ N- , we have
-
-
E, N, r, u,/N. v, U,. n, N ~ . (2.7) (2.7) and n, of the the tip tip potential U,. The streamer streamer velocity velocity is is independent independent of of N The Nand nc of potential U t. The latter latter fact fact opens opens up up aa tempting yet unused unused possibility to test test The tempting but but yet possibility to E, being being experimentally the the theoretical theoretical concept concept of of the the maximum maximum tip tip field field Em experimentally constant. For this, it is sufficient to measure the electron density right constant. For this, it is sufficient to measure the electron density right of a fast streamer, in which U, = U, EJ rapidly decreases behind the tip behind the tip of a fast streamer, in which Ut = Ua - Eavl rapidly decreases with the the channel (Eav is the the average average channel channel field). field). The The constant constant data data with channel length length I1 (E av is indicate the the constant constant values values of of E,. On the the other other hand, decrease on n, on nc will will indicate Em. On hand, the the decrease in with decreasing in electron electron density density with decreasing streamer streamer velocity velocity could could become become aa strong strong r,, which which still still has has argument to to support support the the hypothesis hypothesis of of constant constant tip tip radius argument radius rrn' advocates. advocates. From (2.6) (2.6) and and (2.3), (2.3), the the velocity velocity of of aa fast fast streamer streamer is is proportional to its its From proportional to tip potential, its radius radius is is proportional to the at tip potential, because because its proportional to the potential potential at E, == const. const. The The velocity velocity V s, becomes lower than than the electron drift drift velocity Em becomes lower the electron velocity 5 V,(E,) 4 Xx 10 lo5 mjs at at U U,t ~ M 5-8 5-8 kV. kV. At At lower lower voltages, voltages, the streamer Ve(E ~Z 4 m!s the streamer rn ) R moves slower slower than than the the drift drift electrons, electrons, so so that that the formulas become become invalid. invalid. moves the formulas The analysis analysis of of equations equations presented in [10] [lo] shows shows that that the the streamer streamer velocity presented in velocity The U,t at at V s, < Vern' Vem. The The electron electron density density at at the the drops with with further further decrease decrease in in U drops
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streamer A long streamer
39 39
wave front front decreases, decreases, all all electrons electrons are are drawn drawn out out of of the tip, and and space space charge charge wave the tip, fills it it up. up. However, final plasma density behind ionization wave does fills However, the the final plasma density behind the the ionization wave does not decrease. decrease. The The tip tip radius radius becomes becomes very very small, small, and and the the streamer streamer stops stops at at not U,. This result agrees with experiments: no streamers with a velocity low U low This result agrees with experiments: no streamers with a velocity . t less than than (1.5-2) (1.5-2) x 10 lo55 m/s mjs have ever been been observed observed in in air air under less have ever under normal normal conditions. conditions.
2.2.3 Current Current and field in the channel behind the tip 2.2.3 As the the streamer streamer develops, develops, its its channel channel is is under which changes changes As under high high potential potential which from potential U point to to aa certain the from the the anode anode potential U,a at at the the starting starting point certain value value U, U, at at the front potential U between U/ front end, end, close close to to the the tip tip potential U,t (the (the difference difference between U , and and U U,t of of about EmD.x « about E,Ax << U U,t is is due due to to aa small small potential potential drop drop in in the the tip). tip). The The channel channel is point x along is electrically electrically charged, charged, since since the the potential potential at at any any point along it it is is higher higher than U o ( x )created created by by electrode electrode charges charges than the the unperturbed unperturbed potential potential of of space space Uo(x) in the the absence absence of of streamer. streamer. Current Current must must flow flow through through the the channel channel to to in supply supply charge charge to to the the new new portions portions of of the the growing growing streamer. streamer. When When setting setting ourourselves the the task task of of estimating estimating this this current current and and the the current current through through the the external external selves circuit (this (this is is the the streamer streamer current current to to be be measured), measured), we we must must first first find find the the circuit channel charge, charge, for for it it is is the the time time variation variation of of this this charge charge that that produces produces the the channel current. current. Suppose Suppose aa streamer streamer has has started started from from an an anode anode of of small small radius radius rP a', . Let us us examine examine the the stage stage when when the the streamer streamer length length becomes becomes I» 1 >> rraa (l (I isis Let much r ) . We We can can then then neglect neglect the the time time much larger larger than than the the channel channel radius radius r). variation variation of of the the anode anode charge, charge, because because its its capacitance capacitance is is small, small, and and take take the the external external current current to to be be close close to to the the current current ii,a entering entering the the channel channel through through its its base base at at the the anode. anode. Besides, Besides, aa streamer streamer conductor conductor can can be be regarded regarded as as being being solitary, solitary, and and the the unperturbed unperturbed potential potential U U,o far far from from the the anode anode can can be be ignored. ignored. Assume Assume first first that that the the channel channel is is aa perfect perfect conductor. conductor. From From aa well-known well-known electrostatics electrostatics formula, formula, the the capacitance capacitance of of aa long long solitary solitary conductor conductor is is C = 27rc 2 7 ro~llln(llr). ~ l / l n ( l / Its Its ~ ) . charge charge is is Q Q= = CU, CU, because because aa perfect perfect conductor conductor is is C under under only only potential potential U. U . Introduce Introduce now now the the concept concept of of capacitance capacitance per per unit unit length , which is frequently used in electroand radiolength of of the the conductor, conductor, C C1, which is frequently used in electroand radio1 engineering engineering to to analyse analyse long long lines. lines. The The average average capacitance capacitance per per unit unit length length - 5.56 5.56 Xx 10lo-”11 In (llr) F/m ln(I/r) = ln(I/r)
C 27rco 27rq c1=-=-C1 = I = In (llr)
I
(2.8)
has has aa nearly nearly constant constant value value which which only only slightly slightly varies varies with with I and and r.r. CalcuCalculations C1 (x)practically practically coincides coincides with with the the lations show show that that the the local local capacitance capacitance C 1 (x) average average value value from from (2.8) (2.8) along along the the whole whole length length of of aa long long conductor, conductor, except except for for its its portions portions lying lying close close to to its its ends. ends. But But even even at at the the ends, ends, the the local local capacitance capacitance is is less less than than twice twice the the average average value. value. This, This, however, however, does does not not concern concern capacitances capacitances of of the the free free ends ends which which are are much much larger larger (see (see below). below). As an an approximation approximation justifiable justifiable by by calculations, calculations, we we shall shall use use the the capacicapacitance tance per per unit unit length length from from (2.8) (2.8) and and apply apply itit to to aa real real streamer streamer channel. channel. If If aa
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The process in aa long spark spark The streamer-leader streamer-leader process
40 40
channel must have have aa longitudinal longitudinal channel possesses possesses aa finite finite conductivity, conductivity, then then it it must potential gradient and U = U(x), when current flows through it. The The = U ( x ) , when current flows through it. potential gradient and charge per unit channel length has the form charge per unit channel length has the form
.(x) ~ % C j, [U(x) [ U ( x )-- Uo(x)] Uo(x)] = T(X) =
27Tco[U(x) - U()] Uo] 27r&O[U(X) In(ljr) Wllr)
(2.9)
which channel raises raises its its potential potential which allows allows for for the the fact fact that that the the local local charge charge of of aa channel relative refinement should should be be Uo(x).A similar similar refinement relative to to unperturbed unperturbed potential, potential. Uo(x). introduced looks like like introduced in in formula formula (2.3), (2.3), which which generally generally looks
U 2Em rm , (2.10) (2.10) U,t -- Uo(l) UO(l) = = 2Emrm. as varies along along its its length, length, aa charactercharacter(2.7). If If the the channel channel radius radius r varies as well well as as in in (2.7). istic value may be substituted into (2.9), because the capacitance varies with r istic value may be substituted into (2.9), because the capacitance varies with only only logarithmically. logarithmically. Now elongates by by dl, d/, its its new new Now turn turn to to channel channel current. current. When When aa channel channel elongates portion 1 parameters of of the the front front E will will denote denote parameters portion acquires acquires charge charge Tt ~ [d/;d lindex index ; channel by local local current current ir it over over = I.1. This This charge charge is is supplied supplied directly directly by channel end, end, xx = time development, we we have have = d/jV dl/V,.s' Therefore, Therefore, at at any any stage stage of of streamer streamer development, time dt dt = . 27Tco[Ut - Uo(l)]Vs It=T/Vs = In (ljr) ,Ut~Ut·
(2.11) (2.11)
The current current at at the the tip tip is is defined defined mainly mainly by The by the the tip tip potential potential and and streamer streamer velocity. velocity. At At the the anode, anode, the the current current is is .
dQ
= dQ
tJ’ /
~ ( xdx dx ) = = T(X)
t 1
(2.12) C1j [[U(x) U ( x )Uo(x)] dx (2.12) C - Uo(x)] dx so should be where, Q Q is is the the total total channel channel charge. charge. Strictly, Strictly, Q where, Q should be supplemented supplemented by by = 27Tcorm[Ut 2 7 r ~ ~ r ~--[ U U,([)], , but the tip tip charge charge Qt Q, = Uo(l)], but it it is is relatively relatively small small in in aa long long the streamer. streamer. and itii at at the the opposite opposite ends ends of Currents i,a and Currents of aa streamer streamer channel channel do do not not generally coincide. coincide. Of Of course, course, their their values values may generally may be be very very close close or or differ differ conconsiderably, depending depending on on particular particular conditions. conditions. For siderably, For example, example, if if the the electrode electrode is raised raised during during the the streamer streamer development, development, the voltage is voltage the potential potential and and charge charge distributed along along the the channel channel increase. increase. Some Some of of the the anode anode current current is is used used to to distributed supply an an additional additional charge charge to to the the old old channel channel portions, that only only the the supply portions, so so that remaining current current reaches reaches its its front front end: end: ii,a > it. il. But remaining But if if the the electrode electrode voltage voltage is decreased, the ‘excess’ charge of the old channel goes back is decreased, the 'excess' charge of the old channel goes back to to the the supply supply through the the anode anode surface, surface, so so that that the the current current decreases decreases nearer nearer to to the the through anode (positive current is created by charges moving away from anode (positive current is created by charges moving away from the the if. anode): i,a < if. anode): A long long streamer streamer can can develop develop at at constant constant voltage A voltage when when the the electric electric field field in in the channel, channel, E(x, E ( x .t), t ) ,does does not not vary vary much much with with time. time. The The potential potential at at any any point point the of the the existing existing channel channel U(x) U ( x )= =U U,a -- f6' E ( x )dx ( xvary ) slightly of E(x) dx and and ~ T(X) vary slightly with with time, which which means means that that current current does does not not branch branch off off on on the the way way from from the the time, 1,. la
dt, =ili'
Q= = Q
0
Jt
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A long streamer
41 41
anode case, the anode to to the the channel channel tip. tip. In In this this case, the anode anode current current is is close close to to the the end end U,t which current (2.11) including current defined defined by by (2.11) including potential potential U which may may be be much much lower lower than U,. Many average channel channel field than Ua' Many experiments experiments have have shown shown that that the the average field for air air in must exceed aa certain about 55 kV/cm must exceed certain minimum minimum value value of of about kV/cm for in normal normal conditions conditions (see (see sections sections 2.2.6 2.2.6 and and 2.2.7) 2.2.7) to to be be able able to to support support aa long long positive positive 600 kV kV at at the the anode anode and and the the streamer streamer length length streamer. For For instance, instance, if if Ua, == 600 streamer. is I1 ;::::: 1m, 1 m, nearly nearly all all voltage voltage drops drops in in the the channel channel and and U U,t « << Ua, U,, but but currents currents is and i{il still still do do not not differ differ much. much. ii,a and To To get get aa general general idea idea about about the the orders orders of of magnitudes, magnitudes, let let us us consider consider aa variant realistic and by some variant which which seems seems quite quite realistic and is is manifested manifested by some calculations calculations (section This is constant applied and (section 2.2.6). 2.2.6). This is the the variant variant with with constant applied voltage voltage and slowly average field field in slowly varying varying average in the the channel, channel, when when the the current current along along it it For example, at changes little little and, and, therefore, therefore, can can be be evaluated evaluated from from (2.11). changes (2.1 1). For example, at 1= cm, Vs, == 1.7 m/s, and and U ;::::: U as in in the the I = 1m, 1 m, rr == 0.1 0.1 cm, 1.7 x 10 lo66 m/s, Ul1 = U,t ;::::: 34kV, 34 kV, as illustration in section 2.2.2 (with U, > 500 kV), we have ln(l/r) = 6.9, illustration in section 2.2.2 (with Ua 500 kV), we have In(ljr) = 6.9, 12 7 F/m, T{ C/m, and A. Streamer C C1 =8 8 Xx 1010-12F/m, r1== 2.7 2.7 Xx 10lOP7C/m, and ii,a == i{i, == 0.46 0.46A. Streamer j = currents currents of of this this order order of of magnitude magnitude (as (as well well as as much much higher higher or or much much lower lower currents) have have been been registered registered in in many many experiments. experiments. These These values values can can also also currents) be of possible possible streamer streamer velocities velocities be obtained obtained from from calculations calculations with with the the account account of from from 10 lo55 to to 10 lo77 m/s m/s in in air, air, which which have been found found in in some some experiments experiments to to be have been be even higher higher [4]. even [4]. In aa simple simple model model of of potential potential and and current current evolution evolution in in aa developing developing In streamer channel, the the latter latter can can be be represented represented as as aa line line with with distributed distributed streamer channel; parameters: the the capacitance capacitance C1 C 1 and and resistivity resistivity R1 Rj = = (7rr2ei-Lenetl per unit unit (xr2epene)-' per parameters: length. The The electron electron density density n,(x. ne(x, tt)) should should be be calculated calculated in in terms terms of of the the length. plasma plasma decay decay kinetics kinetics (see (see section section 2.2.5). 2.2.5). Estimations Estimations show show that that self-induction self-induction effects effects are are not not essential essential in in streamer streamer development development [4]. [4]. Then, Then, the the process process is is described described by by the the following following equations equations for for current current and and voltage voltage balance: balance: [JT d r 8i di -+--0, at+ax=O' at
dx
aU dU . j, - - = zR1, -ax=iR dX
7 = C1(U - Uo). T=Cj(U-Uo)·
(2.13) (2.13)
A boundary boundary condition condition in in the the set of equations at x x= = I1 is the equality equality A set of equations (2.13) (2.13) at is the
4
= c1[U1 -
UO(4l Vs
(2.14) (2.14)
equivalent to to (2.11). Formula (2.12) automatically follows follows from and equivalent (2.1 1). Formula (2.12) automatically from (2.13) (2.13) and Another boundary condition may be the setting of anode potential, (2.14). (2.14). Another boundary condition may be the setting of anode potential, since U ( 0 ,t)t ) == Ua(t). U a ( t ) .Equations Equations (2.13) (2.13) and and (2.14) (2.14) will will be be used used in in the the next next since U(O, section section to to evaluate evaluate the the heating heating of of aa streamer streamer channel. channel. Illustrations Illustrations of of streamer streamer development development calculations calculations will will be be given given in in sections sections 2.2.6 2.2.6 and and 2.2.7 2.2.7 plasma decay A complete after after aa discussion discussion of of the the plasma decay mechanism. mechanism. A complete set set of of equations for for aa long long line, line, generalized generalized by by taking taking self-induction into account, account, equations self-induction into will be be applied applied in in section 4.4 to to the the treatment treatment of return stroke. will section 4.4 of aa lightning lightning return stroke. of longitudinal longitudinal field E c in in the the channel Equality (2.11) (2.11) allows channel Equality allows evaluation evaluation of field E, is still as high high as as that that behind the the streamer tip, where where the the electron electron density density is behind streamer tip, still as
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42 42
The The streamer-leader streamer-leader process in a long spark
behind the the tip tip is is conduction conduction created created by by an an ionization ionization wave. wave. The The current current behind current i, = Jrr~encf.LeEc' By equating this expression to (2.11) and using using current il = 7rrfnencpeEc.By equating this expression to (2.11) and and (2.10) with V = V" we find (2.6) (2.6) and (2.10) with Ut, = U,, we find
/3Em
E c i'::j In (nmlno) In (fIrm)
i3
=
4k 2k _ 1 i'::j 2.5.
(2.15)
For the denominator denominator of of (2.15) (2.15) is is For aim a 1 m streamer, streamer, the the product product of of logarithms logarithms in in the close of aa streamer streamer channel channel in in to 100. 100. Therefore, Therefore, the the field field in in the the front front end end of close to normal kV jcm from from section section 2.2.2). 2.2.2). normal density density air air is is E E,c i'::j M 4.2 4.2 kVjcm kV/cm (Em (E, = = 170 170 kV/cm Within not contradict contradict the the average average Within the the theory theory accuracy, accuracy, this this value value does does not measured kV jcm necessary necessary to to support support the the streamer. streamer. measured channel channel field field of of 55kV;cm There therefore electrons electrons are are lost lost in in There is is no no ionization ionization in in such such aa weak weak field, field, therefore attachment processes. attachment and and electron-ion electron-ion recombination recombination processes. Current than that that of of the the tip tip adjacent adjacent to to near the the channel channel end end is is lower lower than Current i,il near the tip length =Q Qtlrm is larger larger than than TT J r , is the channel, channel, because because the the charge charge per per unit unit tip length TTtt = in end effects effects for for long long conducconducin the the channel. channel. This This is is aa typical typical consequence consequence of of end tors, charge density density at at the the free free tors, well-known well-known from from electrostatics. electrostatics. The The surface surface charge end its lateral lateral surface. surface. In In our our end of of aa conductor conductor is is much much higher higher than than on on its simple been replaced replaced by by aa hemisphere hemisphere of of simple model, model, in in which which aa channel channel tip tip has has been radius the average average charge charge radius ,rYm and and charge charge Qt Q, written written after after formula formula (2.12), (2.12), the M 2m:o[Ut 2 7 r ~ ~ -[ UVo(f)]. U~O ( l ) ]It It. is is In (l/rm) times per unit unit length length is is TT,t i'::j per In (fIrm) times larger larger than than T, T, at the the channel channel end end (see (see (2.9)). (2.9)). The The tip tip current current i,it much at much exceeds exceeds il. it. This This does does not because the the charge charge Q Q of of aa long long channel channel not affect affect the the total total charge charge balance, balance, because Q,. is much much larger larger than than the the tip tip charge charge Qt. is Note that that current current perturbation perturbation in in the the tip Note tip region region has has aa local local character. character. It It cannot be be detected detected by by current current registration registration from from the cannot the anode anode side. side. The The streamer streamer the charge charge of here makes makes use use of of its its own own resources resources -- the here of the the ‘old’ 'old' tip tip has has moved moved on on into the the gap gap with with the the elongating elongating streamer. streamer. It It is into is the the charge charge overflow overflow that that it. If If aa current current detector detector were creates current current it. creates were placed placed at at the the site site of of aa newly newly born portion portion of of the the channel, channel, it it would would register born register current current ii M i'::j it for for aa very very 9 At i'::j = rrm/Vs M 10lop9 s; then short period period of of time time D.t short ml Vs i'::j s; then the the current current would would decrease decrease and evolve evolve as as the the solution solution of of equations equations (2.13) (2.13) and to i,il and to and (2.14) (2.14) indicates. indicates.
2.2.4 Gas Gas heating heating in a streamer streamer channel 2.2.4
streamer process process is is accompanied accompanied by by current current flow flow and, A streamer and, hence, hence, by by Joule Joule heat heat was mentioned mentioned above, above, the the viability viability of release. As was release. of aa plasma plasma channel channel depends depends primarily on on temperature, temperature, so so this this issue issue is is of of principal primarily principal importance. importance. The The initial initial heating of of aa given given gas gas volume volume occurs occurs when when aa streamer streamer tip heating tip with with its its high high current current the channel channel develops, develops, the and field field passes passes through through it. it. As the and the gas gas is is heated heated further further by streamer streamer current current flowing flowing through through it. it. Let Let us us evaluate evaluate both both components components of of by released energy. energy. released 1 cm33 per per second second is is jjE E= The energy energy released released in in 1cm The = aE2, aE 2 , where where jj = = aE is is the the is the the plasma plasma conductivity conductivity in current density density and and aa is in aa given given site site in in aa given given current
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A long streamer
43 43
3 moment of of time. time. The The energy energy released released in in 1cm 1 cm3 as aa result result of of ionization ionization wave wave moment as passage passage is is
s 2 Js
J
2 W = aE2 w == aaEE 2 dt dt = aE dx/Vs dx/Vs
(2.16) (2.16)
--x to to +00 +xbut but actually actually over over the the where the the integrals integrals are are formally formally taken taken from from -00 where ionization wave wave region. region. The The principal principal contribution contribution to to energy energy release release is is made made ionization by aa thin thin layer layer behind behind the the wave wave front front where where the the electron electron density density and and field field are are by ~ ~ E i / using using 2, high. The The integral integral of of (2.16) (2.16) was was found found rigorously rigorously to to be be foE~/2, high. equations for for this this wave region [4]. [4]. This This value of value has has the the physical physical meaning meaning of equations wave region electrical energy at maximum maximum field. field. The The contribution of the the region region electrical energy density density at contribution of (nm/no)times, smaller before before the the wave wave front front is is In In (nm/no) times, or or an an order order of of magnitude, magnitude, smaller than is as front, than this this value. value. Although Although the the field field there there is as high high as as that that behind behind the the front, the is of of n, and the conductivity (J0 is the electron electron density density is of the the order order of nm and the conductivity is (nm/no)times smaller (section In In (nm/no) times smaller (section 2.2.2). 2.2.2). Therefore, Therefore,
Wx oEk/2 ~ x 2.6 x 10- 3 J/cm J/cm33 W ~ efoE~/2
(2.17) (2.17)
where = 170 170 kV. where the the numerical numerical value value corresponds corresponds to to E, Em = kV. The fact that the density of energy release in a gas is of of the the same same order order of of The fact that the density of energy release in a gas is magnitude as the energy density of of the the electric electric field field is is quite quite consistent consistent with with magnitude as the energy density electricity electricity theory. theory. When When aa capacitor capacitor with with capacitance capacitance C is is charged charged through through resistance R to to voltage voltage V of aa constant constant voltage voltage supply, half of the work work U of supply, half of the resistance 2 QV = done by by the the supply supply is is stored stored by by the the capacitor capacitor as as electrical electrical QU = CV CU2 done energy, and and the the other other half half is is dissipated dissipated due due to to resistance, resistance, irrespective irrespective of of its its energy, value. The The value value of of R determines determines only only the the characteristic characteristic time time of of the the capacitor capacitor value. charging, RC. Something like this this is is valid valid for the case case in in question question but, but, of of Something like for the charging, course, without without both both energies being rigorously rigorously equal equal to to each other, because because course, energies being each other, this situation situation is is much much more more complicated. complicated. Indeed, Indeed, according according to to the the results results this C,t = = Q/ Q/U 27re0rm, of section 2.2.2, of section 2.2.2, the the tip tip capacitance capacitance is is C Vt, % ~ 27rfor m, volume volume V, , E Ut/rm, Vt x ~ 4rrLI3, 47rr~/3, and and field field E Em ~ Vt/r so that that the the energy energy dissipation dissipation per per unit unit m, so tip tip volume volume is is W F~Z CtU:/2Vt Ct vf /2 Vt z ~ eOEk foE~ (we (we have have ignored ignored the the unessential unessential term term Uo(l)). Vo(l)). Joule heat heat is is released released directly directly in in aa current current carrier carrier gas, gas, or or an an electron electron gas. gas. Joule Then electrons electrons give give off off their their energy energy to to molecules molecules in in collisions. collisions. An An appreciable appreciable Then portion of of electron electron energy energy (even (even most most of of it it in in aa certain certain range range of of E E/I N) is used used N ) is portion for the excitation excitation of relaxing vibrations vibrations of nitrogen molecules. molecules. Some for the of slowly slowly relaxing of nitrogen Some energy is is used used for for ionization ionization and and electron electron excitation excitation of of molecules, molecules, about about energy W = ~ 100 eV per per pair pair of of charged charged particles particles produced, produced, i.e., i.e., nneW ~ 10- 3 J/cm3 J/cm 3 at at U’ 100 eV , ~ ’= ne F~Z 1014 10 14 cmP3. cm- 3 . But But even even without without the the account account of of these these ‘losses’, 'losses', the the gas gas temperatemperan, ture rise rise in in the the wave wave front front region region appears appears to to be be negligible: negligible: AT 6.T < Wlcv W / Cv = ~ 3 3 K. K. ture Here, Here, cv Cv = = qkBN ~kBN = = 8.6 8.6 x loP4 10- 4 J/(cm3/K) J/(cm 3 /K) is is the the heat heat capacity capacity of of cold cold air air and and kB kB is is the the Boltzmann Boltzmann constant. constant. Let us us see see what what subsequent subsequent gas gas heating heating can can provide provide by by the the moment moment it it Let is somewhere somewhere in in the the middle middle of of aa long long streamer streamer channel. channel. We We multiply multiply the the is
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spark The streamer-leader streamer-leader process in a long spark
44
second by ii and whole channel second equation equation of of (2.13) (2.13) by and integrate integrate over over the the whole channel length, length, assuming, the anode. U,a is is applied applied to to the anode. assuming, for for simplicity, simplicity, that that constant constant voltage voltage U After by-part integral After taking taking aa by-part integral in in the the left left side side of of the the equation, equation, we we substitute substitute ai/ox ailax from from (2.13) (2.13) and and i(l) i(1)== ii,l from from (2.14), (2.14), followed followed by by simple simple transformatransformaresult, we tions. tions. As As aa result: we have have
fSo' 1
I
f'
+
2
+
[y
I d 0-2C j U dx U,i, = = 0 i2 R R Ij dx dx + d x + [C1U Cij UIUO(l) U I U O (]/ )V Vs ]s Uaia dt -2- -- C dt
(2.18) (2.18)
which describes power balance balance in which describes the the power in the the system; system; here, here, Uo(l) Uo(1) is is unperturbed unperturbed potential of the external x == l.1. The power The input input power potential of the external field field at at the the streamer streamer tip tip point point x Uai used for Uai,a into into aa discharge discharge gap gap is is used for Joule Joule heat heat release release in in the the channel channel (the (the first first term on on the the right), right), for for increasing increasing the the electric electric energy energy stored stored in in its its capacitance capacitance term (the to the (the second second term), term), and and for for the the creation creation of of new new capacitance capacitance due due to the channel channel elongation third term). elongation (the (the third term). Joule Joule heat heat associated associated with with the the ionization ionization wave wave is is not represented represented here. here. The The field field burst burst and and the impulse current current that that make make up up not the tip tip impulse W W calculated calculated above above are are absent absent from from equations equations (2.13) (2.13) and and (2.18). (2.18). Having Having integrated period of integrated equality equality (2.18) (2.18) over over the the period of time time from from the the moment moment of of channel has acquired length length l,I , we we channel initiation initiation to to the moment t the channel the moment the channel has acquired get balance in get the the equation equation for for the the energy energy balance in the the system system at at the the moment moment t:t :
UQ a
= Kd,s + II0 Cdu
2 -
2
U5(x)] dx
+ \/
Cj
ul _ C
2
U U (l)) l j
lOt
(2.19)
where where charge charge Q Q is is given given by by (2.12). (2.12). The The energy energy input input into into the the channel, channel, UaQ, U,Q, is is used right), partly partly stored used to to create create capacity capacity (the (the last last term term on on the the right), stored in in this this capacity braces ( )1 (&Is). dis ). The The braces ) t indicate indicate capacity (the (the integral) integral) and and partly partly dissipated dissipated (K process. In the time time averaging the averaging of of the the process. In case case of of aa long long channel, channel, much much of of the the applied potential UI, is so the the tip tip potential is small small applied voltage voltage drops drops across across its its length, length, so most time, as potential U U,,av of of about about most of of the the time, as compared compared with with average average channel channel potential U,.a . Then, Then, the the last last term term in in (2.19) (2.19) can can be be neglected. neglected. U Q If we compare compare the the left left side side of of (2.19) (2.19) with with the substituted expression expression for for Q Ifwe the substituted from (2.12) (2.12) and and the the integral integral in in the the right side of of (2.19), (2.19), we we can can conclude conclude that that from right side the difference between these values cannot be be much much smaller smaller than than their their own own the difference between these values cannot values the same same order order of of magnitude. magnitude. Therefore, Therefore, the the energy energy values but but rather rather have have the KdlS dissipated in in the the channel channel is is equal, equal, in in order order of of magnitude, magnitude, to to the the gained gained K dis dissipated electrical electrical energy, energy, which which is is in in agreement agreement with with aa similar similar situation situation discussed discussed above. above. The average average energy energy dissipated dissipated per unit channel channel length length is is W W,,jav ~ xC CIj Uiv/2 The per unit U~v/2 and per unit unit channel volume is and the the average average energy energy contributed contributed per channel volume is I
C1U~v
W~--
2JTr~v
(2.20) (2.20)
where rav is the the average average channel channel radius. radius. With With the formation of of every every new new where ray is the formation portion of the the channel, channel, its its radius radius was approximately proportional to the was approximately proportional to the portion of
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A long streamer
45 45
tip potential owing that the the maximum maximum tip tip potential owing to to the the fact fact that tip field field remained remained approxiapproxiSo we we have have Uav/r Uav/r,,av i=::j M E,. Substituting this expression and and mately constant. constant. So Em. Substituting this expression mately (2.8) (2.8) into into (2.20), (2.20), we we find find ~ Em co 2
/
W
i=::j
In (f/r m ) < W.
One can can see see that that subsequent subsequent heating of the channel gas gas adds adds little little to the initial initial One heating of the channel to the heating by by an an ionization ionization wave wave passing passing through through the channel site. site. heating the particular particular channel To heating due To conclude, conclude, gas gas heating due to to streamer streamer development development is is negligible negligible if if the the gap gap voltage voltage remains remains constant. constant. Higher Higher voltage voltage does does not not change change the the situation situation because because the the energy energy dissipated dissipated in in the the channel channel grows grows in in proportion proportion with with the the channel cross cross section section and and the the air air volume volume to to be be heated. heated. Specific Specific heating channel heating remains volume remains unchanged, unchanged, since since it it is is determined determined by by aa more more or or less less fixed fixed volume density density of of electric electric energy. energy. 2.2.5 2.2.5 Electron-molecular Electron-molecular reactions and plasma decay in cold air
Electron to attachment molecules and Electron loss loss in in cold cold air air is is due due to attachment to to oxygen oxygen molecules and dissociative dissociative recombination. recombination. The The main main attachment attachment mechanism mechanism in in dry dry air air at at moderate fields fields is is aa three-body three-body process moderate process
+
0 22 + + ee + O 0 22 O
kat k,,
= (4.7 (4.7 -- 0.25 0.257) 1 ) xX
i=::j
°
+ 0 22,, 2+
+O 0; ----+
10~31 cm /s, cm6/s, 6
16 /N X x 10 10'6V.cm2 1y == E E/N y .cm 2
(2.21)) (2.21
where kat kat is the rate K. In where is the rate constant constant at at T = = 300 300K. In higher higher fields, fields, the the dominant dominant rate constant process is process is dissociative dissociative attachment attachment O O2 e ----+ -+ 0- + 0 with with the the rate constant 2+e
°
-9.42 h 12.717 -9.42 -- 12.7 log k,a == { logk -10.215.7h -10.21 - 5.7,'~
at 1y < 99 at 9. at 17 > 9. at
(2.22) (2.22)
In high fields kY/cm at In not not excessively excessively high fields of of E < 70 70 kV/cm at 11 atm, atm, air air is is ionized ionized at at the the rate constant constant kkii == Vi/ q /N N rate
logkii = = -8.31 -8.31 -- 12.7 12.7,'~ at 1y < 26. 26. logk h at
(2.23) (2.23)
Since Since the the rate rate of of electron electron loss loss through through attachment attachment is is proportional proportional to to electron electron density through recombination to y12, n~, the density ny1,e and and that that through recombination is is proportional proportional to the latter latter is is unimportant beginning of unimportant at at the the beginning of ionization. ionization. The The equality equality kkii == k a, valid valid at at 1y i=::j M 12 12 determines determines the the minimum minimum field field mentioned mentioned above, above, which which is is necessary necessary to E i i=::j to initiate initiate the the growth growth of of electron electron density density in in unperturbed unperturbed air; air; Ei E 30 30 kYat kV at p == 11 atm atm and and room room temperature. temperature. Oxygen possessing aa lower N 2 are Oxygen molecules molecules possessing lower ionization ionization potential potential than than N2 are ionized the ionization threshold. Electrons ionized in in fields fields not not much much exceeding exceeding the ionization threshold. Electrons recombine with with 0; at usually termed recombine at the the rate rate constant constant 3, 0, usually termed aa recombination recombination coefficient: coefficient:
ot
0; +e----+ + e -+ 0+0, 0 + 0, ot
Copyright © 2000 IOP Publishing Ltd.
/? M 2.7 x 10-7(300/Te)'/2cm3/s
(2.24) (2.24)
46 46
The The streamer-leader streamer-leader process in in a long spark
where where TT,e is is electron electron temperature temperature in in Kelvin Kelvin degrees. degrees. However, However, complex complex ions ions are are more more effective effective with with respect respect to to electron-ion electron-ion recombination. recombination. The The most most important ions, while while in in atmosphere atmosphere saturated saturated by by important ions ions in in dry dry air air are are 0; ions, water water vapour, vapour, as as in in thunderstorm thunderstorm rain, rain, H H 330+ 0 +(H ( H220h 0 ) 3cluster cluster ions ions are are more more important. important. For For these, these, the the recombination recombination coefficients coefficients
ot
ot 0; + + ee ----+ O O2 02. 2 + 02,
6 3/s D = 1.4 1.4 x 1O10-6(300/T,)'/2 cm3/s /3;:::: (300/Te )I/2 cm
+
+
H + e ----+ H H + 4H 4 H220, 0, H 330+(H 0 + ( H20h 2 0 ) 3+ ---f
(2.25) (2.25)
3/s 6 B 6.5 6.5 Xx 1O10-6(300/T)'12 cm3/s /3;:::: (300/T)1/2 cm
(2.26) (2.26) are are an an order order of of magnitude magnitude larger larger than than for for simple simple ions. ions. ions are are formed formed from from simple simple ions ions in in the the conversion conversion Complex 0; ions Complex reaction reaction
ot
0;
+0 2 +0 2
--f
0;
+ 02,
k = 2.4 x 10-30(300/Te)'/2cm6/s. (2.27)
Chains of of hydration hydration reactions reactions lead lead to to the the production production of of H H30'(H20)3 ions. A A Chains 30+ (H 20h ions. typical chain chain looks looks like like this: this: typical 3/s 9 1.5 Xx 10lop9 cm3/s ot 0) + 02, k == 1.5 cm 0; + + HH20 -+ 0i(H O;(H20) 02> 20 ----+ 3/s Ot(H H OH k == 3.0 3.0 x 10lo-'' cm cm3/s O i ( H 20) 0)+ + HH20 H30' +O H + 02, 02? 20 30+ + 3/s = 3.1 3.1 x 10- cm H kk = cm3/s H30+ -tH H20 + (M) (M) ----+ H H30+(H20) + (M), (M), 30+ + 20 + 30+(H 20) + 3/s 2.7 x 10lop9 cm cm3/s H H30+(H20) + HH20 -t (M) (M) ----+ H H30+(H20)2 (M). kk == 2.7 30+(H 20) + 20 + 30+(H 20h + (M), 2
10
X
--+ +
2
9
X
+
X
+
9
3/s cm3/s H (M) H3O+(H20)2 -t H H 2200 + i(M) ----+ -+ H H 330+(H 0 + ( H220)3 0 ) 3+ -t (M), (M), k == 2.6 2.6 Xx 10-9 cm 30+(H 20h + (2.28) (2.28)
(M is is any any molecule, molecule, kk correspond correspond to t o pp == 1atm, 1 atm, T == 300 300 K); K); here, here, aa hydrated hydrated (M ion replaces replaces aa 0; ion. ion. ion Another, similar similar chain chain begins begins with with the the production production of of an an H H 220+ 0 + ion ion in in Another, ionization of of water water molecules molecules by by electron electron impact. impact. Then Then comes comes the the conversion conversion ionization reaction reaction
ot
+
+
H2200 --+ -+ H H 330+ 0 ++ OH, OH, H 220+ 0 ++ H H
3/s 9 kk == 1.7 cm 1.7 Xx 10lop9 cm3/s
producing an an H H30+ ion, followed followed by by the the reaction reaction chain chain of of the the type type (2.28). (2.28). producing 30+ ion, The production production of of complex complex ions ions is is accompanied accompanied by by their their decay. decay. For For an an The 0; ion, ion, this this is is the the reaction reaction
ot
0; + + O0 22 ot 0; + i-O 0 2 2+ -t02, 0 2 , ot + --+
6 3/s. 3.3 Xx 1O10p6(300/T)4exp (-504O/T) cm cm3/s. k == 3.3 (300/T)4 exp (-5040/T)
(2.29) (2.29)
It is is greatly greatly accelerated accelerated by by gas gas heating, heating, but but in in cold cold air air the the reaction reaction effect effect It is negligible. negligible. The The same same is is true true of of other other complex complex ions, ions, including including hydrated hydrated ions. ions. is
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A long streamer
47 41
In aa cold cold streamer streamer channel, channel, simple simple positive ions turn turn to to complex complex In posItIve ions 8 ions very very quickly, quickly, for for the the time time T,eonv , x 1010-8-10-7s. It is is these these ions ions that that ions ~ _10- 7 S. It determine the rate of electron-ion recombination in cold air, except for determine the rate of electron-ion recombination in cold air, except for aa very very short short initial initial stage stage with with t :::;; 6T r,,,,. eonv . If the ionization rate is too low and if if the the detachment-decay detachment-decay of of negative If the ionization rate is too low and negative ions is slow, as in a cold streamer channel, the plasma decay is described ions is slow, as in a cold streamer channel, the plasma decay is described by by the equation equation the
3 = -vane - pn,2 dt
(2.30) (2.30)
where va is electron electron attachment attachment frequency. frequency. Its Its solution solution at at initial initial electron electron where Va is density equal equal to the plasma plasma density density behind the ionization ionization wave, wave, nn,, is density to the behind the e , is n (t) e
-
ne exp (-vat) -,-----;-::-----;:----:-=:-:-,----...::.....c..-,--------,-,1 + (,8ne /va )[1 - exp (-vat)]
(2.31)) (2.31
where where the the time time is is counted counted from from the the moment moment the the streamer streamer tip tip passes passes through through aa particular point point of of space. space. particular According to (2.15), wehaveE andE/ N ~ x 4.2kV/cm 4.2kV/cmandE/N x 1.7 1 . X7 1010-16V/cm ~ 16 V/cm Accordingto(2.15), wehaveE ~ for aa streamer streamer channel channel just behind the the tip tip at at p = = 1atm. 1 atm. The The electron electron for just behind 7 attachment S-I va ~ x 1.2 1.2 Xx 10 lo7 sC1 and and the the characteristic characteristic attachment frequency frequency from from (2.21) (2.21) is is Va attachment attachment time time is is T 7, = v;1 v;’ ~ x 0.8 0.8 Xx 10- 7 . Over Over this this time, time, most most simple simple oj 0; a = ions ions in in dry dry air air turn turn to to complex complex 0; ions. ions. Electrons Electrons recombine recombine with with them them 7 3 with cm /s corresponding with the the coefficient coefficient ,8 ,8 ~ x 2.2 2.2 Xx 10lop7 cm3/s corresponding to to electron electron temperatempera4 ture / N. The ture TTe x 11 eV eV == 1.16 1.16 xx 10 lo4K K at at the the above above value value of of E E/N. The initial initial electron electron e ~ 14 3 x 10 1014 cmP3 is so so high high that that the the parameter parameter ,8n @nc/va x 22 determining determining density nn,e ~ density cmis e /va ~ the the relative relative contributions contributions of of recombination recombination and and attachment attachment is is larger larger than than unity. This This means means that that at at an an early early decay decay stage stage with with t < T ra x 10lop77 s,s, electrons electrons unity. a ~ are lost lost primarily primarily due due to to recombination, recombination, with with attachment attachment playing playing aa lesser lesser role. role. are 7 Later, s, the Later, at at t > 2T 2raa ~ x 22 Xx 10lO-’s, the electron electron density density decreases decreases exponentially, exponentially, as as is is inherent inherent in in attachment, attachment, but but as as if if starting starting from from aa lower lower initial initial value value nj nl = = ne/(l nJ(1 + ,8n @ ne Tca~) a~ x) 0.3n 0.3n,; nee ~ x nl nl exp exp (-vat). (-vat). e; n The The plasma plasma conductivity conductivity decreases decreases by by two two orders orders of of magnitude, magnitude, as as compared with with the the initial initial value, value, over over t ~ x 3 3 Xx 10-7 S.s. At At the the streamer streamer velocity velocity compared 6 V mis, Vs, ~ x l0 lo6 m/s, this this occurs occurs at at aa distance distance of of 30 30 cm cm behind behind the the tip. tip. A A micromicrosecond later, later, the the conductivity conductivity drops drops by by six six orders orders of of magnitude. magnitude. The The streamer streamer second plasma in in cold cold humid humid air air decays decays still still faster faster because because of of aa several several times times higher higher plasma of an an rate of of recombination recombination with with hydrated hydrated ions ions and and due due to to the the appearance appearance of rate additional attachment attachment source source involving involving water water molecules. molecules. These These estimations estimations additional indicate indicate aa low low streamer streamer viability. viability. It It is is only only very very fast fast streamers streamers supported supported by by megavolt megavolt voltages voltages that that are are capable capable of of elongating elongating to to I1 ~ x 1m 1 m in in cold cold air air without without losing losing much much of of their their galvanic galvanic connection connection with with the the original original electrode. electrode. This is is supported supported by by experiments experiments with with aa single single streamer streamer and and aa powerful powerful This streamer [4]. streamer corona corona [4].
ot
+
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48 48
The streamer-leader streamer-leader process in a long spark
Note that where Note that aa streamer streamer plasma plasma has has aa longer longer lifetime lifetime in in inert inert gases, gases, where attachment attachment is is absent absent and and recombination recombination is is much much slower. slower. This This makes makes it it possible to heat by flowing possible to heat the the plasma plasma channel channel by flowing current current for for aa longer longer time time after after the the streamer streamer bridges bridges the the gap gap (the (the estimations estimations of of section section 2.2.4 2.2.4 do do not not extend to these these conditions). extend to conditions). Such Such aa process process sometimes sometimes leads leads to to aa streamer streamer the formulation (leader-free) breakdown [17]. (leader-free) gap gap breakdown [ 171. Still, Still, the formulation of of the the streamer streamer breakdown problem is justified for hot air and is related to lightning breakdown problem is justified for hot air and is related to lightning (see (see section section 4.8 4.8 about about dart dart leader). leader). 2.2.6
Final streamer length
When When aa streamer streamer starts starts from from the the smaller smaller electrode electrode (anode) (anode) of of radius radius ra, Y,, to to which Va, » Eir a is propagates in which high high voltage voltage U >> Elya is applied, applied, it it propagates in aa rapidly rapidly decreasing decreasing external but then external field. field. It It is is first first accelerated accelerated but then slows slows down down after after it it leaves leaves the the the voltage region region of of length length rY,a where where it it senses senses aa direct direct anode anode influence. influence. If If the voltage is is too too low, low, the the streamer streamer may may stop stop in in the the gap, gap, without without reaching reaching the the opposite opposite electrode (say, (say, aa grounded grounded plane plane placed placed at at aa distance distance d). d). The The higher higher is is Va' U,, electrode the longer is the distance the streamer can cover; at a sufficiently high voltage, the longer is the distance the streamer can cover; at a sufficiently high voltage, it the gap. to estimate it bridges bridges the gap. In In order order to estimate the the sizes sizes of of the the streamer streamer zone zone and and leader leader cover cover in in aa long long spark spark or or lightning lightning -- aa task task important important for for their their theory we need that would theory -- we need aa criterion criterion that would allow allow estimation estimation of of maximum maximum streamer No direct measurestreamer length length under under different different propagation propagation conditions. conditions. No direct measurements of of this this kind have been been made made for for single single long long streamers streamers in in air, air, because because ments kind have there however, is there is is always always aa burst burst of of numerous numerous streamers. streamers. This, This, however, is quite quite another matter (see below). below). So So we we shall shall use indirect experimental experimental results another matter (see use indirect results and physical considerations, and invoke invoke physical considerations, theory, theory, and and calculations. calculations. It has has been established experimentally experimentally that that streamers streamers comprising comprising aa It been established streamer burst burst are are able able to to cross cross an an interelectrode interelectrode gap gap of of length length dd only only if if the the streamer relation which varies varies with relation E E,,av = Val U,/dd exceeds exceeds aa certain certain critical critical value value E E,,cr which with the the kind kind of of gas gas and and its its state. state. Under Under normal normal conditions conditions in in air, air, this t h s critical critical value value is wide range ~ OJ-10m. is E E,,cr ~ e 4.5-5kV 4.5-5kV in in aa wide range of of dd = 0.1-10m. The The data data spread spread does does not the measurement not exceed exceed the measurement error. error. Bazelyan Bazelyan and and Goryunov Goryunov [18] recommend recommend E,,cr = = 4.65 4.65 kV/cm kVjcm for for positive streamer, averaged averaged over over various various the value the value E positive streamer, measurements. the voltage bridge aa measurements. Therefore, Therefore, the voltage necessary necessary for for aa streamer streamer to to bridge is U,,, or more. more. For For example, example, aa gap gap of of I1 m m length length gap of of length length dd is Vamin == E,,d Ecrd or gap requires about about 500 500 kV (Ec, e 10 kV/cm for negative streamer in air). requires kV (E ~ 10 kV Icm for negative streamer in air). cr At moment of voltage Va to the At the the moment of crossing crossing aa gap, gap, all all voltage U , is is applied applied to the streamer, is also the average field in the streamer. If a gap is long streamer, so so E Ea" is also the average field in the streamer. If a gap is long av E,,av can can be be identified identified with the average average channel channel field. field. Indeed, Indeed, in in criticritienough, E enough, with the cal with E a streamer crosses a gap at its limit parameters. cal conditions conditions with E,,av = =E Ecr, a streamer crosses a gap at its limit parameters. cn It approaches approaches the the opposite opposite electrode electrode at at its its lowest lowest velocity velocity corresponding corresponding to to It the minimum excess of of the the tip tip potential potential VI U, ~ e VI U , over over the the external external potential, potential, the minimum excess AUlI == VI U , -- Vo(d) Uo(d)~ = 5-8 5-8 kV, kV, below the streamer streamer practically stops. In In D..V below which which the practically stops. U o ( d )= = O. 0. If If aa gap gap is is so so long long (say, (say, 1m) 1 m) that that the case case of of aa grounded grounded electrode, electrode, Vo(d) the
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A long streamer
49
t1U, in the the channel. channel. Therefore, Therefore, A U , :::: =U U,t « << Ua' U,, nearly nearly all all applied applied voltage voltage drops drops in
E which aa streamer streamer is is still still capable capable E,,er can can be be treated treated as as the the lowest lowest field field limit, limit, at at which
of of propagating. propagating.?t This does not not bridge bridge aa gap gap This interpretation interpretation remains remains valid valid when when aa streamer streamer does but according to to (2.6) (2.6) and and (2.10), (2.10), the the but stops stops somewhere somewhere on on the the way. way. Indeed, Indeed, according is determined determined only only by by streamer to move move on, on, is streamer velocity, velocity, and and hence hence its its ability ability to - Uo(I), the potential, A t1U :::: Ut, Uo(l), and and the tip tip potential potential excess excess over over the the external external potential, U ,t = is where aa long long streamer streamer stops, stops, we we No matter matter where is independent independent of of the the latter. latter. No shall potential value value at at this this point, point, shall have have t1U A U t, « << Ua' U,, though though the the external external potential Uo(lmax), field limit limit in in the the channel channel and and Uo(lmax),may may be be high. high. Generally Generally the the average average field are interrelated interrelated as as the the streamer streamer length length at at the the moment moment it it stops, stops, ,I'max, are E ~ Ua - Uo(lmax) er~
,
max
'
,~ Ua - Uo(lmax) max~
E
er
.
(2.32)
In practice, we we must must know know not not only only In order order to to be be able able to to use use these these relations relations in in practice, inaccessible to to the but also potential Uo(lmax) the easily easily registered registered gap gap voltage voltage Ua, but also potential Uo(Imax)inaccessible even by by calculation. calculation. For For aa measurement. measurement. In In most most cases, cases, it it is is hard hard to to estimate estimate even in addition addition to to the the anode anode particular particular streamer, streamer, the the external external field field is is determined, determined, in that have have emerged emerged in in the the gap gap charge, charge, by by the the whole whole combination combination of of charges charges that and its its vicinity. vicinity. Especially Especially important important is is the the charge charge of of all all other other streamers streamers and that were were formed formed together together with with the the one one under that under study. study. Consequently, Consequently, the the field Uo(x), U o ( x ) ,in in which which the the streamer streamer is is moving, moving, represents field represents aa self-consistent self-consistent field. An An outburst outburst of of hundreds hundreds of of streamer streamer branches field. branches is is characteristic characteristic of of air; air; they fill fill up up aa space space comparable comparable with with 'max' l,,,. It It is is this this maximum maximum length, length, rather rather they than the the small small anode anode radius, radius, that that will determine the will determine the external external field field fall fall than along the the gap gap length. length. Estimations Estimations of of aa self-consistent self-consistent field U o ( x )involve field Uo(x) involve along considerable difficulties difficulties and and errors errors (we (we shall shall come come back back to to this this when when considerable evaluating the the size size of of the the streamer streamer zone zone of of aa leader). leader). So So in in reality, reality, critical critical evaluating field E E,,er can can be be evaluated evaluated only only from from experimental experimental data field data that that relate relate to to aa situation with streamers bridging a discharge gap. Then, the potential situation with streamers bridging a discharge gap. Then, the potential Uo(I,,,) is is known known reliably reliably because because it it coincides coincides with Uo(lmax) with the the potential potential of of the the electrode, usually usually the the grounded grounded one: one: Uo(lmax) Uo(lmax)= U o ( d )= = Uo(d) = 0. O. But But if if it it is is electrode, known that that Uo(lmax) Uo(lmax)« << Ua, U,, as as in in the the case case of of aa long known long streamer streamer moving moving in in aa sharply non-uniform non-uniform field, field, the the criterion criterion of of (2.32) (2.32) for for aa definite definite streamer streamer sharply length will be extremely simplified: I , x U,/Ecr. length will be extremely simplified: 'max ~ Ual Eer . The existence existence of of critical critical field field E,, has aa rather The Eer has rather clear clear physical physical meaning. meaning. The The reason for the appearance of a minimum average field in a channel is its its finite finite reason for the appearance of a minimum average field in a channel is The average average quantities quantities E E,,av and and E E,,cr describe, describe, to to some some extent, t The extent, the the actual actual field field strengths strengths in in the the channel even even when when the the external external field field is is extremely extremely non-uniform, non-uniform, changing changing by by several several orders orders of of channel magnitude along along the the gap gap far far from from the the conductor. conductor. For magnitude For example, example, if if we we close close the the gap gap with with such such aa thin wire wire that that short-circuiting short-circuiting current current does does not not change change the thin the electrode electrode voltage, voltage, aa short short time time later, after after the the current current along along the the wire wire is is equalized, equalized, the later, the actual actual gap gap field field will will become become constant constant Eav. along its its length length and and exactly exactly equal equal to to E along av .
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50
The streamer-leader streamer-leader process in a long spark
resistance. A channel channel must must conduct conduct current current necessary necessary to to support the motion motion resistance. support the of the the streamer streamer tip. tip. This This is is the the current current which which supplies supplies charge charge to to aa new new portion portion of of the the channel channel produced produced at at its its tip tip front. front. The The nature nature of of the the streamer streamer process process is is of such just behind such that that the the current current itil just behind the the tip tip is is proportional proportional to to its its velocity velocity (see (see support this formulas (2.1 1) and (2.14)). Local formulas (2.11) and (2.14)). Local field field E, Ec necessary necessary to to support this current current is is defined by by (2.15) derived from the channel channel resistivity resistivity per per unit unit length length for for aa still defined (2.15) derived from the still dense plasma. plasma. The The value value of of E, E c in (2.15) slightly slightly depends on varying varying streamer dense in (2.15) depends on streamer parameters, such tip radius, radius, and parameters, such as as length, length, tip and velocity, velocity, and and is is largely largely determined determined E, at at the the by the the gap gap gas gas composition, composition, which which predetermines predetermines maximum maximum field field Em by tip and and the the slope slope of of the the vi(E) v,(E) curve curve (the (the latter latter was was taken taken into into account account in in tip (2.15) (2.15) by by the the k parameter). parameter). The The calculated calculated value value of of E E,c == 4.2kV/cm 4.2kV/cm for for air air appeared to to be be surprisingly surprisingly close close to to the the measured measured value value E E,,cr == 4.65 4.65 kV/cm. kV/cm. appeared One should should not not give give too too much much importance importance to to this this coincidence coincidence of of the the One values, but but the the agreement agreement in in the the order order of of magnitude magnitude is is definitely definitely not not values, accidental. accidental. Because of of the the plasma plasma decay decay and and conductivity conductivity decrease, decrease, the the current current Because support in in other other channel channel portions portions may may require require aa stronger stronger field field than than E E,.c . support For this this reason, reason, decay decay processes processes appreciably appreciably affect affect the the value value of of Em Ecr, as as is is For indicated by by experiments. experiments. An An important important mechanism mechanism of of electron electron loss loss in in cold cold indicated air in in aa relatively relatively low low field field E E,,cr is is the the attachment attachment in in three-body three-body collisions collisions air 2 (section 2.2.5). 2.2.5). Here, Here, the the attachment attachment frequency frequency is is Va v, :::::: M N 2,, so so the the convenconven(section tional similarity similarity principle principle E rv N for for field field E E,,cr is is violated: violated: the the reduced reduced field field tional E,,/N does not remain constant and the value of E,, decreases more rapidly E / N does not remain constant and the value of E decreases more rapidly cr cr than than density density N [19]. [19]. When When aa streamer streamer propagates propagates through through heated heated air, air, the the critical field field becomes becomes lower lower not not only only due due to to aa lower lower density density but but as as aa result result critical of aa direct direct temperature temperature action. action. This This was was found found from from measurements measurements at at various various of and T, T , up up to to 900 900 K K [19, [ 19,201. The reason reason is is clear: clear: on on gas gas heating, heating, the the action action of of p and 20]. The attachment and and recombination recombination becomes becomes weaker weaker (section (section 2.2.5). 2.2.5). In In electroelectroattachment positive gases, gases, in in which which there there is is no no attachment, attachment, the the value value of of E E,,cr is is lower lower positive than in in cold cold air, air, other other things things being being equal. equal. For For instance, instance, in in nonpurified nonpurified than nitrogen with with an an oxygen oxygen admixture admixture up up to to 2%, 2%, the the field field is is E E,,cr :::::: M 1.5 1.5kV/cm nitrogen kV/cm at p == 1atm. 1 atm. In In inert inert gases gases where where the the attachment attachment is is absent absent and and the the recombirecombiat nation has has aa lower lower rate rate than than in in molecular molecular gases, gases, E E,,cr is is much much lower, lower, about about nation 0.5 kV/cm [21,22]. [21,22]. 0.5kV/cm The The channel channel field field does does not not vary vary much much in in time time along along its its length length because because of of the compensation compensation due due to to countereffects. countereffects. On On the the one one hand, hand, the the conductivity conductivity in in the an old old channel channel portion portion is is lower lower than than in in aa new new one one because because of of the the plasma plasma an decay. On On the the other, other, that that old old portion portion was was produced produced by by aa faster faster ionization ionization decay. wave at at aa higher higher tip tip potential potential corresponding corresponding to to aa larger larger channel channel cross cross section. section. wave result, the the resistivity resistivity per per unit unit length length R R1 = ( ~ r ~ e p ~ n does , ) ~ not vary As aa result, = Crrr2eJ-lene)-1 does not vary 1 much along along the the channel. channel. Of Of course, course, itit grows grows in in time time because because of of electron electron much loss, but but at at the the same same time, time, the the streamer streamer velocity velocity decreases decreases together together with with the the loss, channel current. current. For For this this reason, reason, the the time time variation variation of of the the channel channel field field channel E ( x ,t)t ) == RR1I(x, (x,t)i(x, t ) i ( x .t)t ) is is much much slower slower than than that that of of any any of of the the cofactors. cofactors. E(x, N
Copyright © 2000 IOP Publishing Ltd.
A long streamer
51 51
We shall illustrate illustrate this this by by giving giving aa particular particular analytical analytical solution solution to to the the We shall set of of equations equations (2.13), (2.13), (2.14), (2.14), (2.6), (2.6), (2.10), (2.10), and and (2.31), (2.31), which which is is not not far far set from the the actual actual result result (see (see below). below). Assume Assume the the channel channel field field and and capacitance capacitance from per length to to be be constant, constant, with with the the current current along along the the channel channel being being the the per unit unit length same: E(x: t ) == const, const, C Cl(x, t ) = const, and i(x, t ) = i(t). Neglect potential same: E(x, t) (x, t) = const, and i(x, t) = i(t). Neglect potential 1 Uo(x), inessential to to aa long long streamer streamer in in aa sharply sharply non-uniform non-uniform field, field, and and Uo(x), inessential suppose that that the the plasma at point point x decays exponentially exponentially starting starting from from the the suppose plasma at x decays moment tt,x of of its its production (as is is inherent inherent in in attachment attachment without without recombirecombimoment production (as nation). We shall have nation). We shall have U ( X )== Ua, - EEx, X, U(x)
U Uit == U U,a - EEI, l,
V AUt, Vss == A U,,
C,V,U, = C1AUr CIAU: i == C 1 VsUt = is, according according to to (2.6) (2.6) and and (2.10), (2. lo), equal equal where the the nearly constant coefficient coefficient A where nearly constant A is, to to vim A = const. A == 2(2k Em In 2(2k _- 1) l)Em In (nrn/no) (n,/no) ~ const. Vim
(2.33) (2.33)
yields II ~ -- exp The The integration integration of of dl dl = =V V,s dt dt yields x Irnax[l lmax[l exp (-AEt)], ( - A E t ) ] , II, rnax ~ = Uaa/ /E,E , tt,x == t(l) I. But R 1(x, t(1) at at x == 1. But the the requirement requirement R1 (x, t) t ) == R 1I(t) ( t ) involved involved in in the the initial initial assumptions be met met only value of assumptions can can be only for for one one value of the the channel channel field: field: 5 E= =v va/2A = (AUt)min/2Vem~a x 1.2kV/cm. Here, V,, 4 x 105m/s is a/2A = (b.Ut)rnin/2VernTa ~ 1.2kV/cm. Here, Vern ~ X 10 m/s is electron drift velocity at maximum tip field E,, (Aut),, x 8 kV and electron drift velocity at maximum tip field Em, (b.Ut)rnin ~ 8 kV and 7 ~ 0.85 ss is be T r,a == v;l vi' x 0.85 Xx 10lop7 is the the attachment attachment time. time. The The relation relation for for E can can be interpreted as follows. The potential difference (A U,),,, necessary to interpreted as follows. The potential difference (b.Ut)rnin necessary to provide min ~ Vern must be gained in provide the the minimum minimum streamer streamer velocity velocity V Vsmin x V, , must be gained in s field VernTa. A similar field E along along the the plasma plasma decay decay length length V,,T,. similar treatment treatment of of the the streamer process will be offered next section streamer process will be offered in in the the next section when when discussing discussing the the streamer motion in a uniform field. The order of magnitude of streamer motion in a uniform field. The order of magnitude of the the 'critical' 'critical' field field E is is correct. correct. Therefore, Therefore, the the assumptions assumptions underlying underlying the the particular particular the solution the main main idea. solution solution are are not not meaningless, meaningless, so so the solution illustrates illustrates the idea. The been confirmed The existence existence of of aa critical critical field field has has been confirmed quantitatively quantitatively by by numerical models the streamer process. Let numerical models of of the streamer process. Let us us discuss discuss the the calculations calculations obtained the above obtained from from aa simple, simple, evident evident model. model. We We mean mean the above set set of of equations equations (2.13) (2.13) and and (2.14) (2.14) supplemented supplemented by by expressions expressions (2.6), (2.6), (2.10), (2. lo), and and (2.33), (2.33), which which define define the the streamer streamer velocity velocity and and local local channel channel radius, radius, together together with with (2.31) (2.3 1) for for the the moment moment tt,x of the plasma plasma decay, decay, in in which which the the time time is is counted counted off off from from the of its its production at point x. x. The production at point The streamer streamer development development in in air air from from aa spherical spherical anode anode of of radius radius rraa == 5cm 5 cm at at U U,a == 500kV 500 kV is is demonstrated demonstrated in in figure figure 2.5.t 2.5.1 7 The calculations calculations were were made made with with T The 7, = 0.85 x s and recombination = 0.85 X 10s and recombination a 3 coefficient {3 coefficient /3 == 2 2 Xx 10- 7 cm cm3/s. The general general tendency tendency in in the the behaviour of /s. The behaviour of principal parameters parameters is principal is quite quite consistent consistent with with the the qualitative qualitative picture picture above. above.
t The was made N Shneider. The numerical numerical simulation simulation was made in in cooperation cooperation with with M MN Shneider. This This type type of of equation, equation, but was solved [23]. but with with aa constant constant channel channel radius, radius, was solved in in [23].
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The streamer-leader streamer-leader process in a long spark
52
5 00""'--------------, 5M) _. __. __. U
10
o
1 - 1=10.85 ns 23.37 ns 2 -- 23.37 53.5ns 3 - 53.5 ns 44 - 100ns IOOns 5310ns 5-
400 400
2
~ 300
.-m
J28.s v
!l :>-"
6
i
L
,g 'E ~
8
200
U
4
'1:
i
UJ
0 iIi 100 100
2
...... _- .. o00~~..:'20.,...-~4....L0::--~5:'-:0~----:8J....0~-1-'00 40 60 80 1 20 X, x,
o0~~-::'20:--~4.,.0:--'~6:':0~--:8':-0~-1~00
em cm
x , em cm x.
07
5
10 I
-t
«U
4
.~I: 04. ~
I:
e
~
iIi
. - -_ _ 5
0.3 0.2 0,1
01 0.1
o0
20
40 50 40 60 x , em cm x.
80 80
1 100
0, 00!-===2:':0==40:=::::;6;0=....-8::':0:--'"~1000
x, em
Figure 2.5. Streamer Streamer propagation propagation in in air air from from aa spherical spherical anode anode of of 5cm 5cm radius radius at at Figure 500 kY. kV. The The distributions distributions of of potential potential U, U , current current I, I , field field E, E , and and electron electron density density nnee 500
along the the channel channel at at various various moments moments of of time time until until the the streamer streamer stops. stops. along
Note that that the the nonmonotonic nonmonotonic character character of of some some current current distributions distributions when when Note the potential potential at at aa given given point point xx grows grows with with time time is is associated associated with with aa slight slight the time time decrease decrease in in capacitance capacitance (2.8) (2.8) of of the the elongating elongating channel. channel. The The streamer streamer 7 acquires mls acquires its its maximum maximum velocity velocity 10 lo7 mjs very very soon, soon, over over 10- 8 s;s; then then it it steadily steadily I, = 0.94 0.94 m. m. decelerates and and stops stops at at lmax decelerates = It was was found found from from (2.32), (2.32), with with the the account account of of Uo(lmax) U,,(lmax)and and (b.Ut)min, (AUt),in, It that the the actual actual average average field field in in the the channel channel at at the the moment moment of of zero zero velocity velocity that
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A long streamer
53 53
was Ecr == 4.9 be was E,, 4.9 kVjcm. kV/cm. From From aa simplified simplified criterion, criterion, it it was was found found to to be E,, = Ua/l U,/I,,,max == 5.3kVjcm. 5.3 kV/cm. The The agreement agreement with the experimental experimental value value of of E cr = with the 4.65 yielded 4.65 kVjcm kV/cm is is quite quite satisfactory. satisfactory. Calculation Calculation with with Ua, = = 250 250 kV kV yielded I, = 0.39 0.39m and at at UUa, = = 750 750 kV kV it it was 1.42m. Equation (2.32) (2.32) is is satisfied satisfied lmax = m and was 1.42 m. Equation at the same E cr == 4.9 4.9 kVjcm, kV/cm, i.e., i.e., the the constant constant value value of of the the average average channel channel at the same E,, field by calculations the final process field is is confirmed confirmed by calculations of of the final stage stage of of aa streamer streamer process When one one takes takes into into account account only only recombinarecombinafor various various streamer streamer lengths. lengths. t When for tion, to 1.25 m; with with no of electron electron tion, aa streamer streamer elongates elongates much much more, more, to 1.25 m; no account account of (Ec, = 1.7kV/cm), as the qualitative picture losses, it it elongates elongates to to 33 m m (E = 1.7 kVjcm), as the qualitative picture losses, cr suggests. suggests. 2.2.7 2.2.7 Streamer in a uniform field and in the 'absence' ‘absence’ of electrodes
So we have have dealt So far, far, we dealt with with aa streamer streamer which which starts starts from from aa high-voltage high-voltage elecelectrode, trode, to to which which it it remains remains galvanically galvanically connected, connected, and and is is supplied supplied by by current current from the electrode. from aa voltage voltage source source through through the electrode. Such Such are are typical typical experimental experimental designs partly, conditions the streamer designs and, and, partly, conditions in in the streamer zone zone of of aa positive positive leader, leader, in in which tip possessing possessing aa high high positive potential act which the the leader leader channel channel and and its its tip positive potential act as as the the electrode. electrode. However, However, aa situation situation may may arise arise when when aa streamer streamer is is initiated initiated in in the the body body of of aa gas gas gap, gap, where where the the external external field field is is sufficiently sufficiently high. high. This This kind kind of with aa high-voltage high-voltage of streamer streamer develops develops without without aa galvanic galvanic connection connection with source. to be be present present in negative leaders. Note that in negative leaders. Note that source. Such Such streamers streamers seem seem to lightning lightning propagating propagating from from aa cloud cloud down down to to the the earth earth most most often often carries carries aa negative negative charge, charge, while while that that going going up up from from an an object object on on the the earth earth is is positive. positive. In may take In some some situations, situations, aa streamer streamer may take its its origin origin from from the the electrode electrode vicinity vicinity and begin its travel being being connected to it, but later because of off because of and begin its travel connected to it, but later it it may may break break off the plasma plasma decay portion. If the external If the external field field is is still still strong strong the decay in in an an old old channel channel portion. at at some some gap gap space space length, length, the the streamer streamer will will move move on, on, having having 'forgotten' ‘forgotten’ about about its with the behaviour is its former former connection connection with the electrode. electrode. This This behaviour is characteristic characteristic of of the streamer the streamer zone zone of of aa leader. leader. Consider when there uniform electric Eo at at some some Consider aa simple simple case case when there is is aa uniform electric field field Eo distance 1 and distance from from the the electrode electrode and and aa fairly fairly long long conductor conductor of of length length land along the x-axis. This This may may be metallic rod rod in in laboratory laboratory vector vector Eo Eo along the x-axis. be aa metallic conditions, plane or conditions, or or aa plane or rocket rocket going going up up to to charged charged clouds, clouds, or or aa dense dense plasma entity this way polarized by plasma entity created created in in this way or or other. other. The The conductor conductor is is polarized by the vectors of the external external field field to to form form aa charged charged dipole. dipole. The The vectors of the the dipole dipole and and external body of E,,,sum in in the the body of aa perfect perfect external fields fields are are summed. summed. The The total total field field E conductor conductor drops drops to to zero, zero, since since an an ideal ideal conductor conductor is is always always equipotential. equipotential. In the symmetry points take the potential potential of In the symmetry condition, condition, all all its its points take the of the the external external unperturbed point of unperturbed field field at at the the middle middle point of the the conductor. conductor. Sometimes, Sometimes, the the field the conductor body is field in in the conductor body is said said to to be be pushed pushed out out into into the the external external t The the account (lmax) at m The calculation calculation using using aa simplified simplified formula formula without without the account of of Va Uo(lmax) at Il,,max = = 0.39 0.39m 5.7 instead instead of of 4.9kV/cm. 4.9 kV cm. gives an an error: error: 5.7 gives
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54 54
The streamer-leader process in a long spark The streamer-leader
x
-
Eo
U(x)
Figure 2.6. The The potential distribution along along aa conductive conductive rod rod in in aa uniform uniform electric electric Figure potential distribution of aa conductor. conductor. field. Broken Broken line, line, the the potential in the the absence absence of field. potential in
space. The The dipole dipole charge charge enhances enhances the the field field (E (E,,, at the ends of of the the space. Eo) at the ends sum > Eo) polarized body polarized body (figure (figure 2.6). 2.6). The problem problem offield of field redistribution redistribution by charge can can be be solved solved The by polarization polarization charge rigorously by by numerical numerical methods methods for for any any geometry, geometry, but but simplified simplified evaluaevaluarigorously tions are are also also possible. In the the close close vicinity vicinity of of aa charged charged dipole dipole 'tip' ‘tip’ of of tions possible. In << 1, the longitudinal field field varies varies nearly in the the same same way as the the radius rY «I, radius the longitudinal nearly in way as field of of aa sphere sphere of of identical identical radius. radius. Therefore, Therefore, the the external external field field perturbation field perturbation by polarization charges is is attenuated attenuated at at aa distance distance rY from from each each of of the the two by polarization charges two conductor ends. ends. Let Let us us take take the conductor middle middle point the coordinate coordinate conductor the conductor point to to be be the origin. The The end end potentials potentials of of aa polarized polarized conductor conductor differ differ from from that that of of origin. an unperturbed unperturbed one, one, U U.o == --Eox at the the same same points by .6. AU x Eol/2. The an Eox at points by U~ Eol/2. The absolute strengths strengths of of the the total total field field at at the the conductor conductor ends ends rise rise to absolute to E, x .6.U A U /r /r ~ x Eol/2r, and the the field field increases increases with with increasing increasing l/r. This Em ~ Eol/2r, and I/r. This [24]. estimate fits fits fairly fairly well well the the numerical numerical evaluation evaluation in in [24]. estimate At >> r,r, ionization ionization processes and streamers streamers may may arise arise at at the the ends ends of of aa At I1 » processes and polarized conductor even even at at aa relatively relatively low low external external field field Eo (figure 2.7(a)). 2.7(a)). polarized conductor Eo (figure Ionization waves waves run run in in both directions, leaving leaving plasma channels behind, in Ionization both directions, plasma channels behind, in much the the same same way as with with aa streamer streamer starting starting from from aa high-voltage high-voltage electrode. electrode. much way as If their their velocity velocity Vs, appreciably appreciably exceeds exceeds the the electron electron drift drift velocity, condiIf velocity, the the conditions of of streamer streamer travel travel from from the and negative negative ends ends will will not not differ differ the positive positive and tions much. The The total total charge charge of of developing developing streamers streamers is is zero zero at at any any moment moment of of much. time. This This could could not not be be otherwise, otherwise, because none of of them is connected connected with with time. because none them is the electrode electrode and, and, through through it, it, with the high-voltage high-voltage source. source. Charges Charges do do not not the with the escape the the gap gap but are only only redistributed the streamer streamer current. current. A A streamer streamer but are redistributed by by the escape
Figure 2.7. Excitation Excitation of of streamers streamers of of both both signs signs from from the the ends ends of of aa conductor conductor in in aa Figure uniform field. The The charge charge distributions distributions per per unit unit channel channel length length are are shown shown schematischematiuniform field. ( a ) with with active active plasma, plasma, (b) (b) with with plasma decay in in the the older older channel channel portions. cally: (a) plasma decay portions. cally:
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A long streamer
55 55
travelling along the travelling along the vector vector Eo Eo is is charged charged positively, positively, while while aa counterpropacounterpropagating acquires aa negative negative charge. charge. Naturally, Naturally, the the current current and and the the gating streamer streamer acquires charge separation occur due due to to electron drift. As in in aa streamer charge separation occur electron drift. streamer starting starting from electrode, the for plasma and charge separafrom an an electrode, the work work done done for plasma production production and charge separaEo. Ifit If it tion is is done done at at the the expense expense of of the the power power source source creating creating external external field field Eo. tion is aa conventional conventional high-voltage high-voltage source, source, current current flowing flowing through through the the streamer streamer is during its its propagation propagation is is due due to to the the variation variation in in the the charge charge induced induced on on the the during electrode surfaces surfaces when when the the value value and and distribution distribution of of polarization polarization charges charges electrode in the the gap gap bulk bulk change. change. This This current current takes takes away away some some of of the the source source power power in which is eventually eventually used used for for the the streamer streamer development. development. which is streamers develop, the total total length polarized conductor conductor As streamers develop, the length of of aa polarized increases, increasing the potential potential difference pushed out out of the plasma plasma increases, increasing the difference t:.u AU pushed of the channels. the other hand, the the plasma plasma in in the the old, old, central central channel decays, channels. On On the other hand, channel decays, so that from one one half conductor to so that the the charge charge overflow overflow from half of of the the conductor to the the other other becomes streamer cannot cannot elongate elongate any becomes more more difficult. difficult. Finally, Finally, aa streamer any more, more, because the the gain gain in in length length at at the the tip tip is is lost lost due due to to the the plasma plasma decay decay at at the the because ‘tail’. What What one one observes observes now now is is aa pair pair of of detached detached plasma plasma sections sections of of limited limited 'tail'. length going going away away in in both both directions. directions. At At the the front front ends, ends, they they have have aa limited limited length potential difference difference t:.U AU with with respect respect to to the the external external potential. potential. The The process process is is potential stabilized. probably go until there there is stabilized. It It may may probably go on on until is the the external external field. field. We should like to to emphasize emphasize that that this this issue issue is is of of principal principal importance. We should like importance. A streamer needs the the external external field charge redistribution redistribution in in the the created created streamer needs field for for charge plasma, i.e., i.e., behind behind the the tip tip but but not not in in front. its own plasma, front. The The streamer streamer creates creates its own field that contributes to gas ionization in the channel field that contributes to the the gas ionization in the tip tip region, region, while while the the channel field provided an external external source source is support the field provided by by an is necessary necessary to to support the current current to to the the tip, without without which which the the streamer streamer could could not not move move on. on. tip, When When the the streamer streamer plasma plasma in in the the old old channel channel near near the the starting starting point point has has decayed completely, completely, two two galvanically galvanically disconnected disconnected streamers streamers continue continue to to decayed move move in in opposite opposite directions. directions. Now Now the the polarization polarization effects effects of of each each of of the the conductive polarization effects conductive sections sections are are added added to to the the earlier earlier polarization effects of of the the whole channel. As aa result, result, there there are are four four charged charged regions regions with with alternating alternating whole channel. As polarity (figure (figure 2.7(b)). Nevertheless, there there are are still only two two ionization ionization polarity 2.7(b)). Nevertheless, still only waves moving moving only forward, away away from the channel centre, trying trying to to elongate elongate waves only forward, from the channel centre, the streamer. No return return ionization ionization waves waves arise in its its central central portion portion which which the streamer. No arise in has lost conductivity because because of of aa smooth charge distribution towards the the has lost conductivity smooth charge distribution towards ends. ends. The The charge charge is is 'smeared' ‘smeared’ along along more more or or less less extended extended 'semiconducting' ‘semiconducting’ regions regions and and cannot cannot create create aa sufficiently sufficiently strong strong field field to to initiate initiate ionization. ionization. High-voltage High-voltage engineers engineers are are familiar familiar with with this this phenomenon. phenomenon. They They can can somesometimes times decrease decrease an an electric electric field, field, alternating alternating in in time, time, at at the the site site of of its its local local rise between aa sharp by coating the dielectric rise between sharp metallic metallic edge edge and and an an insulator, insulator, by coating the dielectric at their their boundary boundary with with semiconducting material. at semiconducting material. of fast 'electrodeless' streamers described by by the the The parameters parameters of The fast ‘electrodeless’ streamers are are described same (2.14). The The role role of of U here is is played played - U same formulas formulas (2.6), (2.6), (2.10), (2.10), and and (2.14). U,t U.o here by the the quantity quantity t:.U ~ E where I1 is is the the length of aa channel channel section with AU x Eo1/2, length of section with by ol/2, where
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56
The The streamer-leader streamer-leader process in a long spark
preserved conductivity. tip preserved conductivity. Here, Here, b"U A U also also represents represents the the excess excess of of the the streamer streamer tip potential over potential over the the external external one. one. In In particular, particular, as as the the length length I increases increases and and the the field at at the the conductor conductor ends ends becomes becomes as as high as E, 150-170kV/cm for high as Em M ~ 150-170 kV/cm for field normal density Em ceases. As II and normal density air air (section (section 2.2.2), 2.2.2), the the growth growth of of E, ceases. As and b"U AU increase further, further, the the streamer streamer tip tip radius radius Y,f ,m , not not E,, increases because because increase Em' increases AU ~ z E,Y,. This is is accounted accounted for for by the self-regulation self-regulation mechanism mechanism discussed discussed b" Emfm' This by the in in section section 2.2.2. 2.2.2. For understanding of For the the understanding of the the streamer streamer process process in in the the streamer streamer zone zone of of aa leader, the field uniform and the leader leader, where where the field is is nearly nearly uniform and the leader channel channel acts acts as as aa high voltage voltage 'electrode', high ‘electrode’, it it is is essential essential that that the the 'electrode' ‘electrode’ and and 'electrodeless' ‘electrodeless’ situations the positive positive and situations should should be be strictly strictly equivalent, equivalent, provided provided that that the and the the negative streamers negative streamers are are identical identical and and the the external external field field is is uniform. uniform. Let Let us us mentally cut, at at the the centre, centre, aa plasma conductor developing developing in in both both directions directions plasma conductor mentally cut, from this this centre centre and and discard, discard, say, say, the the negatively negatively charged charged half. half. Let Let us us now now from replace it it by by aa plane plane anode anode under under zero zero potential potential and and assume assume aa negative negative replace potential to to be be applied applied to to aa remote remote plane plane cathode. cathode. A A cathode-directed cathode-directed potential streamer produced produced at at the the anode anode by by aa local local inhomogeneity, inhomogeneity, whose whose field field streamer initially supported supported ionization, ionization, will will be be identical identical to to aa positive positive streamer streamer in in the the initially electrodeless case. case. Indeed, Indeed, in in both both cases, cases, the the conductor conductor potential potential U coincides coincides electrodeless with the the external external potential, potential, U(O) U ( 0 ) == Uo(O). Uo(0). The The charge charge pumped pumped from from the the with negative half half into into the the positive positive one one will will now now be be supplied supplied by by the the source source current current negative from the the anode. anode. Here, Here, the the principle principle of of mirror mirror reflection reflection in in aa perfectly perfectly from conducting plane, plane, well-known well-known from from electrodynamics, electrodynamics, reveals reveals itself itself in in every every conducting detail. According According to to this this principle, principle, the the distributions distributions of of charge, charge, current current and and detail. field in in half-space half-space do do not not change change if if the the plane plane is is replaced replaced by by the the mirror mirror field reflection reflection of of half-space half-space charges. charges. These considerations considerations were were used used in in the the calculations calculations and and representation representation of of These results on on streamer streamer development development in in aa uniform uniform field field U U,o == --Eox from the the results Eox from point xx == 00 towards towards lower lower potential potential (figures (figures 2.8 2.8 and and 2.9).t 2.9).t The The solution solution point was derived derived from from the the same same set set of of equations equations (2.13), (2.13), (2.14), (2.14), (2.6), (2.6), (2.10), (2.10), was (2.31) and and the the same same plasma plasma decay decay characteristics characteristics as as in in section section 2.2.6. 2.2.6. The The (2.31) calculations show that a streamer does not develop if the external field is calculations show that a streamer does not develop if the external field is lower than a certain minimum value. The values of Eomm do not differ lower than a certain minimum value. The values of Eo min do not differ E,,cr which which determines determines the the streamer streamer much from from the the critical critical channel channel field field E much length Eomin ~ M 7.7kV/cm. 7.7 kV/cm. One One length in in aa non-uniform non-uniform field field calculated calculated with with (2.32): (2.32): Eo may probably probably use use for for estimations estimations the the experimental experimental value value E E,,cr ~ x 55 kV/cm kV/cm as as aa may Eomin (section (section 2.4.1). 2.4.1). If If the the uniform uniform external external field field slightly slightly exceeds exceeds realistic Eo realistic the minimum, minimum, the the excess excess tip tip potential potential is is small, small, the the streamer streamer velocity velocity is is low, low, the Eomn. This and the the channel channel field field isis close close to to the the unperturbed unperturbed external external field field Eo and min' This situation is is illustrated illustrated in in figure figure 2.8. 2.8. situation is appreciably appreciably higher higher than than Eo Eomin' however, If Eo is however, the the tip tip potential potential is is much much If higher than than the the external external potential, potential, and and the the streamer streamer develops develops aa high high velocity velocity higher Numerical simulation simulation was was made made in in cooperation cooperation with with M MN N Shneider. Shneider. tt Numerical
Copyright © 2000 IOP Publishing Ltd.
A long streamer
57
0.7 -50
. ~
0.6
5 0.5
-100
1-150
------. U o
'i
l-t=22.15ns 2· 94.5 ns 3 - 275 ns 4· 495 ns 5 - 679.5 ns
tij
-200
i- 0.4
.~
~ 0.3
j
'-I.l
·250
0.2 0.1
-3001 ' . ...0~.... ' . 25-....J3L..0 ' . ' ~3 ' " 1 -3000L..-........-1 20-... ... 5 ......J 0 5 IO 15-.... 15 20 25 30 35 40 x,cm x, cm '
0.20r-------------.
0.0 0t:::::::;:::t::-"'- ""5::::::;j2"'0-20:::~30""""'35-"'40....l 5 10 1 x,cm x, cm 10r--------------,
0.15
..: i
~ ::l u
2 0.\0
0.05
' 0
x,em x, cm
2 20L..-........-1... 0~15-.... .... 20-... 25-....J30........ .J3... 5 ......J 0 5 10 15 20 25 30 35 400
x, em cm
Figure 2.8. An Eo =7.7kV/cm, An air air streamer streamer in in aa uniform uniform field field Eo =7.7 kV/cm, slightly slightly exceeding exceeding the the critical with calculated potential U, critical minimum, minimum, with calculated distributions distributions of of potential U , current current II,, field field E and and electron potential counted electron density density ne' ne. Dashed Dashed line, line, applied applied field field potential counted from from the the streamer streamer origin. origin. The The oppositely oppositely charged charged streamer streamer running running in in the the opposite opposite direction direction is is not not shown. shown.
(figure portions at but 2.9). The The current current in in well-conducting well-conducting portions a t the the tip tip is is high high but (figure 2.9). decreases high current, much decreases towards towards the the channel channel centre. centre. Owing Owing to to the the high current, much positive positive charge charge is is pumped pumped into into the the tip tip region region even even at at an an early early stage; stage; in in the the tail, tail, however, however, where where the the conductivity conductivity has has decreased, decreased, the the current current is is low. low. As aa result, positive charge pumped out result, positive charge pumped out of of it it is is not not reconstructed; reconstructed; moreover, moreover,
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58 58
The The streamer-leader streamer-leader process in a long spark 7.5
-200 5.0
...>
E -400
Ii '"§ B 0
cJc
--_
-600
Q.
u
---I_. U ----_. o
'5u
5 iii
-800 -800
-1000 -1000 0
20 20
40
2.5
..c
h
1 - 1=194.1 ns 2- 360 ns 482.1 ns 3 -- 482.1 ns 4 - 576.6 576.6ns 4653.1ns 5 -- 653.1
~
'~"
u
.4
0.0
\ 5
60 60
80 80
100 100
20
40
x,em x, cm 2.5 2.5
60
80
100
cm x, em 14
7 5
2.0
12
E
<:
...~
1.5
-0
i
~::l
10
"ii <;::; u
1.0
'Eu
u
~
'iii 0.5
0.0
6
4
0
x, x, em cm
0
20
40
3
4
5
60
80
100
x, cm x,em
Figure 2.9. 2.9. An air air streamer streamer in in aa uniform uniform field field Eo Eo =10 =10 kVjcm kV/cm with with the the charge charge distribudistribuFigure tion instead of of the the nnee curves curves similar similar to to those those in in Figure Figure 2.8. 2.8. tion T7 instead
the the tail tail becomes becomes negatively negatively charged. charged. The The calculated calculated charge charge distributions distributions in in figure 2.9 were were found found to to be be exactly exactly as as those those represented represented schematically schematically in in figure 2.9 the 2.7(b) in in terms terms of of double double polarization polarization of of the the the right-hand right-hand side side of of figure figure 2.7(b) whole of its its conducting conducting sections sections individually. individually. At At whole channel channel and and each each of Eo ;::::: zz Eo Eomin min and and low low current, current, the the polarization polarization effect effect of of the the conducting conducting section section Eo is 2.8). is very very weak weak (figure (figure 2.8).
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The principles of of a leader process
2.3
59 59
The principles principles of a leader process process
This section section is is aa key key one one for for the the understanding understanding of of aa long long spark spark and and the the first first This lightning component. component. We We shall shall try to answer answer the the question question why simple strucstruclightning try to why aa simple tureless plasma channel channel has no chance chance to acquire aa considerable considerable length length in in cold cold tureless plasma has no to acquire air of of atmospheric atmospheric pressure. The reader see what what is is necessary for aa spark spark to to pressure. The reader will will see necessary for air become long and and have long lifetime lifetime and and how how Nature realizes this this possibility. become long have aa long Nature realizes possibility.
2.3.1 The necessity of gas heating 2.3.1 Section plasma channel Section 2.2 2.2 dealt dealt with with the the development development of of aa simple simple plasma channel -- aa streamer -- which which has has no no additional additional structural structural details. details. The The theoretical constreamer theoretical considerations by siderations concerning concerning the the streamer streamer process process are, are, in in general, general, supported supported by experiment, indicating indicating that that the the streamer streamer gas gas is is cold cold and and the the channel channel field field is is experiment, too low low for for ionization ionization to to occur. occur. In In these these conditions, conditions, the the plasma in too plasma produced produced in the tip by an ionization ionization wave decays later. later. Electrons Electrons are are lost lost due due to to recombirecombithe tip by an wave decays nation (it exists exists in in any any gas) gas) and and attachment attachment inherent inherent in in air air as as an an electronegative electronegative nation (it gas. Losing Losing its its conductivity conductivity and, and, hence, the possibility to use current from from an an gas. hence, the possibility to use current external source, source, aa streamer streamer eventually eventually stops stops its its development, development, unless unless it it encounencounexternal ters aa strong ters strong field field on on its its way way (section (section 2.2.7). 2.2.7). Sometimes, Sometimes, the the streamer streamer lifetime lifetime can be longer by steady voltage voltage rise, this possibility possibility is, is, naturally, naturally, can be made made longer by aa steady rise, but but this limited. Even Even at at aa megavolt megavolt voltage voltage of of aa laboratory laboratory generator, generator, an an air air streamer streamer limited. can become only several several metres metres long. long. Voltages Voltages of of aa few few dozens dozens of of megavolts megavolts can become only inducing the streamer inducing lightning lightning discharges discharges are, are, at at best, best, capable capable of of increasing increasing the streamer length but not not to the kilometre length to to several several tens tens of of metres metres but to the kilometre scale scale characteristic characteristic of At high however, the of lightning. lightning. At high altitudes, altitudes, however, the air air density density is is low low and and aa streamer streamer may cover red sprites may cover aa longer longer distance. distance. This This probably probably accounts accounts for for vertical vertical red sprites above above powerful powerful storm storm clouds clouds dozens dozens of of kilometres kilometres above above the the earth's earth’s surface surface [25], travel downwards. [25], which which were were found found to to travel downwards. The way of plasma decay The only only way of preventing preventing or, or, at at least, least, slowing slowing down down air air plasma decay in by increasing temperature in in aa low low electric electric field field is is by increasing the the gas gas temperature in the the channel channel to to several several thousands thousands of of Kelvin Kelvin degrees degrees and, and, eventually, eventually, to to 5000-6000 5000-6000 K K or or more. by more. In In aa hot hot gas, gas, electron electron loss loss through through attachment attachment is is compensated compensated by accelerated recombination slows accelerated detachment detachment reactions, reactions, and and recombination slows down. down. The The mechanism of associative associative ionization ionization comes comes into into action, action, and and electron electron mechanism of impact because the the gas impact ionization ionization is is enhanced enhanced because gas density density decreases decreases on on heating. heating. These plasma channel to support These processes processes make make it it possible possible for for aa plasma channel to support itself, itself, or, or, at at least, to to approach approach this this condition, condition, in in aa relatively relatively low low field. field. A hot hot spark spark looks looks least, like aa hot arc or or aa glow glow discharge discharge column column after after contraction contraction [26]. [26]. We We shall shall not like hot arc not discuss here the details details of of these these processes processes (for (for this, this, see see section section 2.5). 2.5). It It suffices suffices discuss here the to take take for for granted granted the the statement statement that that gas gas heating heating does does maintain maintain plasma plasma to conductivity, making the viable. conductivity, making the spark spark viable. It that an potential U at It follows follows from from section section 2.2 2.2 that an increase increase of of potential at the the streamer streamer front front does does not not contribute contribute to to gas gas heating. heating. Total Total energy energy release release grows grows as as U 22, ,
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The streamer-leader The streamer-leader process in a long spark
but the streamer streamer cross cross section section 7rrk also increases increases as as V U22. .So the the released released energy energy but the 11T~ also )2, which defines the heating, remains low. To density proportional to density proportional to (V/r (U/rm ) 2 , which defines the heating, remains low. To m increase the the channel channel temperature temperature considerably, considerably, it it is is necessary to accumulate accumulate increase necessary to this, the aa much much higher higher energy energy in in aa much much narrower narrower plasma plasma column. column. For For this, the funcfunctional relation providing providing the must be be violated. tional relation the low low V U //rr m m ratio ratio must violated. This This is is impossible in in aa primary ionization wave wave but possible in in aa differently differently impossible primary ionization but becomes becomes possible organized problem by organized channel channel development. development. Let Let us us try try to to approach approach this this problem by considering the final result and estimate estimate voltages voltages and and plasma plasma channel channel radii, radii, considering the final result and at which which the the gas gas temperature temperature would become sufficiently sufficiently high. high. This This can can be be at would become done done in in terms terms of of the the general general energy energy considerations considerations discussed discussed in in section section 2.2.4. 2.2.4. Suppose there is with characteristic Suppose there is aa charged charged space space with characteristic size size R in in the the front front region of aa developing developing plasma channel. Its Its capacitance capacitance is is C C~ x 1rI~oR m o R with region of plasma channel. with C1 x x0 per unit length along the channel axis. If the tip potential is U, C ~ 1l"co per unit length along the channel axis. If the tip potential is V, j the energy energy dissipated dissipated per per unit length of of aa new of the system including including the unit length new portion portion of the system the channel channel and and the the space space being being charged charged is is C C1 U22/2, / 2 ,provided spark develdevelthe provided the the spark j V ops per unit portion is ops steadily. steadily. Since Since the the capacitance capacitance per unit length length of of aa new new system system portion is independent independent of of its its radius radius R or or of of any any other other geometrical geometrical dimension, dimension, we we are are free to to assume assume any any nature, size, and and volume of this charged space. space. (The (The free nature, size, volume of this charged specific capacitance of the central portion of a long system does vary specific capacitance of the central portion of a long system does vary with with total length and and radius only logarithmically, logarithmically, as as is is clear clear from from formula formula total length radius but but only (2.8)) The The dissipated dissipated energy energy includes includes all all expenditures expenditures for for the the creation creation of of aa (2.8).) new channel portion and space charge. Attribution of this energy to the new channel portion and space charge. Attribution of this energy to the various expenditures is a special problem which requires details of the various expenditures is a special problem which requires details of the process to be specified. But But we can estimate estimate the the upper limit of of air air mass mass process to be specified. we can upper limit that can be heated to the necessary temperature, say, to T = 5000K. For that can be heated to the necessary temperature, say, to = 5000 K. For this, let us assume that all energy has been used heating an air column of this, let us assume that all energy has been used heating an air column of initial radius ro. This will be an estimate of the upper radius limit. This initial radius roo This will be an estimate of the upper radius limit. This temperature will will lead lead to to considerable considerable thermal thermal gas gas expansion, expansion, because because aa hot hot temperature channel, as as will will be shown later, later, develops develops much much more slowly than than aa cold cold channel, be shown more slowly streamer channel. channel. Current Current must must have have enough enough time time to to heat heat the the gas, gas, because streamer because it is is eventually eventually the the released released Joule Joule heat heat of of current current that that does does the the heating. heating. If If it the heating is not not high enough, pressure in the the gas gas space space is is equalized, equalized, the heating rate rate is high enough, pressure in so that that the gas of of aa thin channel becomes less dense. dense. The The air air heat capacity so the gas thin channel becomes less heat capacity so energy energy calcucalcudoes not not remain constant within temperature range, remain constant within aa wide wide temperature range, so does lation should be made in terms of specific enthalpy h( T , p ) . Therefore, the lation should be made in terms of specific enthalpy h(T,p). Therefore, the expression to maximum radius radius ro rOmax of aa cold cold air air column column that that can can be max of be expression to aa maximum heated is heated to to temperature temperature T is 2 7rrim,,p0h(T) x T & U ~ / ~ . 1l"r6maxPoh(T) ~ 1l"C o V~/2.
(2.34) (2.34)
T ) is is specific specific enthalpy enthalpy for for air air at at p == 11 atm atm and and po is its its density density at at Here, h( T) Po is Here, p == 11 atm atm and and To To == 300K. 300K. With tip tip potential potential V U,t == 11 MV and and T == 5000 5000 K, K, when when h(5) h ( 5 ) == 12 12 kJ/g, kJ/g, an an With air column column that that can can be heated must must have have an an initial initial radius radius less less than air be heated than yomax = 0.054 0.054cm. The maximum maximum radius radius due due to to thermal expansion will will be ro max = cm. The thermal expansion be
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The The principles of of a leader process
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less than than rmax ro [ p o /p(5)]1/2 p (5 ) ]' I 2 M 0.26 cm, where 5) is is the the air air density density at at less rmax == romax[Po/ ~ 0.26cm, where p( p(5) = 5000 5000 K K and and pressure atm. A A channel channel of of this thickness has observed T = pressure 11 atm. this thickness has been been observed U,t M 100 MV, MV, characteristic characteristic of of very powerful in laboratory laboratory spark spark leaders. leaders. At At U in ~ 100 very powerful be two lightning, the radius radius estimated must be lightning, the estimated from from formula formula (2.34) (2.34) must two orders orders of of magnitude A lightning magnitude larger. larger. A lightning leader, leader, however, however, has has aa temperature temperature higher higher than 5000K 5000K and and h"", h T 22 approximately approximately (h(10) (h(10) = = 48kJ/g), 48kJ/g), so so that that the than the radius does does not grow as as much much as as U U,t and and remains as small small as as several several centicentiradius not grow remains as metres. It It may may seem seem surprising, surprising, but leader channel channel is is thinner thinner than than aa streamer streamer metres. but aa leader channel the same tip potential channel at at the same tip potential (their (their radii radii are are established established due due to to different different reasons: radius follows the heating heating conditions, reasons: aa leader leader radius follows the conditions, while while aa streamer streamer the lateral radius radius is is such such that that the lateral field field is is too too low low for for intensive intensive ionization). ionization).
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2.3.2 2.3.2 The necessity of a streamer accompaniment accompaniment
The The existence existence of of aa long long spark spark and and lightning lightning are are due due to to two two main main mechanisms, mechanisms, even presence of very high even in in the the presence of aa very high voltage voltage source. source. One One is is the the mechanism mechanism of of current be heated. current contraction contraction in in aa thin thin channel channel which which can can practically practically be heated. The The other radial field other is is the the attenuation attenuation of of aa very very strong strong radial field that that arises arises at at the the lateral lateral sursurface high potential potential relative face ofa of a very very thin thin conducting conducting channel channel under under aa very very high relative to to the the earth. mechanism, because earth. We We shall shall begin begin with with the the second second mechanism, because it it opens opens the the way way for the first one. one. In In reality, reality, the the tremendous tremendous value of U /r /rM 10-100 MV/cm MV/cm is is for the first value of ~ 10-100 not the field near the Nor does not the field scale scale near the channel channel tip tip of of radius radius r.r . Nor does the the value value of of U/[rln(l/r)] MV/cm, which Er at U/[rln (Ilr)]~ 1-10 l-lOMV/cm, which is is somewhat somewhat less, less, determine determine field field E, at the lateral be behind the the tip, tip, as as could could be the lateral surface surface of of aa channel channel of of length length II behind suggested Er = suggested from from formula formula (2.9) (2.9) and and the the Gaussian Gaussian theorem, theorem, E, = r/(27rEor). 7/(27rq,r). This be valid valid only This would would be only for for such such aa simple simple structureless structureless channel channel as as aa streamer, streamer, but but its its lateral lateral field field cannot cannot maintain maintain aa high high strength strength for for aa long long time. the channel time. Lateral Lateral ionization ionization expansion expansion would would immediately immediately increase increase the channel radius. be heated to the radius. On On the the other other hand, hand, aa channel channel cannot cannot be heated to the necessary necessary high temperature unless its high temperature unless its radius radius is is small. small. This This is is the the reason reason why why aa single single simple heated. simple channel channel cannot cannot be be heated. A long-living megavolt voltages have aa long-living spark spark requiring requiring megavolt voltages will will inevitably inevitably have we mean complex reader has, has, no no doubt, that we complex structure. structure. The The reader doubt, guessed guessed that mean the the streamer production -- the streamer zone zone in in front front of of aa leader leader tip tip and and its its production the leader leader cover cover representing 2.2). The The representing aa thick thick charged charged envelope envelope around around the the channel channel (figure (figure 2.2). space the same space charge charge of of aa streamer streamer zone zone and and leader leader cover, cover, having having the same sign sign as as that of potential, greatly reduces the the field that of the the channel channel potential, greatly reduces field at at the the channel channel surface. surface. Roughly, to the by space the huge huge potential potential Roughly, owing owing to the field field redistribution redistribution by space charge, charge, the U now now drops drops across across aa much much longer longer length length R of of the the streamer streamer zone zone and and the the charge than across charge cover cover radius, radius, rather rather than across aa length length nearly nearly as as short short as as the the channel channel moderate magnitude magnitude U //RR but but not radius radius r.r . In In this this case, case, the the field field scale scale is is aa moderate not U /r, has R / r , because because even even aa laboratory laboratory spark spark has R of of about about aa metre metre long. long. Indeed, Indeed, the the radius radius of of aa streamer streamer zone zone and, and, hence, hence, of of aa leader leader cover cover is is defined by the maximum distance when they defined by the maximum distance streamers streamers may may cover cover when they travel travel
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The The streamer-leader streamer-leader process in a long spark
away from from the the leader leader tip. tip. We We already already know know (sections (sections 2.2.6 2.2.6 and and 2.2.7) 2.2.7) that that the the away average average field field necessary necessary for for streamer streamer development development in in atmospheric atmospheric air air must must be be at at least E E,,cr ~ x 55 kV/cm. kV/cm. Since Since streamers streamers stop stop at at the the end end of of the the streamer streamer zone, zone, the the least voltage drop drop along along the the zone zone length length R is is about about 6. AU, x EcrR E,,R (cf. (cf. formula formula (2.32)). (2.32)). voltage Us ~ About as as high high voltage voltage drops drops outside outside the the streamer streamer zone, zone, because because the the field field there, there, About i.e., in in aa zero-charge zero-charge space, space, drops drops from from about about E E,,cr to to zero, zero, as as for for aa solitary solitary i.e., sphere of of radius radius R. R. Hence, Hence, we we have have U ~ x 26.U 2AUss and and R == U/2E U/2Ec,. At sphere cr . At x 11MV, MV, the the streamer streamer zone zone radius radius is is R ~ x 11m, m, in in agreement agreement with with laboratory laboratory U ~ measurements. measurements. It follows follows from from both both calculations calculations and and measurements measurements that that the the current, current, It field, electron electron density, density, and and conductivity conductivity of of aa heated heated leader leader channel channel are are field, generally generally comparable comparable with with respective respective parameters parameters of of aa fast fast streamer. streamer. If If they they are somewhat somewhat larger, larger, the the difference difference is is not not orders orders of of magnitude. magnitude. So So the the heating heating are time time to to achieve achieve aa much much higher higher gas gas temperature temperature must must be be much much longer. longer. This This explains explains why why aa leader leader propagates propagates much much more more slowly slowly than than aa fast fast ionization ionization wave. wave. The The capacitance capacitance per per unit unit length length of of aa leader leader system system (the (the channel channel plus plus aa charged charged cover) cover) will will be be described described by by the the same same formula formula (2.8) (2.8) if if II is is substituted substituted by by leader leader length length L and and the the conducting conducting channel channel radius radius rY by by cover cover radius radius R, R , the the actual actual radius radius of of aa charged charged volume. volume. This This follows follows directly directly from from electrostatics. electrostatics. Similarly, Similarly, the the current current iiL at the the leader leader channel channel front front is is related related to to the the tip tip L at potential L by and leader leader velocity velocity V VL by the the same same expression expression (2.11)t (2.1 l)t potential U and . lL
27f6 0 UVL
= In (L/ R) .
(2.35) (2.35)
For aa laboratory laboratory leader leader of of length length L ~ x 10 10m and R ~ M 1 1 m, m, the the logarithmic logarithmic For m and values are are several several times times smaller, smaller, while while the the linear linear capacitance capacitance is is larger larger than than values in aa streamer streamer with with rY ~ in x 10lo-’1 cm. cm. The linear linear capacitance capacitance of of aa conventionally conventionally semispherical semispherical streamer streamer zone zone The is C C1 M 27reo, like the capacitance of a streamer tip. The tip current flowing is ~ 27f60, like the capacitance of a streamer tip. The tip current flowing 1 into the the streamer streamer zone zone into it = 27r&OU v,
(2.36) (2.36)
( L IR) R )higher higher than than iiL, again like like in in aa streamer. streamer. But But since since the the is by by aa factor factor ofln of In (L/ is L , again leader logarithm logarithm is is closer closer to to unity, unity, currents currents iiL and itit do do not not differ differ as as much much as as leader L and for aa streamer. streamer. If If the the current current along along aa leader leader channel channel does does not not vary vary much, much, as as in in aa for fairly short short leader leader at at constant constant voltage, voltage, the the tip tip current current will will not not differ differ much much from from fairly experimental current current i in in the the external external circuit. circuit. A A typical typical laboratory laboratory leader leader experimental 4 has i ~ x iiL x itit ~ M 1 1A, A, U U~ x 11 MV, MV, and and from from (2.36) (2.36) V VL M 2x 2x 10 lo4 m/s which whch is is mls has L ~ L ~ 4 VL x (1-2.2) (1-2.2) xx 10 lo4 mjs [27,28]. [27,28]. close to to numerous numerous measurements, measurements, in in which which V m/s close L ~ Formula (2.36) (2.36) or or (2.35) (2.35) permits permits the the estimation estimation of of any any of of the the three three parameters parameters Formula Here and and below, below, the the external external field field potential potential Uo(x) Uo(x)isis omitted omitted for for brevity. brevity. It It is is indeed indeed small small in in tt Here laboratory leaders leaders normally normally observed observed in in aa sharply sharply non-uniform non-uniform field. field. laboratory
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-- i, U, useful in U , or or V VL from the the other other two. two. This This is is especially especially useful in studying studying L -- from lightning leaders leaders when actual data data are are very very scarce. scarce. lightning when actual Thus, Thus, aa key key condition condition for for aa long-term long-term spark spark development development is is the the formation formation of aa thick thick space-charge space-charge cover cover around around it, it, having same sign sign as as the the channel channel of having the the same potential. The charge charge reduces reduces the field on on the channel surface, surface, depriving depriving the the potential. The the field the channel channel of of its its ability ability to to expand expand due due to to ionization. ionization. It It is is only only aa channel channel with with aa channel charge cover cover small cross cross section section that that can can preserve the ability ability to to be heated. A charge small preserve the be heated. also contributes contributes somewhat somewhat to the linear linear leader leader capacitance, capacitance, because it is is also to the because it than by by the the small small now determined determined by by the the much much larger larger cover cover radius now radius R rather rather than An increase increase in in linear linear capacitance capacitance is is accompanied accompanied by by an an channel radius radius r.r. An channel increase increase in in the the energy energy input input into into the the channel. channel. If Nature were to make If Nature were aa living living being being and and decided decided to make aa spark spark or or lightning lightning travel possible, it by organizing travel as as large large aa distance distance as as possible, it would would do do this this by organizing the the streamer zone zone and and charge charge cover. cover. In In actual actual reality, reality, everything everything happens happens streamer automatically: automatically: the the huge huge voltages voltages that that create create long long sparks sparks produce produce numerous numerous streamers 2.10). streamers at at the the front front end end (figure (figure 2.10). This reminds reminds us us of of aa high high voltage electrode creating, creating, under under suitable suitable This voltage electrode conditions, aa multiplicity multiplicity of of streamer streamer corona corona elements. elements. This This kind kind of of corona corona conditions, can be be registered in laboratory laboratory experiments. experiments. can registered in Currents of of all all streamers streamers starting starting from from aa leader leader tip tip are are summed summed up, Currents up, heating the spark spark channel. channel. This This total total current current charges charges the the region in front front of of heating the region in the the old the tip, tip, neutralizing neutralizing the the charge charge of of the old tip, tip, and and when when aa new new tip tip is is formed, formed, the the spark spark elongates elongates by by aa length length of of about about the the tip tip length, length, as as in in aa
2.10. Photograph Photograph of of aa positive positive leader leader in in aa rod-plane rod-plane gap gap of9 of 9 m m length length at at 22 MV; MV; Figure 2.10. the electronic electronic shutter shutter was was closed closed at at the the moment moment of of contact contact of of the streamer zone zone and and the streamer the the the plane. plane.
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The streamer-leader streamer-leader process in a long spark
single streamer. streamer. Part Part of of the streamer zone zone appears appears to tip, single the streamer to be be behind behind the the tip, transforming to to aa new new cover cover for for the the newly newly born leader portion. But this this transforming born leader portion. But does the streamer meanwhile the does not not decrease decrease the streamer zone zone length, length, because because meanwhile the zone zone has has moved moved forwards forwards together together with with the the tip. tip. Note Note only only that that if if there there are are many many streamers streamers they they are are very very close close to to one one another, another, and and they they travel travel in in aa self-consistent self-consistent field field close close to to the the critical critical field field (sections (sections 2.2.6 2.2.6 and and 2.2.7). 2.2.7). Such Such streamers have aa low the leader [4], so so that that the leader current current is, is, streamers are are slow slow and and have low current current [4], indeed, indeed, aa sum sum of of numerous numerous low low streamer streamer currents. currents. 2.3.3 2.3.3 Channel contraction contraction mechanism mechanism
The mechanism mechanism of of current current contraction contraction in in the the front front region region of of aa leader leader channel channel The is not quite clear, clear, especially especially quantitatively. quantitatively. One One may may assume assume the the existence existence of of is not quite ionization-thermal ionization-thermal instability. instability. This This effect effect looks looks like like the the one one leading leading to to glow glow discharge contraction contraction [26], [26], but but it it has has its its own own specificity specificity [4]. [4]. The The instability instability is is discharge associated with with the dependence of of electron electron impact impact ionization ionization frequency frequency on on associated the dependence v,(E,N f ( E / N), N ) , where where ff(E/ ( E / N) N) field and and molecular number density: density: vJE, field molecular number N)) == N Nf(E/ is aa rapidly rapidly rising rising function function at at small small E / N (figure (figure 2.4). 2.4). This This is is the the ionization ionization is component of of the the instability. instability. Its Its thermal thermal component component is is due due to to the the fact fact that that component rapidly equalizes equalizes in in small small volumes volumes at at aa moderate heating the gas gas pressure the pressure p rapidly moderate heating const, aa more site proves less dense, dense, and and the the rate. With With p '" NT == const, rate. more heated heated site proves less N , determining determining the ionization frequency, frequency, increases increases there. there. reduced field field E / N, reduced the ionization As was mentioned above, above, numerous numerous streamers streamers start start from from the front end end As was mentioned the front of aa developing developing leader. leader. The The frequency frequency of of streamer streamer emission emission has has been shown of been shown 9 S-l experimentally lo9 sC1 at at aa typical typical laboratory laboratory spark spark current current of of 1A 1A experimentally to to exceed exceed 10 [29]. Younger Younger streamers streamers have have not lost their their conductivity conductivity yet. yet. The The streamers streamers at at [29]. not lost so close close to to each each other other that that they they form form aa continuous continuous conductconductthe leader leader tip tip are are so the Current itit flows flows along along tthis h s and and the the initial initial leader leader ing channel channel of of radius radius rrSum. ing sum ' Current channel. It It is is external external current current relative relative to to the the tip, tip, because it is is created created by channel. because it by the the of charges charges exposed exposed and and displaced displaced by by the the streamer streamer zone zone whole combination of whole combination bulk. This This current current is is practically practically independent independent of of the the tip tip conductivity. conductivity. In In bulk. terms of of electric electric circuit circuit theory, the streamer streamer zone zone acts acts as as aa current current source source terms theory, the R -+ + oc) m) relative relative to to the the (an electric electric power power generator generator with with an an inner inner resistance resistance n (an M t:i.Us/i AU,/i,t ;:::0 M U/2i U/2it, where A U is leader tip. tip. Its Its actual actual value is very very large: large: R , n;:::o where t:i.U leader value is t s, is the voltage voltage drop drop across across the the streamer streamer zone. zone. At At U;:::o U M 1MV 1 MV and and i ;:::0 x 1A, 1 A, the the the R ;:::0 ZZ 0.5 0.5MR. matter what happens to to the the leader leader tip tip or or its its short short value is n value is MD. No No matter what happens front portion, portion, the the current current there there does does not not change. change. What What changes changes is is the the electric electric front field, because because it it depends depends on on the the conductivity conductivity and and radius radius of of the the region loaded by field, region loaded by current (in (in aa glow glow discharge, discharge, the the field field is is fixed fixed and and the the current current can can vary vary during during current the instability instability development). development). the Suppose the the current current density, density, whose whose average average cross cross section section value is Suppose value is j = = itC71T~um)-I, it(7rr;,J1, has increased, increased, for for some some reason reason or or other, other, in in aa thin current has thin current << rr,,,. Then the energy density density jjE E and and gas gas column of of radius radius ro « the released released energy column sum ' Then will also also increase. increase. The The gas gas density density N will will become become smaller smaller temperature T will temperature N
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of a leader process The principles of
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and and E /I N larger larger due due to to thermal thermal expansion. expansion. As As the the ionization ionization frequency frequency is is aa steep function function of of reduced reduced field, field, it it will grow much much faster faster than than E / N. N . So, So, the steep will grow the and conductivity conductivity 0"a '" nn,/N will rise. As a result of this electron density density nnee and electron N will rise. As a result of this / e long long chain chain of of cause-effect cause-effect relationships, relationships, the the current current density density j == O"E aE in in the the fluctuation region region will will become still larger, larger, etc. etc. The The process with fluctuation become still process may may begin begin with any the chain. particular fluctuaany link link in in the chain. In In any any case, case, the the current current density density in in aa particular fluctuation will be rising without without limit limit until all current current itit accumulates accumulates there. there. tion region region will be rising until all At the the initial initial stage stage of of instability instability development, development, the the perturbed current density density At perturbed current does within aa does not not exceed exceed much much an an average average value. value. But But as as current current concentrates concentrates within small cross cross section section 7fr~, the gas heating heating rate rate there there rises rises sharply. sharply. The The instability instability small the gas now now develops develops very very quickly, quickly, acquiring acquiring an an explosion-like explosion-like character. character. The The manifested better better in thinner column with high acceleration acceleration effect effect is is manifested in aa thinner column with high density density current. current. A perturbation perturbation region, region, however, however, cannot be infinitely this sets A cannot be infinitely thin, thin, and and this sets aa limit to the rate of limit to the rate of instability instability development. development. The The matter matter is is that that nonnonuniformities of electron electron density density nnee are are dispersed dispersed by by diffusion, diffusion, which which is is uniformities of ambipolar at at very very high high density density values. The characteristic characteristic time time for for perturbaperturbaambipolar values. The tion dispersion dispersion is is Tamb Tamb = = ri/4Da, where D a, == p+Te is an an ambipolar ambipolar diffusion diffusion tion r~/4Da, where /-l+Te is (p+ is is ion ion mobility mobility and and T T,e is is electron electron temperature temperature in in volts). volts). In In coefficient (/-l+ coefficient addition to to charge charge diffusion, diffusion, non-uniformity non-uniformity dispersion dispersion is is due due to to heat heat addition conduction with with aa characteristic characteristic time time Tth Tth = = r~/4"" r i / 4 ~where , K is is thermal thermal diffusivdiffusivconduction where", ity. mechanism appears plasma, ity. The The former former mechanism appears to to be be more more effective effective in in initial initial air air plasma, 2 2 since Da :::::; /s (/-l+ /s. V, V, T since D, x 4 cm cm2/s (p+ :::::; x 2 2 cm cm2/s T,e :::::; GZ 2 2 eV) eV) is is an an order order of of magnitude magnitude 2 larger /s. If x 0.3 0.3 cm cm2/s. If aa non-uniformity non-uniformity takes takes less less time time for for dispersion dispersion larger than", than K :::::; than for for development, development, i.e., i.e., if if Tamb Tamb is is smaller smaller than the instability instability lifetime lifetime Tins' qns, than than the the the latter latter is is suppressed suppressed at at its its origin. origin. The scale scale for for Tins qns is is the the characteristic characteristic time time of, of, say, say, gas gas temperature temperature The doubling perturbed plasma plasma column, temperadoubling in in aa perturbed column, as as compared compared with with initial initial temperature To. To. This This time time is is pocpTo/jE, where j E is is the the power of Joule Joule heat heat release release ture pocpTo/jE, wherejE power of and ccp is specific heat at constant pressure with the account of thermal and is specific heat at constant pressure with the account of thermal p expansion. But But this this is is not not all. all. The The higher higher the the instability instability development development rate expansion. rate is, the steepness the ionization is, the the greater greater is is the steepness of of the ionization frequency frequency dependence dependence on on reduced field ET, reduced field E / N '" E T , i.e., i.e., on on gas gas temperature. temperature. For For instance, instance, if if aa 10% 10% increase in in T raises ionization rate rate by by 20%, 20%, the the instability instability will, will, generally, generally, increase raises the the ionization double rate. This double its its rate, rate, as as compared compared with with aa 10% 10% increase increase in in the the ionization ionization rate. This circumstance brings the the theoretical Ci == dd In In vii vilddIn In (E ( E//N) N ) into into the theoretical circumstance brings the factor factor Vi formula for for Tins qns[26], [26], which which characterizes characterizes the the qVi ((E/ E /N) N ) function function steepness. steepness. formula This This yields yields the the following following expression expression to to be be used used for for estimations: estimations: N
mi,
e
-
T; lnS
:::::;
Poc pTo = Poc pToO" vijE vij2
(2.37) (2.37)
x 1A 1 A and and conductivconductivFor calculations, calculations, we shall take take laboratory laboratory leader leader current current i :::::; For we shall 14 3 2 ity (n to the cmof a x 10lop2 (R cm)-l cm)-’ corresponding corresponding to the electron electron density density nnee :::::; GZ 10 1014 cmp3 of ity 0":::::; air Vi :::::; air ionization ionization by by aa streamer streamer zone; zone; Ci = 2.5. 2.5. Suppose Suppose the the current current density density in in aa
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The streamer-leader process in a long spark The streamer-leader
perturbation region region is ~ 40 A/cm2 , i.e., the average perturbation is j x 40A/cm2, i.e., somewhat somewhat higher higher than than the average 2 value of of 30 30A/cm2 along aa channel channel with with the the initial initial radius to be be value A/cm along radius taken taken to rsum= = 0.1 0.1 cm. cm. We We shall shall obtain obtain Tins T , , ~x 1O--66 s. s. From From the condition Tamb Ta,b 2 -qTins, 'sum ~ 10-the condition ~ Tins' under which the the instability instability has has aa chance chance to to develop develop further, further, we find that the under which we find that the initial radius radius of of aa column column with with accumulated accumulated leader leader current current must must exceed exceed initial '0 ~3 the upper ~ ro min min M 3 Xx 10lO-33 cm. cm. Taking Taking into into account account the upper limit limit '0 ro max max x 2 5 xX 10we conclude 1O-2 cm cm derived derived from from energy energy considerations, considerations, we conclude that that aa probable probable 2 leader prior to to thermal cm. lop2 cm. For For details, details, the the leader radius radius prior thermal expansion expansion is is about,o about ro '" 10reader is referred to [4], [4],but reservation should should be concerning the result reader is referred to but reservation be made made concerning the result the art. accuracy, be too accuracy, which which cannot cannot be too high high in in the the present present state state of of the art.
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2.3.4 2.3.4
Leader velocity
Streamers the leader Streamers generated generated at at the leader channel channel front front cover cover aa distance distance of of several several metres and and stop. stop. As As was in section section 2.3.2, 2.3.2, such such streamers streamers are are metres was mentioned mentioned in weak and and their propagation is is slow; slow; their velocity is is close close to to its its low low limit limit weak their propagation their velocity of mis, which their lifetime Vs ~ V,s ~ M 105 105m/s, which means means that that their lifetime is is Rj R/Vs x 10lO-55 s. s. This This of V time is so so long long that that the the streamer streamer plasma plasma decays decays considerably. considerably. Only Only young young time is 7 streamers, s streamers, whose whose lifetime lifetime is is about about the the electron electron attachment attachment time time Tra x 10lO-’s a ~ (section preserve good (section 2.2.5), 2.2.5), can can preserve good conductivity. conductivity. A A young young streamer streamer length length is is ItIt M Vs7aa ~ M I1 cm. cm. A dense fan fan of of such such plasma plasma conductors conductors starts starts from from the the ~ VsT A dense registered in channel channel front. front. It It is is this this young young streamer streamer fan fan that that seems seems to to be be registered in photographs as aa bright spot with radius of't of r, '" 1cm 1 cm in in order order of of magnitude magnitude photographs as bright spot with aa radius tip. (figure (figure 2.11) 2.11) and and is is generally generally considered considered as as aa leader leader tip.
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2.11. An An instantaneous instantaneous photograph (0.1 Ils ps exposure) exposure) of of the tip region region of of aa Figure 2.11. photograph (0.1 the tip leader. leader.
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The streamer zone and cover The
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This suggestion suggestion is is supported supported by the fact fact that that the the radius radius of of aa leader leader This by the travelling K is cm [20]. travelling through through air air pre-heated pre-heated to to 900 900K is rr,t ::::::: RZ 10 lOcm [20]. Indeed, Indeed, the the plasma decay slows slows down down and and the the values values of of Tr,a and and It become higher. become higher. plasma decay Thus, necessary condition the tip tip region region Thus, aa necessary condition for for leader leader propagation propagation is is the contraction to very small small radius. radius. This This results results from from instability instability development, development, contraction to aa very taking aa time of about about Tins' qns. Over Over this time, all all short short young streamers taking time of this time, young streamers supplying the leader transform to the leader supplying the leader with with current current transform to the leader channel. channel. Therefore, Therefore, over the time time of providing aa steady propagation of the leader tip, over the of the the process process providing steady propagation of the leader tip, the latter must cover young streamer the latter must cover aa distance distance of of about about its its size, size, i.e., i.e., aa young streamer length. length. Only in in this this case case can can aa new front region be formed formed to to replace replace the the old old one. one. Only new front region be Hence, be evaluated respective parameters parameters as Hence, the the leader leader velocity velocity can can be evaluated from from the the respective as (2.38)
In the the absence absence of of attachment attachment or or if if its its rate rate is is low, low, the the role role of of T7,a is is performed by In performed by the time time of of another another plasma decay process process -- recombination. recombination. The The evaluation evaluation the plasma decay with (2.36) velocity of with (2.36) gives gives aa correct correct order order of of magnitude magnitude for for the the velocity of laboratory laboratory 6 4 "-' 10ss and m/s. We leaders: leaders: at at Tins rins lop6 and ItIt "-' 1cm, 1 cm, we we obtain obtain VL, "-' 10 104m/s. We should should like to note these qualitative probably, questionable like to note that that these qualitative and, and, probably, questionable considerations considerations have not yet been substantiated by aa more more rigorous have not yet been substantiated by rigorous treatment. treatment. Some of of the the above above problems problems of of the the leader leader process discussed in in Some process will will be be discussed more the subsequent this chapter more detail detail in in the subsequent sections sections of of this chapter and and further. further. Here, Here, propagation of our our aim aim was was only only to to give give aa general general idea idea of of the the propagation of aa long long spark spark and, presumably, of the first and, presumably, of the first lightning lightning component. component. A reader reader interested interested exclusively may find this information exclusively in in lightning lightning hazards hazards may find this information sufficient. sufficient. N
2.4
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The streamer zone and and cover
We have have shown shown above above that that aa streamer streamer zone zone plays role in in aa leader leader We plays the the key key role process. It here that process. It is is here that aa space space charge charge cover cover is is formed formed which which stabilizes stabilizes the the leader channel, channel, preventing its ionization ionization expansion expansion which otherwise leader preventing its which would would otherwise exclude plasma heating. heating. A streamer the site exclude plasma streamer zone zone is is the site of of current current generation generation for providing its this section, we shall for heating heating the the leader, leader, providing its long long life. life. In In this section, we shall deal, deal, in some some detail, detail, with with processes processes occurring occurring in in the streamer zone zone and and leader leader in the streamer cover, priorities in relationships among cover, defining defining the the priorities in the the causative causative relationships among leader leader parameters. We parameters. We shall shall show show how how the the process process of of streamer streamer generation generation from from aa leader leader tip tip becomes becomes automatic. automatic. field in a streamer zone 2.4.1 2.4.1 Charge and field The tip of possesses aa very The tip of aa long long leader leader possesses very high high potential: potential: U U,t "-' 1MV 1 MV for for laboratory probably, more more for -1010 MV MV or, or, probably, for lightning. lightning. Streamers Streamers laboratory sparks sparks and and "-' are produced in tip, which which means the field are continuously continuously produced in aa leader leader tip, means that that the field at at its its surface the ionization Ett exceeds exceeds the ionization threshold threshold E Ei z 30 30 kV/cm kV/cm (under (under normal normal surface E j ::::::: N
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The streamer-leader streamer-leader process in a long spark
conditions). would conditions). This This excess excess cannot cannot be be very very large, large, otherwise otherwise aa streamer streamer flux flux would become too become too intensive. intensive. The The excessive excessive charge charge of of the the same same sign sign as as V Ut, introduced introduced into would create would reduce into the the space space would create aa much much stronger stronger reverse reverse field field which which would reduce Et to aa level level close close to to E,. Therefore, the field E, does not not exceed exceed much much E, and Et to Ei . Therefore, the field Et does E i and has the the same same order order of of magnitude. magnitude. An An automatic automatic field field stabilization stabilization is is inherent inherent has in by an in any any continuous continuous threshold threshold process process of of charge charge generation generation by an electrode, electrode, for for example, example, in in aa steady-state steady-state corona. corona. Measurements Measurements have have shown shown that that the the field field near aa corona-forming corona-forming electrode electrode is is stabilized stabilized with with high high accuracy accuracy and and does does near not not respond respond to to voltage voltage rise rise across across the the gap; gap; what what changes changes is is the the corona corona intensity, the site intensity, i.e., i.e., its its current. current. A A leader leader tip, tip, too, too, is is the site of of corona corona formation, formation, with high enough with an an intensity intensity high enough to to support support aa quasi-stationary quasi-stationary state state in in the the tip tip and The field at the corona . and streamer streamer zone, zone, corresponding corresponding to to potential potential V U,. The field at the corona t electrode to be be by by aa factor electrode E E,,in is is shown shown by by stationary stationary corona corona experiments experiments to factor of than Ej, tip radius of 1.5 1.5 higher higher than E,, if if the the electrode electrode radius radius is is about about the the leader leader tip radius r - 1lcm. r'" cm. At Et == E : :; 1cm, E,,in :::::;; =850kV/cm 50 kV/cm and and rY,t := 1 cm, the the leader leader tip tip charge charge qt q, == At E, 47l'cor~ E t :::::;; C is portion of 4mO~:Et % 5 5 xX 10lO-'C is capable capable of of creating creating only only aa small small portion of E,r, = 50 50kV of an an actually actually megavolt megavolt potential potential V U,. The main potential Etf t = kV of . The main potential t source source is, is, therefore, therefore, the the space space charge charge of of the the streamer streamer zone zone and and cover cover surrounding the tip. by charactersurrounding the tip. But But the the value value of of V U,t is is primarily primarily determined determined by characteristics external external relative the tip. This is is the the electrode electrode (anode) (anode) potential potential minus minus istics relative to to the tip. This the voltage voltage drop drop across across the the leader leader channel. channel. Consequently, Consequently, the the charge charge Qs Q, and and the the parameters, are the size size R of of aa streamer streamer zone, zone, as as well well as as respective respective cover cover parameters, are established the proper proper potential potential V The . established such such that that they they correspond correspond to to the U,. The t Q, and and R are are directly directly mechanism by by which leader 'chooses' 'chooses' the the values values of of Qs mechanism which aa leader related to streamer properties. There are are many many streamers streamers present present in in the the related to streamer properties. There zone time. They by the tip at zone at at every every moment moment of of time. They are are emitted emitted by the tip at aa high high frequency frequency (see below), have different different lengths lengths at at any any given given moment moment and and are are at at different different (see below), have stages with their their charges up the the zone stages of of evolution, evolution, with charges filling filling up zone space. space. Every Every single single streamer moves moves in in aa self-consistent self-consistent field field created created by whole combination combination of of streamer by the the whole streamers. The The contribution contribution of of the the leader leader tip itself (or (or of of its its channel) channel) to to the the streamers. tip itself total field field has been shown shown to small. One One exception exception is is the the region total has just just been to be be small. region around the tip with size of of its its radius. radius. around the tip with aa size There are are experimental experimental and and theoretical grounds to to believe believe that that the the field field There theoretical grounds strength in in the the streamer streamer zone, zone, except except for for the the tip tip vicinity, is more more or or less less strength vicinity, is constant and and close close to to the the minimum at which streamers can can grow. grow. This This is is constant minimum at which streamers indicated by measurements which does not vary indicated by measurements of of streamer streamer velocity, velocity, which does not vary along along the streamer the streamer zone. zone. (Attempts (Attempts to to measure measure aa single single streamer streamer in in the the tip tip region, where the high, have region, where the streamer streamer density density is is high, have so so far far failed.) failed.) Experiments Experiments of aa laboratory laboratory leader leader touches the opposite opposite show that that until until the the streamer streamer zone zone of show touches the velocity of electrode, electrode, streamers streamers move move slowly, slowly, at at aa nearly nearly limit limit velocity of about about 10 lo55 m/s. mjs. mn (section (section 2.2.7). 2.2.7).Streamers Streamers This is is possible possible only only in in aa uniform field close close to to Eo This uniform field Eo min can travel travel for for such such aa distance distance R, at which whch the the field field Eo mn still still exists, exists, but but they they can R, at Eo min ml,. stop on on entering entering the the region with E < E,, stop region with Eo min'
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Suppose, the streamer hemisphere with Suppose, for for simplicity, simplicity, that that the streamer zone zone is is aa hemisphere with the the centre tip. The hemisphere changes centre in in the the leader leader tip. The hemisphere changes to to aa cylindrical cylindrical cover cover of of the the same radius R with with the the same same order order of of the the space-charge space-charge density. density. A A thin thin conconsame radius ducting leader leader channel channel goes goes along along the cylinder axis axis as as far far as as the the hemisphere hemisphere ducting the cylinder centre. the zone centre. When When evaluating evaluating the zone parameters, parameters, one one should should take take into into account account the cover the cover charge charge at at the the leader leader end, end, which which also also affects affects the the zone zone field. field. We We can can do this this simply simply by connecting the hemisphere, simulating simulating aa streamer streamer zone, zone, to to do by connecting the hemisphere, another hemisphere hemisphere by by mentally mentally cutting cutting it it out out of of the cover space. space. Let Let us us another the cover assume that that there there is is aa uniform radial field field E = = Eomi, in the the sphere. sphere. As was was assume uniform radial Eo min in mentioned in section section 2.2.7, the the theoretical theoretical limit limit of of Eo is close close to to the the experiexperimentioned in Eo min is mental critical average average field field in in the the streamer streamer channel, channel, below streamer mental critical below which which aa streamer cannot ~ 5kV/cm. cannot propagate. propagate. For For air, air, therefore, therefore, we we have have E ~ xE E,,er x 5 kV/cm. A uniuniform field in sphere geometry corresponds to the space charge density form field in sphere geometry corresponds to the space charge density p= = 2c 2 ~o& / r . If If the the leader leader tip tip is is far far from from the the earth earth and and grounded grounded electrodes, electrodes, Elr. its potential potential is is its (2.39) (2.39)
The sphere sphere charge charge Q Q and and its its surface surface potential UR are The potential V R are
Q=
1:
47Tr 2 pdr = 47Tc oR 2 E = 27Tc oRVt
(2.40) (2.40)
U RR = =V U,t -- E R == V Ut/2. V ER t /2.
For m,, the For example, example, for for V Utt == 1.5 1.5 MV, MV, we we have have R == 1.5 lSm the charge charge of of aa hemihemispherical streamer streamer zone zone equal equal to to Qs Q, == QI2 Q / 2 == 6.2 x 10- 5 C. The The leader leader tip tip spherical 3 charge qt = Q(EtIE)(rtIR?,...., 1OQ is indeed negligible, as compared = Q ( E , / E ) ( r , / R ) 2 lOP3Q is indeed negligible, as compared charge with physical role, the with the the zone zone charge. charge. Its Its physical role, however, however, is is very very important: important: the high Et > E high field field it it creates creates near near the the tip, tip, Et Eij »E >> E,,,en isis capable capable of of generating generating streamers. streamers. As plane, its As the the streamer streamer zone zone approaches approaches the the grounded grounded plane, its length length increases because its boundary potential potential V decreases under the action of increases because its boundary UR decreases under the action of R charge Now it most of charge of of opposite opposite polarity polarity induced induced in in the the earth. earth. Now it is is most of the the voltage voltage V which drops the U,, rather than than its its half, half, which drops across across the the streamer streamer space. space. At At the t , rather moment with the = 0 and the moment of of streamer streamer contact contact with the 'earth', 'earth', potential potential V UR = 0 and the R zone Ls = E is relative to zone length length L, = Vtl U,/E is doubled doubled relative to the the value value of of R from from formula formula (2.39). (2.39). This This is is clearly clearly seen seen in in streak streak pictures pictures of of aa laboratory laboratory spark spark (figure (figure 2.12). with aa grounded plane that that 2.12). It It is is at at the the moment moment of of streamer streamer contact contact with grounded plane the ~ 5 kV/cm the critical critical field field E E,,er x kV/cm was was registered registered experimentally. experimentally. The The measuremeasurements ments make make sense sense only only for for short short (compared (compared with with the the interelectrode interelectrode distance) distance) the channel leaders, voltage drop leaders, when when the the voltage drop across across the channel could could be be neglected. neglected. By By equating potentials of we can equating the the potentials of the the anode anode Va U , and and of of the the tip, tip, we can write: write: E,, U,/L, U,/L,. Eer == Vtl L s ~ VaiLs·
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c.
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The streamer-leader The streamer-leader process in a long spark
Figure 2.12. Streak positive leader rodStreak photograph photograph of of the the positive leader channel channel bottom bottom in in an an air air rod-
of 12 12 m m in in length. length. The The streamer streamer zone zone is is seen seen to to elongate elongate when when approaching approaching plane gap gap of plane the plane cathode. the plane cathode.
2.4.2 Streamer Streamer frequency frequency and number
The number of present in The number of streamers streamers present in aa streamer streamer zone zone at at every every moment moment of of time time is N s, == Qs/qs, Q,/q,, where q, is is the the average average charge charge of of aa streamer. streamer. Both Both charges charges where qs is were [29,30], the the first first from from the the integral integral of of conconwere measured measured experimentally experimentally [29,30], duction through the with as duction current current through the anode anode for for short short leaders leaders with as yet yet small small cover cover of current current through through aa cathode cathode charge and and the the second second from from the the integral integral of charge measurement radius that measurement cell cell with with such such aa small small radius that only only one one streamer streamer could could touch After aa successful touch its its surface surface (with (with good good luck). luck). After successful contact, contact, the the charge charge of of aa conductive through an conductive streamer streamer section section flew flew into into the the cathode cathode and and through an integratintegrating was found be ing circuit. circuit. The The charge charge averaged averaged over over many many registrations registrations was found to to be qs previous section q, = =5 X x 10lo-’’10 C. C. For For the the illustration illustration mentioned mentioned in in the the previous section of of this number of this chapter, chapter, we we find find that that the the number of streamers streamers in in aa streamer streamer zone zone of of length m and ~ 1.2 X 105. Q, == 6.2 6.2 Xx 10lO-55 C C is is N s, M 1.2 x lo5. Similar Similar data data length 1.5 1.5m and charge charge Qs qs can can be derived from from the the calculations calculations presented in figure figure 2.8 2.8 or or from from for qs for be derived presented in aa simple just perform. simple theoretical theoretical treatment, treatment, we we shall shall just perform. A streamer streamer produced leader tip tip crosses crosses the the streamer streamer zone zone over over the the produced by by aa leader 5 = R/ V,s min lop5 s, which which is is by by two orders of of magnitude magnitude larger larger than than time tt,s = time R/ V min M ~ 10s, two orders 7 the s. Therefore, M 10lO-’s. Therefore, all all electrons electrons are are lost lost from from older older the attachment attachment time time T7,a ~ streamer portions comparable R.. Conductivity preserved Conductivity is is preserved streamer portions comparable in in length length with with R Copyright © 2000 IOP Publishing Ltd.
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The streamer zone and cover
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only along along the the length length IsI, "-' Vs, min m i nT ~a a"-' 1cm 1 cm behind streamer tip. During only behind the the streamer tip. During the motion motion of the streamer portions the of the streamer tip, tip, the the charge charge of of the the older older streamer streamer portions flows into into the the new new ones ones located located closer closer to to the the tip tip before conductivity turns turns flows before conductivity to zero. Under Under steady-state steady-state conditions, conditions, when when the the tip tip goes goes far far away away from from to zero. the start, start, the the charged charged portion portion of of length length IsI, moves moves together together with with the the tip, tip, the supporting by its potential over supporting by its charge charge the the minimum minimum excess excess of of tip tip potential over external external potential, x 5kV, 5 kV, necessary necessary for for the the streamer streamer propagation propagation (section (section (~Ul)min::::; potential, (AU,),, 2.2.7). The The conductor conductor of of length length IsI, and and radius radius rr,m ::::; M 0.1 0.1 cm cm carries carries the the charge charge 2.2.7). 21[EO(~UI)min/s "-' 10-9 C qs "-' ~ ~.
(2.41)) (2.41
In (lsi rm)
As for As for the the charge charge accumulated accumulated in in the the streamer streamer tip tip
qst ::::; z 21[fO(~UI)minrm 2 r ~ O ( A U ! ) ~ ::::; M i ~7rfo(~UI)~inl mO(AU,)iin/Em r~ M 5 5 Xx 10lo-"11 C, C, qst Em::::;
which taken into into account account in in the the calculation calculation or or (2.39), (2.39), it it is is by one not taken by one which was was not 5 order , the order less less than than that that distributed distributed along along the the channel. channel. At At N s, ::::; M 10 lo5, the average average interstreamer space space is is R/N;I3, i.e., about about several several centimetres. centimetres. With With this this large large RI Ni 13 , i.e., interstreamer separation of of streamer streamer tips, tips, the the streamers streamers can can really really be considered solitary solitary separation be considered and propagating in and propagating in an an average average self-consistent self-consistent field. field. When reaches the the end When aa streamer streamer reaches end of of the the streamer streamer zone, zone, it it stops stops because because it it enters aa field field lower lower than than Eo Since the the streamer streamer zone zone approaches approaches this this field field Eo ,in. min' Since enters fairly fairly slowly, slowly, at at leader leader velocity velocity V VL an order order of of magnitude magnitude lower lower than than L an streamer velocity velocity Vs,min (for (for laboratory laboratory streamers), streamers), the the streamer streamer loses loses its its streamer conductivity conductivity entirely. entirely. The The ions ions of of its its space space charge charge are are gradually gradually repelled repelled (and near the so that that the the streamer streamer (and diffuse), diffuse), reducing reducing the the field field near the charge charge trace, trace, so trace becomes lifeless by for trace becomes lifeless 'forever'. 'forever'. The The streamer streamer zone zone still still passes passes by for aa time time 4 tL RI VL, "-' 10s, trace, which which is tL = = R/ lop4 s, after after which which the the immobile immobile charged charged trace, is now now behind the tip, becomes becomes aa cover Viable streamers behind the leader leader tip, cover component. component. Viable streamers fly fly 5 across the the streamer streamer zone zone over over time time t s, == R m i n"-' 10lop5 s, an an order order of of across RI/ VVsS min s, magnitude production is magnitude shorter. shorter. Therefore, Therefore, if if the the frequency frequency of of streamer streamer production is the number of viable viable streamers streamers in in aa streamer streamer zone zone is is N 1I "-' vst v,t,, while vv,, number of s' while s , the the number of practically coinciding the number of non-viable non-viable traces, traces, practically coinciding with with the the total total number of N,,-, vst number of charged charged streamer streamer portions, portions, is is N v,tL. Hence, the the streamer streamer L. Hence, generation NVL! generation frequency frequency is is Vv,s "-' N /ItL t L "-' N V L /RR "-' 10 lo99 S-I. s-'. N
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2.4.3
--
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Leader tip current current
The related to The streamer streamer production production frequency frequency is is directly directly related to the the leader leader tip tip current: current: 9 9 itit = With qs = 10C and V = 10 S-I, we get it ::::; 1 A, a value typical ' = qsV qp,. With 9, = lop9 C and v, = lo9 s-l, we get it x 1 A, a value typical s s for while the yet for laboratory laboratory leaders leaders in in the the initial initial stage stage while the streamer streamer zone zone has has not not yet reached value has has been been registered reached the the cathode. cathode. This T h s current current value registered in in many many experiexperiments [27-29], [27-291, and and the the relation relation Vv,s == iLlqs i L / q s has has been confirmed by direct ments been confirmed by direct measurements. piece from measurements. The The streamers streamers were were counted counted by by piece from current current impulses impulses in in small small cathode cathode measurement measurement cells cells after after the the streamer streamer zone zone boundary boundary had had
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The streames-leader streamer-leader process in a long spark
touched touched its its surface surface [29]. [29]. The The counts counts were were integrated integrated over over the the area. area. MeasureMeasurements relation ments made made at at different different currents currents showed showed that that the the relation iL!v lO~lO C remained remained constant. iL/v,s == qs qs :::::: RZ 5 5 x 10-”C constant. It It is is consistent consistent with with measuremeasurements presented here. ments and and the the estimations estimations of of average average streamer streamer charge charge presented here. The The formula formula for for leader leader tip tip current current can can be be given given in in aa conventional conventional form form of of 2 the type Jrr2 ene Ve = when eneve = TTe V ~e,Vwhen ~ , current current is is expressed expressed as as the the number number of of the type i == 7rr charge per unit unit current and electron electron charge carriers carriers (electrons) (electrons) per current column column length length TT,e and . velocity V This formula can also be changed to the phenomenological V,. This formula can also be changed to the phenomenological velocity e expression the result result of expression (2.36) (2.36) describing describing the of the the current current process process without without indicating nature of indicating the the nature of carriers. carriers. Although Although the the current current carriers carriers in in this this case case are we can are electrons, electrons, we can also also speak speak of of 'macroscopic' ‘macroscopic’ carrier-moving carrier-moving charged charged streamer streamer sections. sections. With With what what we we mentioned mentioned at at the the end end of of section section 2.3.2 2.3.2 and and formula have formula (2.36), (2.36), we we have
(2.42) (2.42) Q J R :::::: RZ 1rc mO Ut t is is the linear charge charge in in aa streamer streamer zone zone and and where TT,s == Qs/ the linear where OU T~ = = qsN) q , N 1/ /RR is is the the linear linear charge charge of of 'macroscopic' ‘macroscopic’ carriers. carriers. The The three three T1 expressions for for current current are are equivalent equivalent to to one one another another but but reflect reflect different different expressions aspects process. The aspects of of the the current current process. The second second expression expression in in the the chain chain of of equalities equalities (2.42) (2.42) indicates indicates the the current current origin origin while whle the the latter latter shows shows the the actual actual process of of charge charge transport; transport; the the penultimate penultimate expression expression is is phenomenological phenomenological process and and describes describes the the result result of of travel travel of of the the streamer streamer zone zone as as aa whole. whole. The mechanism mechanism ofleader of leader current current production production just described is is valid valid until until The just described aa streamer polarity (the streamer zone zone touches touches the the electrode electrode of of opposite opposite polarity (the earth earth or or aa grounded grounded object object in in the the case case of of lightning). lightning). Then Then the the situation situation changes changes radically. jump, the radically. In In the the final final jump, the charges charges of of all all streamers streamers 'hitting' ‘hitting’ the the electrode electrode leave through its Nonviable streamers no longer leave the the gap gap through its surface. surface. Nonviable streamers are are no longer produced, and possess an portion of produced, and they they possess an ever-decreasing ever-decreasing portion of total total streamer streamer zone this case, zone charge. charge. The The last last expression expression in in (2.42) (2.42) is is valid valid in in this case, too, too, = T1 T~VS, min, mln, but but the value of of T) T~ is is no no longer longer equal equal to the portion portion VL! VL/ Vs, mln itit = the value to the min of the streamer zone zone charge charge Qs' Q,. In In the the limit, limit, when when the the leader leader tip of the total total streamer tip reaches reaches so the the the electrode, electrode, all all zone zone charge charge will will be provided by moving streamers, streamers, so the be provided by moving = TT,V,. The linear linear charge charge of of the the streamer streamer zone zone in in the the final final current will will be current be it :::::: s V s ' The jump remains the same in in the the order order of of magnitude magnitude as as in in the initial stage. stage. Of Of jump remains the same the initial course, it is is elongated, elongated, looking looking course, the the streamer streamer zone zone has has now now aa different different shape shape -- it more than aa hemisphere. hemisphere. But L,s and and more like like aa cylinder cylinder than But still, still, its its longitudinal longitudinal L transverse R s, dimensions dimensions are are comparable, comparable, and and the the value value of of In ln(L,/R,) transverse (Lsi R s ) which appears appears in in the the denominator denominator of of the the expression expression for for linear linear capacitance capacitance which in is about about unity. unity. Therefore, Therefore, leader leader curcurin cylindrical cylindrical geometry geometry (2.8) (2.8) and and (2.35) (2.35) is rents to the jump and before it rents after after the the transition transition to the final final jump and before it are are related related as as Vss/ /V VL, L, the ratio ratio being being above above 10. 10. the Moreover, the the streamer streamer velocity velocity rapidly rapidly increases increases as as the the streamer streamer zone zone is is Moreover, reduced. The The potential difference between the leader leader tip and the the grounded grounded reduced. potential difference between the tip and
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13 73
electrode remains remains the same, U U,, whereas the the length length L, becomes shorter. shorter. The The electrode the same, L s becomes t , whereas average field field in in the the streamer streamer zone zone E = =U U,/L, rises, together with streamer average L rises, together with streamer / t s velocity (section 2.2.7) 2.2.7) and and leader leader current current it'" it V V,. The final final jump current velocity (section jump current s ' The was shown shown by laboratory leader leader experiments experiments to to rise rise by by aa factor factor of of tens or was by laboratory tens or 2 _10 3 A. A. This hundreds, hundreds, from from about about 1A 1 A to to 10 102-103 This fast fast current current rise rise lasting lasting for for several prelude to to aa still the return return several microseconds microseconds is is aa prelude still higher higher current current of of the stroke. stroke. The The latter latter begins begins when when the the leader leader channel channel reaches reaches the the electrode. electrode. The jump is by the The current current rise rise of of the the final final jump is stimulated stimulated by the fact fact that that fast fast streamers streamers cross aa shorter shorter streamer streamer zone zone much much faster faster than than before, before, so so that that the the zone zone cross plasma is unable decay as as much, much, preserving the streamer streamer conductivity. conductivity. plasma is unable to to decay preserving the At the jump, the the leader be linked the At the end end of of the the final final jump, leader channel channel appears appears to to be linked to to the opposite opposite electrode electrode by by numerous numerous streamer streamer filaments filaments with with current current (for (for details, details, see see [4]). [4]). N
2.4.4 Ionization processes processes in the cover 2.4.4 leader cover cover contains contains aa large large number number of of non-viable non-viable charged charged traces traces of of A leader earlier produced and when aa streamer earlier streamers. streamers. They They were were produced and developed developed when streamer zone passing through through this very low, zone was was passing this site. site. The The axial axial field field in in the the cover cover is is very low, much No ionization ionization can can occur occur in in it, it, so so it it much lower lower than than in in the the streamer streamer zone. zone. No is no interest us. What by the is of of no interest to to us. What is is important important is is the the radial radial field field created created by the leader the only leader surface surface charge charge and and all all cover cover charges, charges, as as in in aa streamer streamer zone, zone, the only difference to aa channel however, aa difference being being in in geometry. geometry. In In contrast contrast to channel cover, cover, however, leader with aa streamer leader tip tip with streamer zone zone is is formed formed as as aa self-consistent self-consistent system system from from the very beginning. The the very beginning. The tip tip pumps pumps into into the the zone zone as as much much charge charge as as necessary necessary to the tip the level providing the production of to maintain maintain the tip field field at at the level E Ei, the production of the the in providing necessary ready-made leader channel channel 'inherits' ‘inherits’ aa ready-made necessary number number of of streamers. streamers. A leader cover. cover. The The charge charge amount amount and and distribution distribution in in the the former former quasi-spherical quasi-spherical zone unsuitable for potentials U(x) zone is is unsuitable for the the cylindrical cylindrical geometry geometry and and channel channel potentials U(x) . different But the inherited charge of dead streamer traces is different from from U U,. But the inherited charge of dead streamer traces is t invariable, there must to make invariable, which which means means that that there must be be aa mechanism mechanism to make the the channel-cover the potential channel-cover system system self-consistent self-consistent and and controllable, controllable, since since the potential distribution distribution U(x) U ( x ) and and linear linear leader leader capacitance capacitance vary vary in in time. time. We We mentioned mentioned in in section section 2.3.2 2.3.2 that that the the 'intrinsic' ‘intrinsic’ charge charge of of aa conducting conducting channel radius rL ~ would create and radius x 0.1 0.1 cm cm would create at at its its surface surface aa huge huge channel of of length length LL and radial field field Er, = U /[rL / [ r LIn In (L/rdl ( L / r L ) ]~ = 1MV/cm 1 MV/cm at at channel channel potential potential radial Era = U x ~ 1MV. the presence presence of 1 MV. This This critical critical situation situation is is unfeasible unfeasible owing owing to to the of aa cover. which cover. The The cover cover charge charge induces induces in in the the conductor conductor an an opposite opposite charge charge which is to be be subtracted is to subtracted from from the the intrinsic intrinsic surface surface charge. charge. As aa result, result, the the field field E E,r created by the resultant charge value. It to imagine, created by the resultant charge has has aa moderate moderate value. It is is hard hard to imagine, be as however, however, that that the the inherited inherited charge charge will will be as large large as as is is necessary necessary for for confining confining the narrow range --Eln E in < E, E r < Ein E in with with the the resultant resultant surface surface field field in in the narrow range ,~ = 50 50 kV/cm« kV/cm << El, 1 MV/cm. No doubt, the the cover cover charge charge will will turn turn out out Eiin Era x ~ 1MV/cm. No doubt, to to be be either either too too large large or or too too small. small. In In the the first first case, case, the the channel channel will will be be charged charged
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74 I4
The streamer-leader The streamer-leader process in a long spark
negatively (if positive), the will become negatively (if the the leader leader is is positive), the field field around around it it will become negative, negative, its its E in . A negative corona be excited module module exceeding exceeding Ei,. A negative corona will will be excited and and introduce introduce into into the the cover much charge necessary to reduce IErl E in . This to reduce IE,I to to Ein. This situation situation cover as as much charge as as is is necessary becomes voltage is becomes feasible feasible when when the the gap gap voltage is constant constant or or decreases, decreases, since since then then the the channel channel 'enters' ‘enters’ the the cover cover with with aa too-large too-large charge. charge. Indeed, Indeed, suppose suppose the the is distributed distributed as as p == 2e 2 ~oE~I I'E / r charge of of aa streamer streamer hemisphere hemisphere of of radius charge radius R is (section potential U (section 2.4.1), 2.4.1), creating creating potential U,t == 2ER (2.39) (2.39) in in its its centre. centre. The The cover cover inherits the same inherits aa charge charge of of the same radial radial density density distribution distribution p == 2e 2 ~oEIr ~ E==/ r ~ EUti ~ Rr U ,and and / R amount amount ~ r’' == 27feo 27reoUt unit length. length. At At point far enough enough 2eo T Ut per per unit point x x far L» R,, this this charge from >> R charge will will create create potential potential from the the channel channel ends, ends, L
5”J’
1 JR JL 2 ~ r dr p dz U'(x) = _1_ 27frpdrdz U’(x) =1,2 2 Y7feo 0 0 [r + (z x)2]] 1/2 Y T E O o o [r2+ ( z - x)
r’ L e = U In Le Le ~ Le M ~ln -In= U,t In27feo 2 ~
~R 0
R
(2.43) (2.43)
where natural logarithmic by the where ee is is the the natural logarithmic base. base. The The potential potential U' U’ is is by the logarithmic logarithmic factor than the which has U,. The excess excess cover cover charge charge which has factor larger larger than the actual actual value value of of U t • The created must be be compensated by introU’ -- U U,t and and must compensated by introcreated excessive excessive potential potential is is U' ducing aa charge charge of of opposite opposite sign. sign. ducing The other other situation, situation, when when the the inherited inherited charge charge is is too too small, small, is is usually usually The feasible feasible if if the the gap gap voltage voltage rises rises appreciably appreciably during during the the leader leader evolution. evolution. U ( x ) at at aa given given point increases, and and the the cover cover must must The channel channel potential potential U(x) The point increases, be charged be charged up. up. of direct direct or or reverse corona display display by by aa 'wire', ‘wire’, such such It is is this this mechanism mechanism of It reverse corona as aa leader leader channel, channel, which leads to to aa self-consistent self-consistent channel-cover channel-cover system. system. as which leads The but not The system system is is controlled controlled and and corrected corrected automatically automatically but not very very quickly. quickly. It the slightest potential distribution It is is sensitive sensitive to to the slightest variation variation in in potential distribution along along the the Ei, being too small compared with El., of the ‘intrinchannel due due to to the the field field E being too small compared with Era of the 'intrinchannel in sic’ channel channel charge. charge. A A slight slight effect effect on on the the cover cover is is sufficient sufficient to to change change the the field field sic' of aa developdevelopvalue, and and even even its its direction, direction, at at the the channel channel surface. surface. The The cover cover of value, ing reverse corona ing leader leader with with aa reverse corona acquires acquires aa double-layer double-layer structure: structure: outside outside is is the charge the streamer new charge the charge inherited inherited from from the streamer zone zone and and inside inside is is the the new charge of of 1.5 MV, MV, opposite sign, sign, introduced introduced by by the corona. For For example, example, at at U, opposite the corona. Ut == 1.5 R == 1.6 1.6m, and L == 10 10m, the linear linear cover cover capacitance capacitance from from (2.43) (2.43) is is m, and m, the C u' == 2 xX 10F/m. It by the the same but C1j = = T'l r’/U’ lo-”11 Fjm. It is is defined defined by same formula formula (2.8) (2.8) but with the effective effective radius radius Reff varying with the radial radial charge charge distribution; distribution; for for R etf varying with the with the pM 1/r, Rre^ = RI R / ee and and for for pp(r) (r) = = const, const, Rre^ = RI R / eel1/2 i 2,, etc. etc. With With U = =U U,, ~ III', etf = etf = t, the corrected steady steady state state charge charge in in the the cover cover is is TL rL == C C1 = 3 Xx 10lo-’ 5 Cjm. Cjm. the corrected j U = Since values of C1 and TL rL gradually gradually decrease decrease with with the the leader leader length, length, Since the the values of C j and E,r ~ M -50 -50 kV/cm. kV/cm. the surface surface field field must must support support aa negative negative corona; corona; hence, hence, E the The actual actual channel channel charge charge (intrinsic (intrinsic charge charge minus minus induced induced charge) charge) is is found found The rLc== 27fe ~ ToE r~==E-8.2 -8.2 , IO-’ 7 Cjm« Cjm << TL' rL.Therefore, Therefore, it it is is easy easy to control to be x 10to control to be TLc rLc and and even even to to reverse reverse its its sign. sign. the value value of of TLc the We have have deliberately deliberately considered considered the the mechanism mechanism of of cover-leader cover-leader selfselfWe so much much detail, detail, because corona neutralizes neutralizes the the regulation in in so regulation because aa reverse reverse corona cover return stroke, cover charge charge in in aa laboratory laboratory spark spark and and lightning lightning during during the the return stroke,
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75 75
A long leader channel
when the channel becomes equal potential (see when the channel potential potential becomes equal to to the the earth's earth’s zero zero potential (see section 4.4). 4.4). section
2.5
A long long leader channel channel
All ionization ionization processes for the the leader leader development development are are localized localized in in All processes responsible responsible for the behind it. the streamer streamer zone, zone, leader leader tip tip and and aa short short channel channel section section behind it. In In the the latter, latter, gas heating heating is is completed completed and and aa quasi-stationary quasi-stationary state state characteristic characteristic of of aa long long gas spark is is established. established. In In this t h s sense, sense, the the rest rest of of the the channel channel plays plays aa minor minor role, role, spark simply part of simply connecting connecting the the operating operating part of the the leader leader to to aa high-voltage high-voltage source. source. High High potential potential and and current current vital vital to to the the ionization ionization and and energy energy supply supply are are transmitted through channel. But But how how much much voltage voltage reaches reaches the the leader leader transmitted through the the channel. tip depends which, in tip depends on on the the channel channel conductivity conductivity which, in turn, turn, is is determined determined by by the the channel channel state. state. For For this this reason, reason, what what is is going going on on in in aa developing developing channel channel is is as process as as important important to to the the leader leader process as the the mechanisms mechanisms described described above. above. 2.5.1 2.5.1 Field and the plasma state There the state There are are no no direct direct experimental experimental data data on on the state of of aa lightning lightning leader leader channel. the information channel. Therefore, Therefore, of of special special value value is is the information derived derived from from laboratory laboratory spark spark experiments, experiments, since since it it can can serve serve as as aa starting starting point point in in lightning lightning treatments. treatments. Here with aa Here we we present present some some values values derived derived from from experimental experimental data data [27] [27] with minimum photographs were minimum number number of of assumptions. assumptions. Streak Streak photographs were taken taken continucontinuously propagating from plane. Pulses ously of of aa leader leader propagating from aa rod rod anode anode to to aa grounded grounded plane. Pulses of risetime were were applied of voltage voltage U U,o with with the the microsecond microsecond risetime applied to to gaps gaps of of various various measuring the the photographs photographs at length d. By By measuring the streamer streamer zone zone length length L L,s in in the at length d. the moment the touched the the moment the zone zone touched the grounded grounded electrode electrode and and assuming assuming the the averaverEcr == 4.65 tip age age zone zone field field to to be be E,, 4.65 kV/cm kV/cm (section (section 2.4), 2.4), one one can can find find the the leader leader tip potential U,t == E,,L, and evaluate evaluate the the average average field field in in the the leader leader channel channel as as EcrL s and potential U E L == (U where L == d -- L EL ( Uoo-- Ut)/L, U , ) / L , where L,s is is the the channel channel length length (table (table 2.1). 2.1). The however, is because one The accuracy accuracy of of E EL evaluation, however, is not not high, high, because one L evaluation, calculates between large calculates aa small small difference difference between large values. values. Besides, Besides, measurements measurements of the moment of the the streamer streamer zone zone length length at at the moment of of contact contact with with the the cathode cathode contain When determining contain errors errors as as large large as as those those of of the the channel channel length. length. When determining the latter the latter from from streak streak pictures, pictures, one one can can hardly hardly take take into into account account all all channel channel parameter derived Leader parameter derived from from experimental experimental data. data. Table 2.1. Leader
d,m
55
10 10
15 15
Uo,MV
Ls,m
L,m
Ut,MV
E L , V/cm
1.3 1.3 1.9 1.9 2.2 2.2
2.3 2.3 3.2 3.2 3.6 3.6
2.7 2.7 6.8 6.8 11.4 11.4
l.l 1.1 1.5 1.5 1.7 1.7
750 750 590 590 440 440
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The streamer-leader streamer-leader process in a long spark The
bending, which which may may increase increase the the length length by by 20-30%. 20-30%. Finally, Finally, the the accuracy accuracy of of bendings, E,,cr is is not not as as high high as as is is necessary necessary for for such such aa delicate delicate operation. operation. Experimental Experimental E researchers know know that that the the streamer streamer zone zone field field varies varies with with air air pressure, pressure, researchers humidity, and and temperature temperature but but they they do do not not know know the the respective respective corrections. corrections. humidity, Nevertheless, the the data data in in table table 2.1 2.1 demonstrate demonstrate aa decrease decrease in in the the average average field field Nevertheless, with increasing increasing channel channel length. length. This This is is also also evident evident from from experiments experiments with with with superlong sparks. sparks. A A voltage voltage of of 3-5 3-5 MV MV is is sufficient sufficient to to create create aa spark spark 100 100m m superlong long or or longer. longer. The The tip tip potential potential necessary necessary for for the the development development of of aa streamer streamer long zone of of several several metres metres in in length length is is l-2MV, 1-2 MV, as as is is clear clear from from table table 2.1, 2.1, theretherezone fore the the average average field field in in such such aa long long channel channel will will be be as as low low as as 200-250 200-250 V V/cm. fore jcm. These These values values are are more more applicable applicable to to older, older, remote remote channel channel sections sections which which have have acquired acquired aa quasi-stationary quasi-stationary state, state, but but the the fields fields close close to to the the tip tip are are much much higher. higher. This This follows follows from from many many experiments experiments indicating indicating aa regular regular increase increase in in average field field with with decreasing decreasing leader leader length. length. More More explicitly explicitly this this was was shown shown average by supershort supershort spark spark experiments, experiments, when when the the channel channel length length was was only only aa few few by dozens of of centimetres centimetres [31]. [31]. The The field field in in aa supershort supershort leader leader and, and, therefore, therefore, dozens the field field at at the the respective respective distance distance from from the the tip tip of of aa long long spark spark may may be be 22the 4kVjcm. 4 kV/cm. At At the the site site of of ionization-thermal ionization-thermal instability, instability, where where current current is is accumulated within within aa thin thin column, column, the the field field was was found found (section (section 2.3) 2.3) to to be be accumulated 20kVjcm 20 kV/cm [4]. [4]. But But far far from from the the tip, tip, itit is is nearly nearly two two orders orders of of magnitude magnitude lower. lower. At x 1A, 1 A, its its velocity velocity is is At aa typical typical experimental experimental leader leader current current of of ii ~ VL, ~ cmjlls, and M 1.5-2 1.5-2cm/p, and the the lifetime lifetime of of aa leader leader section section at at aa distance distance of of 33 m m is at at least least l501lS. 150 ps. This This time time is is long long enough enough for for the the relaxation relaxation from the the tip tip is from processes processes in in the the channel channel to to be be nearly nearly completed completed and and for for aa nearly nearly steadysteadystate state to to be be established. established. The The thermal thermal expansion expansion of of the the channel, channel, very very fast fast at at the the beginning, beginning, is is also also completed completed by by that that time. time. Measurements Measurements made made in in aa 10m gap between between aa cone cone anode anode and and aa grounded grounded plane plane [28] [28] (voltage (voltage 1.61.610 m gap 1.8MV, MV, average average current current about about 1A, 1.8 1 A, and and average average leader leader velocity velocity 22 cm/Ils) cm/ps) showed showed that that the the channel channel expansion expansion rate rate was was 100 l00m/s at first first but but 100llS mjs at loops later mjs. Measurements later the the rate rate dropped dropped to to 22m/s. Measurements made made by by the the shadow shadow techtechnique nique showed showed the the average average thermal thermal expansion expansion radius radius to to be be rL rL ~ = 0.1 0.1 em. cm. According According to to spectroscopic spectroscopic measurements, measurements, the the temperature temperature of of aa channel channel which which has has reached reached the the gap gap middle middle is is 5000-6000 5000-6000 K. K. Some Some other other experimental experimental data on on laboratory laboratory leaders leaders can can be be found found in in [4, [4, 27, 27, 28,31,32]. 28, 31, 321. data The M30003000The air air ionization ionization mechanism mechanism changes changes radically radically at at temperatures temperatures TT ~ 6000 K and Vjcm, 6000K and relatively relatively low low reduced reduced fields fields (for (for example, example, at at E == 450 450V/cm, 2 T K, and ).t In T == 5000 5000K, and p == 1atm, 1 atm, we we have have E // N N == 33 Xx 10- 16 Vjcm V/cm2).t In cold cold air, air, We should should like like to to warn warn against against the the commonly commonly used used postulate postulate that that the the field field in in aa leader leader channel channel t We 2 has Some E / N "" 88 x 1O-16Ycm 10-'6VcmZ. Some authors authors use use itit for for the the calculation calculation of of has aa constant constant value value of of E/N SO00K, K, we we have have E = 1.15 1.15kY/cm, kV/cm, breakdown voltages voltages in in air air gaps, gaps, including including long long ones. ones. At At TT = 5000 breakdown for more more or or less less long long leaders leaders (table (table 2.1) 2.1) and and contradicts contradicts which disagrees disagrees with with experimental experimental data data for which consequence of of air air heating heating is is aa the physics. physics. The The underlying underlying implicit implicit suggestion suggestion is is that that the the only consequence the change in in its its density. density. We We shall shall show show that that this this is is not not the the case. case. change
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A long leader channel
77 71
it is is ionization ionization of of O O2 electrons gaining gaining energy energy in in aa strong strong field, field, but but it molecules by by electrons 2 molecules at rate of unexcited oxygen at the the above above value value of of E / N, N , the the ionization ionization rate of unexcited oxygen and and nitronitromostly gen by electron gen molecules molecules and and atoms atoms by electron impact impact is is negligible. negligible. Electrons Electrons are are mostly produced produced in in the the associative associative ionization ionization reaction reaction
+°+
N 2.8eV ---+ + NO+ NO' N + 0 + 2.8eV
+ e.e. +
(2.44) (2.44)
Due to to aa low low ionization ionization potential potential of of NO (9.3eV), the reaction reaction requires requires aa NO (9.3 eV), the Due small activation activation energy energy and and occurs occurs at at aa large large rate rate constant. constant. Recent Recent data data small [33] [33] give give kk,,, = 2.59 2.59 x 1O-17T1.43exp(-31140/T) 10-'7T'.43exp (-31 140/T) cm cm33 S-l, s-'. ias =
T [K] [K]
(2.45) (2.45)
NO ionization by electron with associative Direct Direct NO ionization by electron impact impact may may compete compete with associative ionization but at temperature higher ~ 10 ionization (2.44) (2.44) but at an an electron electron temperature higher than than T T,e M lo44 K. K. Both that thermodynamiBoth estimations estimations and and kinetic kinetic calculations calculations [34] [34] show show that thermodynamically N, 0, NO and ~ 4000cally equilibrium equilibrium concentrations concentrations of of N, 0, NO and electrons electrons at at T M 40006000 6000 K K (table (table 2.2) 2.2) are are established established for for 20-50 20-50 IlS. ps. During cm, i.e., process During this this time, time, aa leader leader elongates elongates only only by by 20-100 20-100cm, i.e., the the process of thermodynamic equilibrium of establishing establishing aa thermodynamic equilibrium in in the the channel channel seems seems to to occur occur concurrently process of concurrently with with the the transitional transitional process of channel channel formation formation and and heating heating to Although the practically to aa quasi-stationary quasi-stationary state. state. Although the electron electron density density does does not not practically 14 differ ~ 10 cm -3, the differ from from the the density density due due to to streamer streamer generation, generation, nnee M 10'4cm-3, the ~ 3.3 higher than ionization 5000, nn,/N 3.3 Xx 10lo-',5 , is is an an order order higher than ionization degree degree at at T == 5000, e/ N M 6 that in . Therefore, that in the the streamers, streamers, nn,/N M 4 4 Xx 10lop6. Therefore, intensive intensive ionization ionization e/ N ~ occurs during during the evolution of of ionization-thermal ionization-thermal instability instability and and subsequent subsequent occurs the evolution heating to heating to the the final final temperature. temperature. Thus, be subdivided two Thus, aa long long laboratory laboratory leader leader channel channel can can be subdivided into into two unequal parts. First, unequal parts. First, there there is is aa relatively relatively short short (about (about 1m) 1 m) transitional transitional porportion just behind the gas tion just behind the the tip tip where where the gas is is gradually gradually heated heated and and additionally additionally ionized. by aa change the plasma plasma density ionized. This This is is accompanied accompanied by change in in the density and and conconductivity. rest of heated to K, ductivity. Second, Second, there there is is the the rest of the the channel channel heated to 5000-6000 5000-6000K, which reached aa quasi-stationary which has has reached quasi-stationary state. state. The The suggestion suggestion of of an an equilibrium equilibrium electron value electron density density in in this this part part of of the the channel channel generally generally leads leads to to aa correct correct value of the measured be of its its radius, radius, rY ~ x 0.13 0.13 cm, cm, close close to to the measured thermal thermal radius radius [28]. [28]. It It can can be Table 2.2. Equilibrium Equilibrium air air composition composition at at p == 11 atm. atm.
T, K T,K N,, 10 N 1 0 ' ~cmc m3- ~ 13 ne, ne, 10 1013cmcm-33 17 No,~10 cm~3 N 10" , cm-3 N 1 0,16' ~cmcm-33 NN,~10 N 1 016~ '~ cm-3 ,3 NNo,~10 cm5 lop5 n ee//N, N , 1018
4000 4000
4500 4500
5000 5000
5500 5500
6000 6000
1.79 1.79 0.63 0.63 4.70 4.70 0.25 0.25 1.62 7.62 0.35 0.35
1.60 1.60 1.70 1.70 4.90 4.90 1.15 1.15 4.54 4.54 1.06 1.06
1.48 1.48 4.90 4.90 4.60 4.60 3.61 3.67 2.73 2.73 3.31 3.31
1.35 1.35 11.2 11.2 4.35 4.35 9.92 9.92 1.67 1.67 8.30 8.30
1.27 1.27 21.4 21.4 3.81 3.81 20.6 20.6 1.03 1.03 16.8 16.8
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streamer-leader process in a long spark The streamer-leader
derived from from the the relation relation for for current current ii == 7rr2eneMeE 7rr2en,p,E if if we electron derived we take take electron 2 22 mobility be Me ~ 1.5 N- 1 cm (V. S)-I, ii ~ V/cm and mobility to to be pe w 1.5 xX 10 1022N-’ cm2(V.s)-‘, w I1 A, A, E ~ M 250 250V/cm and if we we use the values values of of nnee and and N from from table table 2.2, 2.2, corresponding corresponding to to T == 5000 5000 K. K. use the if In reality, reality, with with the and E / N values values characteristic characteristic of of remote remote channel channel In the T and portions, the electron electron temperature temperature T T,e may may differ differ considerably considerably from from the the gas gas portions, the temperature: T,e may may be be as as high high as as 10 l0000K at T = = 5000 5000K. This slightly slightly temperature: T 000 K at K. This shifts the quasi-stationary values values of of nnee relative relative to the thermodynamic thermodynamic equiliequilishifts the quasi-stationary to the brium values corresponding corresponding to to T (as (as in in table table 2.2). 2.2). Stationary Stationary nn,e corresponds corresponds brium values to equality of of forward forward and and reverse in (2.44). (2.44). The The forward forward to the the equality reverse reaction reaction rates rates in reaction rate rate is is independent independent of of T T,, while the reaction rate rate at at reaction the reverse reverse reaction e , while 3 2 4 T ~ 10 K is proportional to the stationary value of will T,e x 104K is proportional to T TL3’*. Hence, the stationary value of nII, e- / . Hence, e will be larger than than that that in in table table 2.2 2.2 by by aa factor factor of of (T ( Tee/T)3/4 / T ) 3 /~4 2. be larger 2. As for for the the less less heated, heated, recent recent channel channel sections, sections, the the difference difference between between As and the electron electron and and gas gas temperatures temperatures is is greater. greater. The The reduced reduced field field E”, / Nand the Te must be higher in in the the unheated unheated channel channel to to provide impact ionization, ionization, T be higher provide impact e must since there there is is no other source source of of electron electron production. production. At temperatures since no other At temperatures 2500K, this is is O O2 ionization by by electron electron impact. impact. As As the the channel channel is is T < 2500 K, this 2 ionization heated further, further, NO ionization requiring requiring lower lower T T,e and and E”, / N begins begins to domiheated NO ionization to dominate, and and only only at at T > 4000-4500 4000-4500KK requiring requiring aa still still lower lower field field does does the nate, the reaction of of (2.44) (2.44) become become important. important. Clearly, Clearly, the the channel channel field field cannot cannot reaction follow the the condition condition E”, / N == const const because because of of the the change change of of ionization ionization follow reactions with different energy energy thresholds. Calculations show show [34] [34] that that the the reactions with different thresholds. Calculations value of E / N drops drops from from 55 to to 1.5 1.5Td heating from from 1000 1000 to to 6000 6000K Td with with heating K value of (figure (figure 2.13). 2.13).
..?
1o4
:/
a u
.... "0
:'E-< "10
~
n
.. ...
........................ ~
. ..
'
o 0
1
22
3 3 4 4 .
55
Temperature, kK Temperature,
6
Parameters of of the the initial initial leader leader channel channel right right behind the tip as aa function function Figure 2.13. Parameters behind the tip as 2 of the gas temperature temperature (model (model calculations calculations of of [34]; [34]; lTd 1 Td == 10lo-’’17 V cm2). of the gas V. cm ).
.
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A long leader leader channel channel
79 I9
2.5.2 Energy balance and similarity to an arc The The older older leader leader portions portions are are similar similar to to an an arc arc in in atmospheric atmospheric air. air. Current Current 1 A and 1A and temperature temperature 5000 5000 K correspond correspond approximately approximately to to the the minimum minimum arc and the equal to only by arc values values and the field field equal to 200-250V/cm 200-250 Vjcm is is only by aa factor factor of of two two or three three stronger stronger than than that that in in aa low low current current arc. arc. So or So it it is is natural natural to to look look at at the the leader leader channel channel as as an an arc arc analogue. analogue. All characteristics characteristics of of aa long long stationary stationary arc arc at at atmospheric atmospheric pressure pressure when when All the plasma plasma is is usually usually quasi-stationary quasi-stationary (maximum (maximum temperature temperature TT,m along along the the the axis, axis, longitudinal longitudinal field field E, E , current current channel channel radius radius ro) ro) are are defined defined only only by by one one 'external' parameter, normally, by current current i. Joule heat released released in the current ‘external’ parameter, normally, by Joule heat in the current channel is carried carried out by heat heat conduction. conduction. Radiation Radiation is for very very channel is out by is essential essential only only for intensive arcs when channel temperature 11 000- 12 000 K. intensive arcs when the the channel temperature exceeds exceeds 11000-12000 K. The The further fate arc cooling cooling providing steadyfurther fate of of the the energy energy depends depends on on the the arc providing its its steadystate. Heat can be conduction through of state. Heat can be removed removed via via heat heat conduction through the the cooled cooled walls walls of the the tube tube containing containing the the arc. arc. It It can can be be carried carried away away by by the the cooling cooling gas gas flow flow or due due to to aa natural natural convection convection if if the the arc arc burns burns in in aa free free atmosphere. atmosphere. Definite Definite or relations among among E, E , T, T , and and ro ro with with current current i are are obtained obtained if if the the heat heat release release relations mechanism mechanism is is known. known. None None of of the the above above mechanisms mechanisms (convection (convection does does not not seem seem to to have have enough enough time time to to develop) develop) are are operative operative in in aa leader leader channel. channel. One One may suggest away from conmay suggest that that heat heat is is carried carried farther farther away from the the channel channel via via heat heat conduction, gradually heating an ever air volume. Strictly, this duction, gradually heating an ever increasing increasing air volume. Strictly, this is is not not aa steady steady state state process, so it is not simple matter process, so it is not aa simple matter to to find find all all relations. relations. However, the the state the channel channel itself itself is is close to aa stationary This is is However, state of of the close to stationary one. one. This due to to aa small temperature variation variation in in the the current channel owing due small and and definite definite temperature current channel owing to to an an exceptionally exceptionally strong strong dependence dependence of of equilibrium equilibrium plasma plasma conductivity conductivity on on its its temperature. temperature. So So one one can can find find the the relation relation between between the the leader leader channel channel temperature temperature TT,m and and the the power power PI PI== iE iE released released per per unit unit length. length. Using Using the the available available experimental experimental values values for for T and and i, one one can can find find E to to see see that that aa fairly fairly low low field field is is sufficient sufficient to to support support plasma plasma in in aa well well developed developed leader leader channel. channel. The electron density in an equilibrium plasma is n ;::::: The electron density in an equilibrium plasma is nee E exp exp (-J (-Zeff/2kT), eff /2kT) , where Zeff Jeff is is an effective ‘ionization 'ionization potential’ potential' of of the the gas. gas. The The relation relation with with where an effective an actual actual ionization ionization potential potential of atoms is is strictly valid for homogeneous an of atoms strictly valid for aa homogeneous gas (the (the Saha For the the temperature temperature range range T ;::::: 4000-6000 K, we we gas Saha equation). equation). For E 4000-6000K, have, have, in in accordance accordance with with table table 2.2, 2.2, Zeff Jeff = = 8.1 8.1 eV eV and and Ien/K Jeff /K = = 94000 94000K, K, which is is close close to to INo JNO = 9.3 eV. eV. Since Jefr/2kT E ;::::: 10, the conductivity conductivity 0 0" '" ne ne which = 9.3 Since Ieff/2kT 10, the is strongly temperature dependent and decreases with radius much more is strongly temperature dependent and decreases with radius much more than the the temperature. temperature. Therefore, Therefore, we we can can use use the the concept concept of of aa current current channel channel than with the radius radius ro ro (figure (figure 2.14). 2.14). By By denoting denoting with aa more more or or less less fixed fixed boundary boundary -- the the and bearing bearing in in mind mind that that the the the channel channel boundary boundary temperature temperature as as TTii and temperature in the the channel channel is is b.T AT = Tij « << TT,,m, we temperature variation variation in = TT,m - T we can can write write an approximate expression for the the channel channel energy energy balance: balance: an approximate expression for AT PI = -27rroX, PI = -27fr oAm(dT) M ;::::: 47rr0X, 47frOAm b.T = = 47rX,AT 47fA mb.T (2.46) (2.46) dr 10 ro ro YO N
(g)
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The streamer-leader The streamer-leader process in a long spark .T
r
Figure nearly the Figure 2.14. Schematic Schematic arc arc channel channel with with nearly the same same distributions distributions of of T and and ao at at the the axis axis of of the the 'old' ‘old’ portion portion of of aa leader leader channel. channel.
where AAm , is is heat conductivity at at T == T T,.m' The The refined factor 44 appears appears instead instead where heat conductivity refined factor of heat conduction uniform distribution of 22 if if the the heat conduction equation equation for for aa uniform distribution of of heat heat sources is is integrated integrated over over the range 00 < r < rD. yo. the range sources An arbitrary arbitrary channel channel boundary should be set such such as as to to allow allow an an adequate adequate An boundary should be set current because current through through the the channel. channel. The The current current should should not not be be too too low, low, because another be too another channel channel will will appear appear 'outside'; ‘outside’; it it should should not not be too high high either, either, because because aa current-free current-free periphery periphery will will arise arise inside inside the the current current channel. channel. Assume, Assume, for for the the sake sake of of definiteness, definiteness, that that the the channel channel boundary boundary conductivity conductivity a-(TJ cr( Ti) is is by by aa factor of of ee less less than than the the axial axial value cr( T T,). With the the exponential exponential dependence dependence value a-( m)' With factor of a-(T), cr(T), this approximately yields yields b.T AT x k T i / I e f . By By substituting substituting this of this approximately ~2 2kT~/Ieff' this expression into into (2.46), (2.46), we we find find the desired relation: relation: expression the desired
k T 2i kT
PI== iE ~ 87rAm-, PI 81l"Am~, Ieff Jeff
T T,m ==
(I) 8 ~ffkPI (*PI) 87rA, k 1l"Am
112 1/2
.
(2.47) (2.47)
Expression (2.47) (2.47) does does not not contain contain the the radius radius ro, ro, its its account account requires requires aa Expression consideration of of the channel environment environment [26]. [26]. The The channel channel temperature temperature consideration the channel T power than p:/ 2 because T,m grows grows more more slowly slowly with with power than Pi12 because the the air air heat heat conductivity conductivity T,m = = 5000K, 5000K, rises rapidly rapidly with with temperature temperature in in the the range range of of interest. interest. At rises At T Wjcm K; we have ~ 130 ~ 1, AAm , == 0.02 0.02 W/cm K; so so we have PI PIx 130 Wjcm. W/cm. For For current current i x 1, for for 130 V/cm. The which this temperature seems to to be characteristic, E x which this temperature seems be characteristic, ~ 130Yjcm. The values of of E and and PI PI are are only only by by aa factor factor of of two two smaller smaller than than those those found found values from experimental experimental evaluations evaluations of of the field in in older older leader leader channel channel portions. from the field portions. It is is possible possible that channel instability instability and and the the necessity to heat an increasincreasIt that the the channel necessity to heat an ing volume require require aa higher higher power power and higher field the given ing air air volume and aa higher field at at the given current. current. This problem problem remains unsolved and and deserves deserves close close study. study. When When applied applied to This remains unsolved to arcs, formula formula (2.47) (2.47) gives gives aa fairly fairly good good agreement agreement with experiments. arcs, with experiments. These considerations considerations of of the thermal balance of aa leader leader channel channel permit These the thermal balance of permit establishing the the 'current-voltage' ‘current-voltage’ characteristic characteristic (eYe) (CVC) E(i) E ( i ) that that we we shall shall establishing need in in the the next next section. section. Heat Heat flow flow from from the the channel channel grows grows with need with temperature temperature
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Voltage Voltage for a long spark
81 81
2 but not not very (JE would would but very rapidly. rapidly. But But the the conductivity conductivity (J0 and and current current ii == 7rr .irr2uE grow very very quickly quickly if if the the field field remained remained constant. constant. As fast fast would the grow would be be the P I == iE, iE,which would set set the the system system out out of of heat growth of of energy energy release release PI growth which would heat balance. The The balance balance is is maintained maintained because the field field drops drops with with rising rising because the balance. current, while the power and temperature temperature do do not not change change much. much. In In an an ideal ideal current, while the power and case with with T T,m = = const, const, which which is is close close to the low low temperature temperature conditions conditions for for case to the aa leader we have x i-I i-' from from (2.47). (2.47). The The arc arc eve CVC is is leader and and low low current current arc, arc, we have E ~ indeed aa descending descending curve, curve, though though its its goes goes down down somewhat somewhat more more slowly slowly indeed because T T,m and and PI P1 slightly slightly rise rise with with current current [26]. [26]. because between the A similarity similarity between the states states and and eve CVC curves curves for for quasi-stationary quasi-stationary leaders cm long leaders and and arcs arcs was was established established in in model model experiments experiments [4]. [4]. Sparks Sparks 77cm long were between rod parameters were were generated generated in in air air between rod electrodes. electrodes. The The circuit circuit parameters were chosen chosen such such that that the the channel channel current current was was stabilized stabilized at at aa level level characteristic characteristic of moment of bridging. The of aa long long laboratory laboratory spark spark at at the the moment of gap gap bridging. The stabilizing stabilizing mode time, aa quasi-stationary mode lasted lasted several several milliseconds. milliseconds. During During this this time, quasi-stationary state state was established under under energy energy supply supply conditions conditions close close to to those those of of the the leader leader was established process. The The eve CVC thus measured is is approximated approximated by expression process. thus measured by the the expression
E
= 32 ++ 52/i 52/i V/cm, V/cm,
~
i [A). [A].
(2.48) (2.48)
The to be than in V/cm at The obtained obtained field field appears appears to be lower lower than in aa leader leader (84 (84 V/cm at 1A) 1 A) and and closer closer to to the the arc arc field. field.
2.6
Voltage for a long long spark
The The problem problem of of minimum minimum voltage, voltage, at at which which aa spark spark can can develop develop to to aa certain certain length length is is of of primary primary importance importance for for high-voltage high-voltage technology. technology. This This quantity quantity characterizes the the electric electric strength strength of of an an air air gap, gap, since since its its bridging characterizes bridging by by aa leader results results in in aa breakdown. breakdown. This This problem problem also also applies applies to to lightning, lightning, because because leader it is is interesting interesting to to know know the the minimum minimum cloud cloud potential at which which aa lightning lightning it potential at discharge is is possible. possible. discharge Experiments show show that that the the leader leader process process has threshold character. character. An Experiments has aa threshold An initial leader leader cannot cannot survive survive in in normal normal air air at at gap gap voltages voltages less less than than 300300initial 400 kV. kV. A leader leader can can only only be be formed formed at at low low voltages voltages in in aa short short gap gap when when it it 400 develops as as aa final final jump from the the very very beginning. beginning. Then Then streamers streamers immediimmedidevelops jump from ately reach reach the the opposite opposite electrode, electrode, and and the the energy energy supply supply mode mode differs differs ately from that that of of the the initial initial stage, stage, with the streamer streamer zone zone isolated isolated from from the with the the from grounded electrode. electrode. The The reason reason for for aa threshold threshold is is easy easy to to understand understand in in grounded 2.2.3 and and formula formula (2.34). (2.34). A leader leader channel channel terms of of the the discussion discussion in in section section 2.2.3 terms has aa minimum possible radius. radius. The The radius radius of of aa cold cold air air column, column, in in which which has minimum possible 2 current can can accumulate, accumulate, is is ra ro > 10lOW2cm. thinner current current channel channel is is cm. A thinner current immediately enlarged enlarged by ambipolar diffusion. diffusion. To To heat column of of such such immediately by ambipolar heat aa column 5000K, the leader leader tip from (2.34) (2.34) must at initial radius radius to to 5000 initial K, the tip potential potential from must be be at 200 kV. kV. If If we consider the the inevitable inevitable energy energy expenditure expenditure for for ionization ionization least 200 least we consider
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The streamer-leader process in a long spark The streamer-leader
and the streamer value will by, at and gas gas excitation excitation in in the streamer zone, zone, this this value will increase increase by, at least, least, aa factor potential in be [4].Therefore, Therefore, the the tip tip potential in the the initia11eader initial leader stage stage will will be factor of of 1.5 [4]. several hundreds of kilovolts kilovolts even even under under favourable favourable conditions. conditions. several hundreds of The the gap U.o applied applied to to the gap drops drops across across the the leader leader channel channel and and is is The voltage voltage U partly transported transported to tip. The partly to the the tip. The general general formula formula is is
+
U0 = EL + Ut, U o=
(2.49) (2.49)
where E is is the the average average field field in in aa leader leader channel channel of of length length L. We We showed showed in in where that in in aa long long channel, channel, most most of of which which is is in in aa quasi-stationary quasi-stationary section 2.5 that section state, is aa more more or or less less definite definite value value varying varying with with current current i. The The channel channel state, E is field requires that field decreases decreases with with increasing increasing current. current. But But current current growth growth requires that C1 U,t V VL the tip tip potential potential determining determining the leader velocity velocity and and current current ii == C the the leader lU L should be raised. At At aa fixed fixed length length L the Uo(i, Uo(i, L function has has aa minimum, should be raised. L,, the L)) function minimum, since the sum of aa falling falling component component and and aa component component rising rising with with i.i. since it it is is the sum of Minimum voltage voltage Uo0min~ (L) * ( Lcorresponds corresponds ) to current current iiOpt(L) optimal for for aa Minimum to opt (L) optimal leader hardly possible, possible, in leader of of length length L. It It is is hardly in the the present present state state of of the the art, art, to to Uomin(L) function theoretically. We shall shall try try to define its its character character find the function theoretically. We to define find the Uomin(L) using semi-empirical using semi-empirical data. data. Many experimental experimental physicists physicists have measured the the leader leader velocity variaMany have measured velocity variaU,. Much Much work work has has been done on on short short leaders leaders tion with applied applied voltage voltage Uo. tion with been done because one neglect the the voltage voltage drop the channel, because one can can neglect drop across across the channel, assuming assuming U U,o ~ = UU,.t . With With the the account account of of this this approximate approximate equality, equality, Baze1yan Bazelyan and and l Z Razhansky ~ aU [35] suggested suggested an an empirical empirical formula: formula: V VL uU:’~, where Razhansky [35] L z t / , where 3 a~ y-l/Z velocity increase with voltage voltage E 1.5 1.5 x 10 lo3 V-1/2 cm cm S-I. s - l . Physically, Physically, the the velocity increase with looks this variation looks quite quite natural natural (though (though this variation is is not not strong). strong). We We also also know know by (2.36). that is defined defined by (2.36). This This gives gives the the relation relation that the the tip tip current current is U AP/3(VL rv iil1/ 3/ )3 )with us. use use the U,t == Ai2l3(VL with A == (C ( C1 ~ UZ/ 3) == - ~(27rEoa)-Z/3. ( /2~7 r ~ ~ a ) -Let Let ~ / ~us the analogy between aa well CVC analogy between well developed developed leader leader and and an an arc arc and and take take the the eye E == b/ b / ii typical typical for for aa low low current current arc. arc. Let Let us us put V A/cm A/cm for for numerical numerical put b == 300 Y calculations the difference between the the tip calculations and and ignore ignore the difference between tip and and channel channel currents. currents. U,o == Lb/i + AiZi3 and after after differentiation differentiation We shall then get U We shall then get Aiz/ 3 and
ar
N
+
iiopt = (3Lb/2A)3/5, (3Lb/2A)3i5, opt =
U = U.Omin mln = = ~A3/5(3bL/2?/5 A3/5(3bL/2)215 = ~$ U U,topt opt
(2.50) (2.50)
If the applied If aa leader leader develops develops under under optimal optimal conditions, conditions, the applied voltage voltage is is shared shared by by the the tip tip and and the the channel channel in in comparable comparable proportions. proportions. The The mode mode with tip potential the limit with aa low low tip potential close close to to the limit admissible admissible from from the the energy energy criteria, criteria, is unprofitable for is unprofitable for aa long long leader, leader, because because it it corresponds corresponds to to low low current current leadleading voltage drop ing to to aa considerable considerable voltage drop across across the the channel. channel. The The long long spark spark paramparameters table 2.3 found found from from (2.50) (2.50)with with semi-empirical semi-empirical constants constants are are quite quite eters in in table reasonable: magnitude for reasonable: these these orders orders of of magnitude for current, current, voltage, voltage, and and velocity velocity meet requirements on meet the the requirements on the the optimal optimal experimental experimental conditions conditions for for long long leader nonlinear, slow leader development. development. Besides, Besides, the the experiment experiment requires requires aa nonlinear, slow dependence breakdown voltage dependence of of minimum minimum breakdown voltage on on the the gap gap length. length. It It is is generally generally known the length multi-metre gap of aa multi-metre gap is is not not aa particularly particularly known that that increasing increasing the length of
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A negative leader
Table 2.3. Long Long spark spark parameters. Table parameters. L,m
UOmin , MV
Ut opt, MV
50 50 100 100 3000 3000
3.3 3.3 4.3 4.3 17 17
2.0 2.0 2.6 2.6 10 10
i,A
1.1 1.1 1.3 1.3
17 17
VL , 104 m!s
E L , V!cm
2.1 2.1 2.4 2.4 4.7 4.1
260 260 170 170 22 22
effective to those effective way way of of raising raising its its electrical electrical strength. strength. This This is is aa key key challenge challenge to those working high-voltage technology. working in in high-voltage technology. The The results results of of extrapolation extrapolation of of formula formula (2.50) (2.50) to to aa lightning lightning leader leader ( L == 33 km) km) also also lie lie within within reasonable reasonable limits. limits. (L necessary for What What is is the the rate rate of of gap gap voltage voltage rise rise necessary for the the optimal optimal mode mode of of spark raised as the spark spark development? development? Clearly, Clearly, the the gap gap voltage voltage must must be be raised as the spark length becomes longer according according to to (2.50), (2.50), where is an an instantaneous instantaneous length becomes longer where LL is leader leader length. length. The The existence existence of of an an optimal optimal mode mode of of spark spark development development has has been confirmed been confirmed experimentally experimentally [36-38]. [36-381. It It has has been been shown shown that that for for aa breakdown to to occur pulse risetime must increase breakdown occur at at minimum minimum voltage, voltage, the the pulse risetime If t f must increase with the the gap gap length length d. d. The The authors authors of[39] of [39] recommend recommend the the following following empirical empirical with formula for for the the evaluation evaluation of of an an optimal optimal risetime: risetime: formula lfopt ::::; lfopt
[ml dd [m]
50d [I!s], [PSI, 50d
(2.51) (2.51)
Generally, Generally, optimal optimal voltage voltage impulses impulses have have aa fairly fairly slow slow risetime. risetime. Their Their values values I!S in modern power power transmission vary vary between between 100 100 and and 250 250ps in modern transmission lines lines with with the the m. We We shall return to insulator insulator string string length length of of 2-5 2-5m. shall return to this this issue issue in in chapter chapter 3, 3, when when considering considering the the diversity diversity of of time time parameters parameters of of lightning lightning current current impulses. minimum electric with aa sharply nonimpulses. The The minimum electric strength strength of of an an air air gap gap with sharply nonuniform field can can be found from from the the formulas formulas [4] [4] uniform field be found 3400 3400 8/d
[kV],, u50% min = =SO % min U 1 + 81d [kV] 1
+
+
U U,,,, min = = 1440 1440 + 55d 55d [kV], [kV], SO % min
2.7
d
1 5m m < 15 15 ~ < dd 15
(2.52) (2.52)
< 30 m m ~ 30
A negative negative leader
Most negative charge because they they are Most lightnings lightnings carry carry aa negative charge to to the the earth earth because are 'anode‘anodebreak down mediumdirected' directed’ discharges. discharges. It It is is always always more more difficult difficult to to break down aa mediumlength between aa negative negative length gap gap between negative electrode electrode and and aa grounded grounded plane. plane. A negative leader higher voltage. voltage. The between leaders leader requires requires aa higher The difference difference between leaders of of different different the streamer while their their channels polarities is polarities is due due to to the streamer zone zone structure, structure, while channels and and voltage drop m voltage drop across across them them are are quite quite similar. similar. Indeed, Indeed, aa gap gap of of about about 100 lOOm long, which an part of voltage drops long, in in which an appreciable appreciable part of voltage drops across across the the channel, channel, is is bridged by leaders both signs voltages [2,3]. bridged by leaders of of both signs at at about about the the same same voltages [2,3].
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The streamer-leader process in a long spark The streamer-leader
The higher tip The streamer streamer zone zone formation formation in in aa negative negative leader leader requires requires aa higher tip potential for for the the same same reason reason as as aa single single anode-directed anode-directed streamer streamer needs needs aa potential higher for its its development. development. Fast Fast streamers streamers with with the the velocity velocity V V,s higher voltage voltage for much higher higher than the electron electron drift drift velocity velocity V, do not exhibit much difference much than the Ve do not exhibit much difference associated with But streamers streamers in in aa leader leader streamer streamer zone zone are are slow: slow: associated with polarity. polarity. But V whether the the components V,s ~ x V V,. It is is of of great great importance importance whether components of of electron electron e • It velocity to the the streamer streamer tip tip are are summed, summed, V V,s + V,, as in in aa cathodecathodevelocity relative relative to Ve , as directed streamer, streamer, or or subtracted, subtracted, V V,s -- V V,, as in in an an anode-directed anode-directed one. one. In In directed e , as the tip tip move the former produced in the former case, case, electrons electrons produced in front front of of the move towards towards it, it, and the the ionization ionization occurs occurs in in aa strong strong field field near the tip. tip. In In the the latter, latter, electrons electrons and near the tend the moving most of their time tend to to 'run ‘run ahead' ahead’ of of the moving tip tip and and spend spend most of their time in in aa lower lower field, under less unfavourable conditions. field, so so that that the the ionization ionization occurs occurs under less unfavourable conditions. The The fact fact that that negative negative streamers streamers generally generally require require aa higher higher field field and and voltage has has been supported by many experiments. experiments. They They show show that that the the voltage been supported by many average critical critical field, field, which defines the the maximum maximum streamer streamer length length in in formula formula average which defines (2.32), is is twice twice as as high high for for an an anode-directed anode-directed streamer streamer as as for for aa cathodecathode(2.32), directed one: one: E,, x 10 kV kV/cm against E,, x 5 kV kV/cm. shall illustrate illustrate directed Ecr ~ fcm against Ecr ~ fcm. We We shall this with with figure figure 2.15 2.1 5 for for aa gap gap of of length length d = = 3m 3 m between sphere of of radius this between aa sphere radius ro em and ro = = 50 50cm and aa grounded grounded plane. plane. The The streamers streamers stopped, stopped, having having covered covered I,,, at at negative negative sphere sphere potential potential U U,c == 1.5 1.5 MV MV (the (the unperturbed unperturbed the distance distance [max the potential at the the stop stop with with the the account account of of the the sphere sphere charge charge reflection reflection in in the potential at the plane is Uo(lmax) Uo(Zmax)x 0.25 U,). Under these these conditions, conditions, cathode-directed cathode-directed plane is ~ 0.25U c). Under streamers practically cross the whole streamers practically cross the whole gap. gap. The propagation mechanism The propagation mechanism and and streamer streamer zone zone structure structure of of aa negative negative leader more complicated than those those of leader are are much much more complicated than of aa positive positive leader leader and and are are still still
+
Figure 2.15. Anode-directed radius at Anode-directed streamers streamers from from aa spherical spherical cathode cathode of of 50 50 cm cm radius at aa negative negative voltage voltage impulse impulse of of 1.8 1.8 MV and and aa 50 50 JlS ps front front duration. duration.
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A negative leader
85
poorly 1930s, when Schonland started started his studies poorly understood. understood. In In the the 1930s, when Schonland his famous famous studies of lightning [41], [41], aa negative found to character of of lightning negative leader leader was was found to have have aa discrete discrete character of elongation, so it stepwise. Streak Streak photographs series of elongation, so it was was termed termed stepwise. photographs exhibit exhibit aa series of flashes, in aa stepwise stepwise manner. flashes, indicating indicating that that the the leader leader propagates propagates in manner. Later, Later, aa similar similar process process was was found found in in aa long long negative negative leader leader produced produced in in laboratory laboratory conditions [42,43]. 142,431. With With every every step, step, aa negative negative leader leader elongates elongates by by dozens dozens conditions [3]; steps steps of of aa hundred hundred of centimetres, centimetres, or or by by several several metres metres in in superlong superlong gaps gaps [3]; of metres have have been been registered registered in in negative negative lightning lightning discharges. discharges. Every Every step step of of aa metres laboratory leader leader is is accompanied accompanied by by aa detectable detectable current current overshoot overshoot which which laboratory quickly time between between two quickly vanishes vanishes during during the the time two steps. steps. Without explanations of Without going going into into theoretical theoretical explanations of this this mechanism, mechanism, based based on let us see what on an an unverified unverified hypothesis, hypothesis, let us see what information information can can be be derived derived from streak photographs of the from streak photographs of the process, process, made made during during laboratory laboratory experiments experiments [44]. These are are naturally naturally more more informative informative than than streak photographs of [44]. These streak photographs of aa stepstepwise lightning lightning leader. leader. It is seen figures 2.16 2.16 and and 2.17 that that in the intervals intervals wise It is seen from from figures in the between between the the steps, steps, the the tip tip of of aa negative negative leader leader slowly slowly and and continuously continuously moves moves on on together together with with its its streamer streamer zone zone made made up up of of anode-directed anode-directed streamers. streamers. The The main events events occur occur near near the the external external boundary boundary of of the the negative negative streamer streamer zone. zone. main It It seems seems that that aa plasma plasma body body elongated elongated along along the the field field arises arises there there and and is is polarized by by the with the polarized the field field (compare (compare with the discussion discussion in in section section 2.2.7). 2.2.7). The The positive plasma plasma dipole dipole end end directed directed towards towards the the main main leader leader tip tip serves as aa positive serves as starting point point for for cathode-directed cathode-directed streamers. They move move towards towards the the tip, tip, starting streamers. They thus elongating elongating the the conducting conducting portion portion of of the the channel channel and and enhancing enhancing the the thus negative field field at at its its end end directed directed to to the the anode. anode. Almost Almost at at the the same same time, time, the the negative an anode-directed anode-directed streamer. nearly mystic mystic plasma body body generates generates an plasma streamer. This T h s nearly picture picture of of streamer streamer production production in in the the gap gap space space is is clearly clearly seen seen in in aa streak streak photograph photograph in in figure figure 2.18. 2.18. Nothing Nothing like like this this has has ever ever been been observed observed with with aa positive positive leader. leader.
picture of negative stepped Figure 2.16. 2.16. A schematic schematic streak streak picture of aa negative stepped laboratory laboratory leader: leader: (1,2) (1,2) secondary cathodecathode- and and anode-directed anode-directed streamers streamers from from the the gap gap interior; interior; (3) secondary secondary secondary (4) main main negative negative leader channel; (5) (5) its its tip; tip; (6) (6) plasma plasma body; body; volume leader leader channel; channel; (4) leader channel; volume of secondary positive and and negative negative leader leader (9) leader flash flash concluding concluding step (8) tip tip of (7), (8) secondary positive (9) leader step development. development.
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The streamer-leader streamer-leader process in a long spark
Figure 2.17. A streak photograph of negative laboratory streak photograph of the the initial initial stage stage in in aa negative laboratory leader. leader. figure 2.16. 2.16. Marking numbers numbers correspond correspond to Marking to figure
The polarized polarized plasma section becomes the starting starting point point not not only only of of The plasma section becomes the streamers but but also also of of secondary secondary leaders leaders which follow them. them. They They are are streamers which follow positive cathode-directed cathode-directed volume volume leader leader grows grows known as as volume volume leaders. leaders. A positive known intensively. Normally, its streamer streamer zone zone almost almost immediately immediately reaches intensively. Normally, its reaches the the so it it looks looks as as if if the the secondary secondary positive leader develops develops main negative leader, so main negative leader, positive leader
The origin origin of of anode-directed anode-directed (1) (1) and and cathode-directed cathode-directed (2) (2) streamers streamers from from Figure 2.18. The the gap gap interior; interior; (3) (3) initial initial flash flash of of aa negative negative corona corona (static (static photograph) photograph) which which trigger trigger the negative leader. aa streak streak photograph photograph regime; regime; (4) (4)arisen arisen negative leader.
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A negative leader
87 87
in negative volume volume leader leader moves moves in the the final final jump jump mode, mode, i.e., i.e., very very quickly. quickly. The The negative of the the main main towards When the the tips tips of towards the the anode anode somewhat somewhat more more slowly. slowly. When into contact, contact, they they form form aa negative negative and and of of the the positive positive volume volume leaders leaders come come into of partial partial charge charge common to the the process process of common conducting conducting channel, channel, giving giving rise rise to result, the the former former volume volume leader leader neutralization neutralization and and redistribution. redistribution. As aa result, negative leader leader tip. tip. This This process process acquires acquires aa potential potential close close that that of of the the main main negative accompanied by by aa rapidly rapidly looks looks like like aa miniature miniature return return stroke stroke of of lightning, lightning, accompanied rising in the the channel channel and and external external rising and and just just as as rapidly rapidly falling falling current current impulse impulse in for aa short short time. time. It It is is circuit. circuit. The The intensity intensity of of the the channel channel emission emission increases increases for hard to say what exactly stimulates this increase the short temperature rise hard to say what exactly stimulates this increase - the short temperature rise or changes the the cover cover charge, charge, or the the ionization ionization in in the the channel channel cover, cover: which which changes (see thereby redistribution along along the the channel channel (see thereby getting getting ready ready for for aa potential potential redistribution section 2.4.4). The negative portion of the plasma dipole turns to a new section 2.4.4). The negative portion of the plasma dipole turns to a new the mechanism mechanism of of step step formation formation negative negative tip tip of of the the main main leader. leader. This This is is the Then the the story story is is repeated. repeated. and and stepwise stepwise elongation elongation of of the the main main channel. channel. Then to draw draw the the conclusion conclusion about about The The picture picture just just described described gives gives no no ground ground to The motion motion of of aa neganegaaa stepwise stepwise character character of of negative negative leader leader development. development. The positive volume volume leaders, leaders, also also contincontintive tive leader leader is is continuous, continuous, but but secondary secondary positive the final result of of their their ‘secret 'secret uous, uous, produce produce aa stepwise stepwise effect. effect. Discrete Discrete is is the final result with imperfect imperfect optical optical instruinstruactivity', activity’, but but only only if if the the observer observer is is equipped equipped with known as as aa step step is is an an instant instant ments. ments. In In other other words, words, what what is is generally generally known process. As As for for gap gap bridging bridging by by aa main main result result of of aa long long continuous continuous leader leader process. of the the channel channel is is created created negative that most most of negative leader, leader, one one should should bear bear in in mind mind that by positive volume volume leaders. leaders. by auxiliary auxiliary agents agents -- by by aa succession succession of of positive This picture picture has has been been reconstructed reconstructed from from streak streak photographs. we This photographs. But But we still do do not not know know how how polarized polarized plasma plasma dipoles dipoles are still are formed formed far far ahead ahead of of the main main leader leader tip. tip. Their Their appearance appearance is is hardly the hardly aa result result of of our our imagination. imagination. making aa volume volume leader leader start start from from aa Steps Steps can can be be produced produced deliberately deliberately by by making place there there aa metallic metallic rod rod several several desired desired site site in in the the gap. gap. For For this, this, it it suffices suffices to to place centimetres rods placed placed in in different different sites sites of of aa centimetres long long (figure (figure 2.19). 2.19). A series series of of rods gap will will create create aa regular regular sequence sequence of of volume volume leaders. leaders. The The work work [45] [45] describes describes gap an experiment experiment with with aa negative negative leader leader 200 200 m m long. long. Its Its perfectly perfectly straight straight trajectrajecan tory was was predetermined predetermined by by seed seed rods rods suspended suspended by by insulation insulation threads threads at at aa tory distance of of 2-3 2-3mm from from each each other. other. A A volume volume leader leader started started from from aa rod rod distance when it it was was approached approached by by the the negative negative streamer of the the when streamer zone zone boundary boundary of main leader. leader. Clearly, Clearly, the the rods rods are are polarized main polarized by by the the streamer streamer zone zone field field to to serve as as seed seed dipoles dipoles instead instead of of natural natural (hypothetical) (hypothetical) plasma plasma dipoles. dipoles. serve There are are many many hypotheses hypotheses concerning concerning the There the stepwise stepwise leader leader mechanism, mechanism, but they they are are so so imperfect, imperfect, lacking lacking strength, strength, and, and, sometimes, sometimes, even even absurd absurd that that but we shall shall not not discuss discuss them them here. here. We We are are not we not ready ready today today to to suggest suggest an an alteralternative model model either. either. Additional Additional special-purpose special-purpose experiments native experiments could could certainly stimulate the theory of this complicated and challenging phenomcertainly stimulate the theory of this complicated and challenging phenomenon. It It would would be be desirable desirable to to take take shot-by-shot shot-by-shot pictures pictures of of aa negative negative enon. leader tip tip region region with with aa short short exposure. exposure. A A sequence sequence of leader of such such pictures pictures would would
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The streamer-leader process in a long spark The streamer-leader
An artificially artificially induced induced step: step: (1) (1) initial initial flash flash of of aa negative corona from from aa Figure 2.19. An negative corona spherical rod spherical cathode; cathode; (2, (2, 3) 3) cathodecathode- and and anode-directed anode-directed leaders leaders from from aa metallic metallic rod 2.5 placed in 2.5 cm cm long, long, placed in the the gap gap interior; interior; (4) (4) leader leader flash flash concluding concluding the the step step developdevelopment; ment; (5) (5) new new streamer streamer corona corona flash flash from from the the tip tip of of the the elongating elongating channe!. channel.
form unambiguous interpretation form aa film film more more accessible accessible to to unambiguous interpretation than than continuous continuous streak of many many details. details. streak photographs photographs with with confusing confusing overlaps overlaps of
References References [l] Lupeiko Lupeiko A A V, V, Miroshnizenko Miroshnizenko V VP P et al1984 a1 1984 Proc. II All-Union Conf Conf Phys. of of [1] Electrical Breakdown of of Gases (Tartu: (Tartu: TGU) TGU) p 254 (in (in Russian) Russian) p 254 [2] V, Gayvoronsky [2] Baikov Baikov A A P, P, Bogdanov Bogdanov 00 V, Gayvoronsky AS A S et al1998 a1 1998 Elektrichestvo 1060 10 60 [3] Gayvoronsky Gayvoronsky A A Sand S and Ovsyannikov Ovsyannikov A G 1992 1992 Proc. 9th Intern. Conf on Atmosph. [3] AG A.I. Voeikov Electricity 3 (St (St Peterburg: Peterburg: A.I. Voeikov Main Main Geophys. Geophys. Observ.) Observ.) pp 792 792 [4] New York: [4] Bazelyan Bazelyan E EM M and and Raizer Raizer Yu Yu P P 1997 1997 Spark Discharge (Boca (Boca Raton, Raton, New York: CRC p 294 CRC Press) Press) p 294 [5] (Washington D.C.) 1481417 148 1417 [5] Loeb Loeb L LB B 1965 1965 Science (Washington [6] [6] D'aykonov D’aykonov M M I and and Kachorovsky Kachorovsky V V Yu 1988 1988 Zh. Eksp. Teor. Teor. Fiz. 9432 94 32 [7] Zh. Eksp. Teor. [7] D'aykonov D’aykonov M II and and Kachorovsky Kachorovsky V V Yu Yu 1989 1989 Zh. Teor. Fiz. 951850 95 1850 VA A 1990 1990 Teplojiz. Teplofz. Vys. Vys. Temperatur 28 1056 1056 [8] Shveigert Shveigert V [8] [9] EM M and and Raizer Raizer Yu P P 1997 1997 Teplofiz. Teplofiz. Vys. Vys. Temperatur 35 35 181 181 (Eng!. (Engl. [9] Bazelyan Bazelyan E trans!': transl.: 1997 1997 High Temperature 35) [10] N 1996 [lo] Raizer Raizer Yu Yu P P and and Simakov Simakov A AN 1996 Piz. Plazmy 22 668 668 (Eng!. (Engl. trans!': transl.: 1996 1996 Phys. Rep. 22 603) Plasma Phys. 603) [11] J. Phys. Chem. Ref Data 4577 [ l l ] Dutton Dutton JJ A A 1975 1975 J. Chem. Rex 4 577 and Loeb Loeb L L B 1935 1935 Physics (N. ( N . Y.) Y . ) 66 125 125 [12] Cravith Cravith A M and [12] [13] Raizer Raizer Yu P P and and Simakov Simakov A AN 1998 Piz. Plazmy 24 700 700 (Eng!. (Engl. trans!': transl.: 1996 1996 [13] N 1998 Plasma Phys. Rep. 24 700) 700)
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Vitello P N 1994 Vitello P A, A, Penetrante Penetrante B BM M and and Bardsley Bardsley JJ N 1994 Phys. Rev. E 495574 49 5574 GV 1996 J. J . Phys. 29 2423 Babaeva N Yu and and Naidis Babaeva NYu Naidis G V 1996 Phys. D: Appl. Phys. Phys. 292423 Ku1ikovsy A 1997 Phys. D: Appl. Phys. 30441 Kulikovsy A AA 1997 J. J . Phys. 30 441 A1eksandrov N L, Baze1yan Fiz. Bazelyan E E M, M, Dyatko Dyatko N N A A and and Kochetov Kochetov II V V 1998 1998 Fiz. Aleksandrov N Phys. Rep. 24 541) Plazmy 24 587 587 (Engl. (Engl. transl. transl. 1998 1998 Plasma Phys. 541) [18] Baze1yan Bazelyan E EN and Goryunov Goryunov A A Yu Yu 1986 1986 Elektrichestvo 11 11 27 27 [18] Nand [19] Aleksandrov L and Bazelyan EM M 1998 1998 J. J . Phys. D: Appl. Phys. 292873 29 2873 [19] A1eksandrov N N Land Baze1yan E S, Baze1yan Bazelyan E E M and Bekzhanov Bekzhanov B B II 1984 1984 Izv. [20] Aleksandrov D S, [20] A1eksandrov D M and Izv. Akad. Akad. Nauk SSSR. SSSR. Energetika i transport 2 120 120 [21] Yu and Izv. Akad. Akad. Nauk [21] Baze1yan Bazelyan E E M, M, Goryunov Goryunov A A Yu and Goncharov Goncharov V VA A 1985 1985 Izv. SSSR. 154 SSSR. Energetika i transport 2 154 [22] Aleksandrov L and Baze1yan Bazelyan E EM M 1999 1999 J. 2636 [22] A1eksandrov N N Land J. Phys. Phys. D: Appl. Appl. Phys. 32 2636 S and Razhansky Razhansky II M M 1986 1986 Zh. Zh. Tekh. Tekh. Fiz. 1110 [23] Gayvoronsky Gayvoronsky A A Sand [23] Fiz. 56 1110 [24] Ko1echizky Kolechizky E E C C 1983 1983 Electric Field Calculation for High- Voltage Voltage Equipment [24] (Moscow: (Moscow: Energoatomizdat) Energoatomizdat) pp 167 167 (in (in Russian) Russian) [25] M, Shneider Nand Novakovsky S.V. Phys. D: [25] Raizer Raizer Yu Yu P, P, Mi1ikh Milikh G G M, Shneider M MN a n d Novakovsky S.V. 1998 1998 JJ.. Phys. Phys. 31 3255 Appl. Phys. 3255 [26] p449 [26] Raizer Raizer Yu Yu P 1991 1991 Gas Discharge Physics (Berlin: (Berlin: Springer) Springer) p449 [27] Gorin Gorin B and Schki1ev Schkilev A V 1974 1974 Elektrichestvo 2 29 29 [27] BN Nand AV 1975’ 1977 1977 Electra 5331 53 31 [28] 'Positive ‘Positive Discharges Discharges in in Air Air gaps gaps at at Las Las Renardieres Renardieres -- 1975' [28] [29] Baze1yan Bazelyan E EM M 1982 1982 Izv. 3 82 [29] Izv. Akad. Nauk SSSR. SSSR. Energetika i transport 382 [30] Zh. Tekh. [30] Bazelyan Bazelyan E EM M 1966 1966 Zh. Tekh. Fiz. 36 365 365 [31] III Symp. on [31] Baze1yan Bazelyan E E M, M, Levitov Levitov V V II and and Ponizovsky Ponizovsky A AZ Z 1979 1979 Proc. 1 11 Inter. Symp. High Voltage Voltage Engin. (Milan) (Milan) Rep. Rep. 51.09 51.09 pp 11 [32] [32] Meek Meek JJ M M and and Craggs Craggs JJ D D (eds) (eds) 1978 1978 Electrical Breakdown of of Gases (New (New York: York: Wiley) Wiley) [33] N 1996 Prikl. Mekh. Mekh. Tekhn. [33] Makarov Makarov V VN 1996 Zh. Zh. Prikl. Tekhn. Fiz. 3769 37 69 [34] N L, N A and Phys. E M, M, Dyatko Dyatko N and Kochetov Kochetov LV. I.V. 1997 1997 J. Phys. [34] Aleksandrov Aleksandrov N L, Baze1yan Bazelyan E Phys. 30 1616 D: Appl. Phys. 1616 [35] Razhansky II M [35] Baze1yan Bazelyan E E M M and and Razhansky M 1988 1988 Air Spark Discharge (Novosibirsk: (Novosibirsk: Nauka) pp 164 Russian) Nauka) 164 (in (in Russian) [36] Nand Brago E EN a n d Baze1yan Bazelyan E EM M 1960 1960 Dokl. Akad. Nauk SSSR 133 133 [36] Stekolnikov Stekolnikov I S, Brago 550 550 [37] [37] Steko1nikov Stekolnikov IS, I S, Brago Brago EN E N and and Baze1yan Bazelyan E EM M 1962 1962 Con! Con$ Gas Discharges and the (Leatherhead, England) England) pp 139 139 Electricity Supply Industry (Leatherhead, [38] Nand Fiz. 32 993 and Steko1nikov Stekolnikov II S 1962 1962 Zh. Tekh. Tekh. Fiz. 993 [38] Bazelyan Bazelyan E E M, M, Brago Brago E EN [39] [39] Barnes Barnes H H and and Winters Winters D D 1981 1981 IEEE Trans. Trans. Pas-90 1579 1579 [40] [40] Gallet Gallet G G and and Leroy Leroy JJ 1973 1973 IEEE IEEE Conf. Conf. Paper C73-408-2 C73-408-2 [41] 1411 Schonland Schonland B B 1956 1956 The The Lightning Discharge. Handbuch der Physik 22 (Berlin: (Berlin: Springer) pp 576 576 Springer) [42] 1421 Steko1nikov Stekolnikov I Sand S and Shkilev Shkilev A AB B 1962 1962 Dokl. Akad. Nauk SSSR 145 782 182 [43] Dokl. Akad. Nauk SSSSR [43] Steko1nikov Stekolnikov II Sand S and Shki1ev Shkilev A AB B 1963 1963 Dokl. S S R 145 1085; 1085; 1962 1962 Intern. Con$ (Montreux) (Montreux) p 466 466 Intern. Con! [44] Nand 1441 Gorin Gorin B BN and Shkilev Shkilev A AV V 1976 1976 Elektrichestvo 6 31 31 [45] [45] Anisimov Anisimov E E I, I, Bogdanov Bogdanov 00 P, P, Gayvoronsky Gayvoronsky AS A S et al1988 a1 1988 Elektrichestvo 11 11 55 55
[14] [14] [15] [15] [16] [16] [17] [17]
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Chapter 3
Available lightning lightning data
Scientific observations observations of of lightning lightning were were started started over over aa century century ago. ago. Much Much Scientific factual information information has accumulated about about this this natural natural phenomenon phenomenon since since factual has accumulated that time. time. Most Most of of it, it, however, has been obtained by by remote remote observational observational that however, has been obtained techniques which can reveal only external external manifestations manifestations of of lightning. lightning. This This techniques which can reveal only is not not the the researchers' researchers’ fault. fault. Even Even aa long long laboratory laboratory spark spark keeps keeps the is the experimenter experimenter at at aa respectful respectful distance: distance: there there have have been been single single and and mostly mostly unsuccessful attempts unsuccessful attempts to to study study the the leader leader interior interior and and the the ionization ionization region region in tip. No No attempts kind have been made made with with lightning. in front front of of its its tip. attempts of of this this kind have yet yet been lightning. Nevertheless, the material is being analysed Nevertheless, the accumulated accumulated material is being analysed and and systematized, systematized, so that that our our knowledge knowledge about about atmospheric atmospheric electricity electricity is is gradually gradually expanding. expanding. so A number number of of carefully carefully written written books books has has made the results results of of field field studies studies A made the of lightning lightning accessible accessible to to specialists. specialists. Among Among them, of great great interest interest is is the of them, of the recent book by Uman [1] by Golde [l] and and the the co-authored co-authored work work edited edited by Golde [2]. [2]. recent book by Uman The nearly all there is The reader reader will will find find there there nearly all available available data data on on lightning, lightning, so so there is no need need to to discuss discuss them them in in this this book. book. We We have set ourselves ourselves aa different different no have set task task -- to to select select the the few few data data available available on on the the lightning lightning discharge discharge mechanism mechanism and to to try try to to build its theory. theory. In In addition, addition, we we shall shall make make aa detailed detailed analysis analysis of of and build its lightning practical point point of lightning characteristics characteristics important important from from the the practical of view. view. The The nature of of hazardous hazardous effects effects of of atmospheric atmospheric electricity electricity on on industrial industrial objects objects nature protection principles principles will will will be be considered considered in in much much detail detail and and lightning lightning protection will be offered. offered. be This because many paramThis task task cannot cannot be be solved solved completely, completely, because many lightning lightning parameters been measured measured or, eters have have never never been or, more more often, often, even even estimated estimated in in order order of of magnitude. magnitude. One One hope hope is is aa method method similar similar to to the the identical identical text text analysis analysis used used in text written in cryptography cryptography to to read read aa text written in in aa dead dead language. language. If If there there is is at at least text written written in related, language, least part part of of the the text in an an accessible, accessible, better, better, related, language, the the task is not not considered considered hopeless. and ingenuity, ingenuity, the researcher hopeless. With With patience patience and the researcher task is we expect has aa chance texts carefully. has chance if if he he compares compares these these texts carefully. In In this this respect, respect, we expect much phenomena much from from long long spark spark studies. studies. Clearly, Clearly, aa spark spark and and lightning lightning are are phenomena 90 Copyright © 2000 IOP Publishing Ltd.
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of different different scales, scales, but it is is also also clear clear that have aa common common nature. nature. For For of but it that both both have this reason, reason, we we shall shall often often compare compare the the parameters parameters of of lightning lightning with those of of this with those long spark. spark. We We should should like like to to emphasize emphasize that will be comparison aa long that this this will be aa comparison rather direct extrapolation, extrapolation, because there is is no complete analogy analogy rather than than aa direct because there no complete between the two between the two phenomena. phenomena.
3.1 3.1
Atmospheric field during during a lightning lightning discharge discharge Atmospheric field
There is is no no strict strict answer answer to to this this physically physically ambiguous ambiguous question. question. It It is is necesnecesThere sary to to specify specify what what part of the the space space between cloud and and the earth is is sary part of between the the cloud the earth meant. meant. One One thing thing is is clear clear -- the the electric electric field field at at the the lightning lightning start start must must be be high enough enough to to increase increase the the electron electron density density by impact ionization. ionization. This This high by impact value Ei = ~ 30kV/cm value is is Ei 30 kV/cm for for normal normal density density air air and and about about 20kV/cm 20 kV/cm at at an an altitude altitude of of 33 km km (the (the average average altitude altitude for for lightning lightning generation generation in in Europe). Europe). Such Such aa strong strong field field has has never never been been measured measured in in aa storm storm cloud. cloud. The The maximum maximum values values were were recorded recorded by by rocket rocket probing probing of of clouds clouds (lOkV/cm, (10kV/cm, Winn Winn et ai, al, 1974 1974 [7]) [7]) and and during during the the flight flight of of aa specially specially equipped equipped aeroplane aeroplane laboratory laboratory (l2kVjcm). (12kV/cm). The The value value obtained obtained by by Gunn Gunn [4J [4] in in 1948 1948 during during his his flight flight on on aa plane plane around around aa storm storm cloud cloud was was about about 3.5 3.5 kV/cm. kV/cm. The The values values between between 1.4 1.4 and and 88 kV/cm kV/cm were were obtained obtained from from some some similar similar measurements measurements [3-9]. [3-91. It It is is hard to to judge judge about about the the accuracy accuracy of of these these measurements, measurements, especially especially those those hard made made in in strong strong fields, fields, because because parts parts of of the the field field detector detector or or the the carriercarrierplane plane parts parts close close to to it it can can produce produce aa corona corona discharge. discharge. In In any any case, case, the the corona corona space space charge charge will will not not allow allow the the strength strength in in the the region region being being measured measured to to go go beyond beyond aa threshold threshold value value (for (for details, details, see see [20]). [20]). There There are are reasons reasons to to believe, believe, however, however, that that aa corona corona on on hydrometeorites hydrometeorites (water (water droplets, droplets, snow snow flakes, flakes, ice ice crystals) crystals) keeps keeps the the field field at at aa level level below below Eii in in the the whole whole of of the the cloud. so, aa field field can can be be enhanced enhanced above above Eii only only in in aa small small cloud, If If this this is is indeed indeed so, volume volume for for aa short short time, time, say, say, as as aa result result of of eddy eddy concentration concentration of of charged charged hydrometeors. hydrometeors. This This enhancement enhancement will will be be reduced reduced to to zero zero by by aa corona corona for for less less than than aa second. second. The The experimenter experimenter has has no no chance chance to to guess guess where where the the field may may be be locally locally enhanced enhanced to to be be able able to to introduce introduce aa probe probe detector detector there. there. field Theoretically, it it is is also also important important to to know know the the average average gap gap field field capable capable Theoretically, of of supporting supporting aa lightning lightning leader. leader. The The field field decreases decreases in in the the charge-free charge-free space space from the the cloud cloud towards towards the the earth. earth. At At the the earth, earth, the the storm storm field field was was found found to to from be be 10-200 10-200 V/cm. V/cm. Such Such aa low low field field did did not not prevent prevent the the lightning lightning development. development. Lightnings Lightnings were were deliberately deliberately produced produced in in numerous numerous experiments experiments described described by by Uman Uman [1,10-16]. [l, 10-161. A rocket rocket was was launched launched from from the the earth, earth, pulling pulling behind behind it m above it aa thin thin grounded grounded wire. wire. A lightning lightning leader leader was was excited excited at at 200-300 200-300m above the the earth's earth’s surface. surface. The The near-surface near-surface field field during during aa successful successful launching launching 60- 100 V/cm. was usually usually 60-l00V/cm. was Strictly, measurements measurements made made at at two two points, points, at at the the earth earth and and in in the the cloud, cloud, Strictly, are are insufficient insufficient for for an an accurate accurate evaluation evaluation of of an an average average electric electric field. field. The The
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Available A vailable lightning data
km 6-
4-
2-
I
0 1.--------0
Figure 3.1. The model of The 'dipole' ‘dipole’ model of the the charge charge distribution distribution in in aa storm storm cloud. cloud.
space between the cloud cloud and and the the earth earth should should be scanned, and and this this must must be be space between the be scanned, done just before before aa lightning done for for some some fractions fractions of of aa second, second, just lightning discharge, discharge, in in the vicinity of its anticipated trajectory. Unfortunately, such attempts have the vicinity of its anticipated trajectory. Unfortunately, such attempts have not been quite not been quite successful. successful. More More successful successful were were measurements measurements made made at at points points on on the the earth's earth’s surface surface separated separated at at aa distance distance of of hundreds hundreds and and thousands of metres metres [17-19]. [17-191. These These have have been been used reconstruct the thousands of used to to reconstruct the charge distribution distribution within storm cloud, cloud, invoking invoking the of direct direct charge within aa storm the results results of cloud probing. The reconstruction procedure procedure and possible errors cloud probing. The reconstruction and its its possible errors are are discussed in in [20]. [20]. Generally, Generally, with simultaneous field field measurements measurements made made at at with simultaneous discussed n points, points, one one can can write write aa closed closed set set of of equations equations for for the the same same number of number of parameters of regions. Its provides the the parameters, parameters, for parameters of charged charged regions. Its solution solution provides for example, the average pre-delineated regions. regions. example, the average space space charge charge densities densities in in pre-delineated Most often, the number of of points is too small, so so the the results obtained only only Most often, the number points is too small, results obtained permit the point charges. permit the construction construction of of simplified simplified models models with with point charges. Very Very common common is the dipole model model with negative charge charge at at 3-5 3-5 km km above above the earth with with the the is the dipole with aa negative the earth same value of the positive charge raised at a double altitude. Sometimes, same value of the positive charge raised at a double altitude. Sometimes, aa small positive point point charge placed at small positive charge is is added added to to them, them, which which is is placed at aa distance distance by 1-2 1-2 km closer closer to to the the earth earth than than the the negative negative charge. charge. All All point point charges charges are are by be located the same vertical line assumed assumed to to be located along along the same vertical line (figure (figure 3.1). 3.1). The The concept concept of of aa cloud cloud filled filled by by charged charged layers layers of of different different signs signs is is based on probe measurements of of charge charge polarity polarity in in hydrometeors. In this based on probe measurements hydrometeors. In this respect, raises no no doubt. the field respect, this this model model raises doubt. But But as as for for the field distribution, distribution, the the measurement measurement error error is is too too large, large, especially especially for for the the space space in in the the cloud cloud between between two point charges. two point charges. Luckily, Luckily, the the descending descending lightning lightning trajectory trajectory lies lies mostly mostly the cloud, particles. For outside of outside of of the cloud, in in the the air air free free from from charged charged particles. For this this part part of the the average model the trajectory, trajectory, the average field field evaluation evaluation in in terms terms of of aa simple simple model makes makes sense. sense.
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We shall shall illustrate illustrate the the procedure of deriving deriving information information from from such such We procedure of measurements. Suppose Suppose we at our our disposal disposal the the values of field field E) El at at measurements. we have have at values of the earth's earth's surface surface just under an an anticipated anticipated charged charged centre centre of of aa storm storm the just under cloud, as as well well as as the the values values for for field field E E22 at at the lower cloud cloud boundary boundary (also (also cloud, the lower under the the charged charged centre) centre) measured measured during during aa plane flight around around the the under plane flight cloud. The The altitude altitude of of the the lower lower boundary, boundary, h, h, is is also also known. known. Assume Assume the the cloud. to be above the the lower lower cloud cloud boundary centre of of the the main main lower lower charge charge q to centre be above boundary at an an unknown unknown distance distance r.r. If If we we ignore ignore the the effects effects of of the the remote remote upper at upper charge and and of of the the additional additional charge charge lying lying under under the the main main one, one, we we can can charge write write El =
2q 47r&o(h
+ Y)2 '
4 E2 = q + 47reor2 47r&o(2h+ r)2 '
(3.1 (3.1))
The The factor factor 22 in in E, El and and the the second second term term in in E E22 are are due due to to the the action action of of charge charge induced reflection). Since induced in in the the earth's earth's conducting conducting plane plane (mirror (mirror reflection). Since in in reality reality El << E2, it is is natural to suggest suggest that that r « << h. h. Then, Then, with the second second term term in in E, « E 2 , it natural to with the E2 ignored, we find E 2 ignored, we find
q = 27r~o(h+ r)2E1,
rM r~
ah, ah,
a = (E1/2E2)1'2.
(3.2) (3.2)
Substituting Substituting the the values values above, above, i.e., i.e., E) El = lOOV/cm, lOOV/cm, E E22 = = 3000V/cm, 3000V/cm, and and = 3km, 3 km, we we shall shall find find q == 6.3C, 6.3 C, aa == 0.13, 0.13, and and r::::; r M 390m. 390m. The The average average h= field boundary and field in in the the region region between between the the lower lower cloud cloud boundary and the the earth, earth, equal potential 'P2 by the equal to to the the lower lower boundary boundary potential p2 ::::; M q/4-rrcor q / 4 7 r ~ ~divided divided r by the cloud cloud Eav ::::; (E,E distance the earth, h, Ea" ( E l E22/2)'/2 / 2 ) 1 1 2::::; M 390V/cm. 390V/cm. Allowance Allowance for for distance from from the earth, h, the E 2 (the reflection by the second second term term in in E2 (the effect effect of of mirror mirror charge charge reflection by the the earth) earth) can hardly be justified, because because our not take can hardly be justified, our model model did did not take into into account account the the effect upper charge effect of of the the upper charge of of opposite opposite sign. sign. This This charge charge is is closer closer to to the the cloud cloud edge edge than than that that reflected reflected by by the the earth earth and and has, has, therefore, therefore, aa greater greater effect E 2 . Its though it effect on on E2, Its consideration, consideration, however, however, simple simple though it may may seem, seem, would require require field would field measurement measurement at at another another point point of of space space and and another another equation new unknown unknown -- the the upper equation for for finding finding aa new the altitude altitude of of the upper charge charge centre centre of of the the dipole. dipole. A would, probably, probably, be be necessary A larger larger scale scale correction correction would, necessary to to account account for for the space the the near-earth near-earth field, the effect, effect, on on the field, of of the space charge charge induced induced by by coronas coronas from pointed grounded branches, high high grass, from pointed grounded objects objects (tree (tree branches, grass, various various buildings, buildings, etc.) [21]. [21]. Estimations Estimations made at the the earth's earth's surface surface show show that that this this charge charge etc.) made just just at reduces the reduces the actual actual field field of of aa storm storm cloud cloud by by half. half. So So one one cannot cannot say say that that lightning moves moves in in an an unusually low electric electric field. field. These These are are just the values values lightning unusually low just the at which superlong at which superlong sparks sparks are are excited excited in in laboratory laboratory conditions conditions (see (see chapter to invent chapter 2). 2). Therefore, Therefore, there there is is no no need need to invent aa special special propagation propagation mechanism for of aa long long laboratory laboratory spark, spark, if if mechanism for lightning, lightning, different different from from that that of we deal breaking down we deal with with average average electric electric fields fields capable capable of of breaking down the the cloudcloudearth gap. gap. earth
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94 94
3.2 3.2
Available lightning data
The The leader of the first lightning lightning component
The in unperturbed unperturbed air, air, so so The leader leader of of the the first first component component oflightning of lightning develops develops in only directly with with laboratory laboratory data data only this this leader leader behaviour behaviour should should be be compared compared directly on be carried out along along two two lines. lines. We We on long long spark spark leaders. leaders. The The comparison comparison can can be carried out can and aa spark spark and, and, second, second, can first first compare compare the the leader leader structure structure in in lightning lightning and their velocities. their quantitative quantitative parameters, parameters, primarily primarily velocities. 3.2.1 3.2.1
Positive Positive leaders leaders
Streak to interpret. interpret. So So we we shall shall begin begin Streak pictures pictures of of aa positive positive leader leader are are easy easy to with is not not frequent. frequent. Many Many with positive positive lightnings, lightnings, though though their their occurrence occurrence is books photographs of of aa descending descending books and and papers papers refer refer to to the the successful successful streak streak photographs positive in 1966 [22]. Its Its schematic schematic positive leader leader taken taken by by Berger Berger and and Fogelsanger Fogelsanger in 1966 [22]. diagram became accessible accessible to to photophotodiagram is is reproduced reproduced in in figure figure 3.2. 3.2. The The leader leader became graphy moved down down in in aa continuous continuous graphy at at 1900m 1900m above above the the earth's earth's surface. surface. It It moved mode, variation. The The average average leader leader velocity velocity mode, without without an an appreciable appreciable intensity intensity variation. over mis, increasing increasing somewhat somewhat as as the the over the the registration registration time time was was 1.9 1.9 x 10 lo66 m/s, leader is much much faster faster than than aa long long leader tip tip approached approached the the earth. earth. Such Such aa leader leader is laboratory spark, spark, whose whose velocity velocity is is 50-100 50-100 times laboratory times lower lower at at minimum minimum breakbreakdown grounded electrode. electrode. down voltage voltage before before the the streamer streamer zone zone contacts contacts aa grounded In the the streak streak picture, picture, the the leader leader tip looks much In tip looks much brighter brighter than than its its channel, channel, but no no signs signs of of aa streamer streamer zone zone can can be identified. We but be identified. We cannot cannot say say how how the the original negative negative looks, looks, but but in in the the published published photograph original photograph the the resolution resolution threshold m. It very large large value value for for aa streamer streamer threshold is is hardly hardly less less than than 50 50m. It is is aa very
leadp channel leader
rl~~~~ 1.9k.m
L-
return
E stroke
t
b1.5 ms ms -A , -..-I 1+---1.5 Figure 3.2. 3.2. Schematic Schematic streak streak picture picture of of aa positive descending lightning Figure positive descending lightning leader leader regisregis[22]. tered on on the the San San Salvatore Salvatore Mount Mount in in Switzerland Switzerland [22]. tered
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The leader of of the first lightning component
T
95
leader channel
lkm
1
t 3ms
Figure 3.3. Schematic Schematic streak streak picture [22] of of aa positive ascending leader leader from from aa 70-m 70-m Figure picture [22] positive ascending tower on on the the San San Salvatore Salvatore Mount. Mount. tower
zone. With With the the data data of of section section 2.4.1, 2.4.1, we we can can calculate calculate the the leader leader tip tip potential potential zone. U,, at which which the the streamer streamer zone zone will will exceed exceed at at least least twice twice the the length length of of about about U t , at loom, threshold value value for for the the measuring measuring equipment. equipment. For For the the average average 100 m, aa threshold streamer zone zone field field E E,s :::::: = 55 kV/cm, kVjcm, the the potential potential derived derived from from (2.39) (2.39) is is streamer U,t :::::: % 100 100 MV. MV. Such Such aa high high value value cannot cannot be be typical typical of of lightning lightning with with average average U parameters. There There is is another another circumstance circumstance preventing preventing streamer streamer zone zone registraregistraparameters. tions -- different different radiation radiation wavelengths wavelengths of of aa hot hot leader leader channel channel and and aa cold cold tions streamer zone. zone. Violet Violet and and ultraviolet ultraviolet radiation radiation from from streamers streamers is is dissipated dissipated streamer by water water vapour vapour and and rain rain droplets droplets in in the the air air much much more more than than long long wavelength wavelength by radiation characteristic characteristic of of aa mature mature channel. channel. At At aa distance distance of of about about aa kilokiloradiation metre between between the the lightning lightning and and the the registration registration site site (closer (closer distances distances are are metre practically unfeasible), unfeasible), aa streamer streamer zone zone may may become become quite quite invisible invisible to to the the practically observer's observer's equipment. equipment. Note Note that that this this is is totally totally true true of of optical optical registrations registrations of of an an ascending ascending positive positive leader. leader. schematic streak streak picture picture of of an an ascending ascending positive positive leader, leader, based based on on 18 18 A schematic successful registrations registrations [22], [22],is is shown shown in in figure figure 3.3. 3.3. All All lightnings lightnings started started from from successful aa 70-m 70-m tower tower on on the the San San Salvatore Salvatore Mount Mount near near the the Lake Lake Lugano. Lugano. The The leader leader does not not exhibit exhibit specific specific features features that that would would distinguish distinguish it it from from aa long long laboralaboradoes tory spark. spark. On On the the whole, whole, it it developed developed in in aa continuous continuous mode mode with with irregular irregular tory short-term short-term enhancements enhancements of of the the channel channel intensity. intensity. Normally, Normally, they they did did not not accelerate the the leader leader development. development. Something Something like like this this has has been been observed observed accelerate in in aa long long laboratory laboratory spark. spark. The The streak streak picture picture in in figure figure 3.4 3.4 demonstrates demonstrates this with with reference reference to to aa positive positive leader leader in in aa sphere-plane sphere-plane gap gap 99 m m long. long. But But this this phenomenon phenomenon has has nothing nothing to to do do with with aa stepwise stepwise elongation elongation of of aa negative negative this leader channel. channel. leader The velocity velocity of of an an ascending ascending positive positive leader leader near near the the starting starting point point is is The 4 104m/s. From some some data data close to to that that of of aa laboratory laboratory spark, spark, about about 22 x 10 close m/s. From 4 [22], m/s for [22], itit was was in in the the range range of of (4-8) (4-8) x 10 104m/s for aa channel channel length length of of 4040100m; but when when the the leader leader tip tip was was at at aa height height of of 500-1150 500-1 150m, it increased increased 100 m; but m, it 5 by nearly nearly an an order order of of magnitude, magnitude, to to 10 105-106 mjs. by -1 06 m/s.
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A vailable lightning data Available
the initial positive leader Figure 3.4. 3.4. A streak streak photograph photograph of of the initial stage stage of of aa positive leader in in aa 9-m 9-m rodrodof the channel. plane gap, displaying displaying short short flashes flashes of plane gap, the channel.
3.2.2 Negative leaders
Negative lightnings occur occur more more frequently frequently than than positive, and their their registraNegative lightnings positive, and registramain distinguishing negative tion is tion is more more common. common. The The main distinguishing feature feature of of the the negative leader leader of of the the first first lightning lightning component component is is its its stepwise stepwise character. character. The The leader leader tip leaves trace in pictures which tip leaves aa discontinuous discontinuous trace in streak streak pictures which look look like like aa movie movie film pictures in film (figure (figure 3.5). 3.5). One One can can sometimes sometimes find find such such pictures in aa sports sports magazine magazine illustrating the the successive successive steps steps in in aa sportsman's sportsman’s performance. performance. The The bright illustrating bright flash behind it flash of of the the tip tip and and the the channel channel right right behind it are are followed followed by by aa dead dead zone zone by another with practically zero with practically zero intensity. intensity. This This is is followed followed by another flash flash showing showing that the the tip tip has has moved moved on on for for several several dozens dozens of of metres. metres. Such Such negative that negative his group back as leader behaviour was leader behaviour was observed observed by by Schonland Schonland and and his group as as far far back as the pause between the 1930s 1930s [24,25]. [24,25]. According According to to their their registrations, registrations, the the average average pause between the JlS, with with aa spread JlS, and the steps steps was was close close to to 60 60ps, spread from from 30 30 to to 100 loops, and the the step step
L////li 1
-I,
leader
--re;
o.5ms-1
t
•
Figure 3.5. 3.5. Schematic Schematic streak streak picture picture of of aa descending descending negative negative leader. leader.
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of the first lightning component The leader of
97 91
1.0 0.8
.S' 0.6 :=: '§ ~
~0.4 0.2 -
o--,..I
o0
Ji a ......-,....5
~-,..----.----
2.0
Ii
5 5,
6.0
4.0
Ii
10
Ii
15
Ii
20
20
i
1
25
25
lOm/s
•
b
Jf p Yp
3.6. integral velocity velocity distributions for descending descending leaders leaders of of the the aand Figure 3 .6. Typical Typical integral distributions for CY- and 25]). f3-types (from [24, P-types (from [24, 251).
length varied varied between between 10 and 200 200 m m with with the the average average value value being being 30 m. The The length 10 and 30 m. duration of of aa step is likely likely to to be be within within several microseconds. The duration step is several microseconds. The available available streak photographs are not good to identify the details details of of aa step. streak photographs are not good enough enough to identify the step. In any case, it hard to to decide whether it it is is similar to aa step the long In any case, it is is hard decide whether similar to step of of the long negative spark spark described described in in section 2.7. negative section 2.7. A A stepwise stepwise negative negative leader leader approaches approaches the the earth earth at at an an average average velocity velocity 5 of _10 6 m/s. Two of 10 105-106m/s. Two descending descending leader leader types types can can be be identified identified in in terms terms of of their their velocity: velocity: slow slow a-leaders a-leaders and and fast fast ;3-leaders. P-leaders. The The former former travel travel at at their their step-averaged step-averaged velocity; velocity; it it varies varies with with the the discharge discharge in in the the range range of of 5 (1-8) 105 mls with value of m/s. The (1-8) xx105m/s with the the average average value of 33 x 10 105m/s. The respective respective ;3Pleader parameters parameters are are 3-4 times higher. higher. This This can can be be seen in figure 3.6 showing showing leader 3-4 times seen in figure 3.6 the integral integral velocity velocity distributions in [24,25]. Usually, ;3-leaders the distributions described described in [24,25]. Usually, P-leaders are are more branched branched and their steps steps are longer. They slow down down when when more and their are longer. They abruptly abruptly slow they approach approach to to the the earth, after which which they they behave behave as as a-leaders. a-leaders. they earth, after An ascending ascending negative negative leader leader also also has has characteristic characteristic steps. Most of the An steps. Most of the 13 13 registered registered leaders leaders ascending ascending from from aa 70-m 70-m tower tower on on the the San San Salvatore Salvatore Mount Mount [22] [22] were were identified identified as as a-leaders. a-leaders. They They have have relatively relatively short short steps steps (5-18 (5-18 m) m) and and aa velocity velocity of of (1.1-4.5) (1.1-4.5) xx 10 lo55 m/s. mjs. Two Two of of the the discharges discharges were were referred referred to to ;3-leaders @-leadersbecause because their their velocity velocity was was (0.8-2.2) (0.8-2.2) xx 10 lo66 mls m/s and and the the m. On step step length length up up to to 130 130m. On the the whole, whole, ascending ascending and and descending descending stepwise stepwise leaders leaders do do not not show show significant significant differences. differences. The The registrations registrations of of ascending ascending discharges discharges from from the the San San Salvatore Salvatore Mount Mount provide provide direct direct evidence evidence for for the the existence existence of of aa streamer streamer zone zone in in aa lightning lightning leader. leader. Registrations Registrations made made at at aa sufficiently sufficiently close close distance, distance, which which became became
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98
A vailable lightning data Available
streamer zone
"'"
-20~s
T 25m
1
-
Figure Figure 3.7. Schematic Schematic diagram diagram of of the the initial initial development development of of aa leader leader ascending ascending from from 70-m tower tower on on the the San San Salvatore Salvatore Mount, Mount, as as viewed viewed from from close close distance distance streak streak aa 70-m photographs. photographs.
possible due to to the the tower tower top top being the only only starting starting point, show streamer streamer possible due being the point, show new step flashes flashes arising arising at at the the moment moment aa new step begins. begins. Streamers Streamers were were initiated initiated not only only from from the the tip tip of of the the main main channel channel but also from from its its branches branches not but also (figure (figure 3.7). 3.7).
3.3
lightning components components The leaders of subsequent lightning
Leaders the first Leaders of of lightning lightning components components following following the first one one are are known known as as dart dart leaders because because of of the the absence absence of of branches. The streak streak photograph in leaders branches. The photograph in bright tip tip looking figure shows the the trace trace of of only only one one bright looking like like aa sketch sketch of of an an figure 3.8 shows arrow or or dart. dart. A A dart dart leader leader follows follows the channel of of the the previous lightning arrow the channel previous lightning 7 m/s. component velocity up lo7 mjs. Averaging Averaging over over many many registraregistracomponent with with aa velocity up to to 44 xX 10 7 107m/s, the minimum minimum values values being being an an tions gives gives the the value (1-2) x 10 tions value (1-2) mis, with with the of magnitude magnitude less less than than the the maximum maximum one one [23,25]. [23,25]. The The dart dart leader leader order of order velocity not vary on the the way way from from the the cloud cloud to to the the earth. earth. velocity does does not vary much much on
t return stroke of ub equent component
/'
schematic streak streak picture of aa dart dart leader. leader. Figure 3.8. A schematic picture of
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The The leaders of of subsequent lightning components
99
Figure 3.9. A streak streak photograph photograph of of aa well-branched well-branched lightning lightning striking striking the the Ostankino Ostankino the components components along along the and B B are are formed formed at at different different Television Tower; Tower; the Television the branches branches A and time. moments moments of of time.
Similar recently been photographs Similar results results have have recently been obtained obtained from from 23 23 streak streak photographs offairly by aa camera with of fairly good good quality quality showing showing dart dart leaders leaders taken taken in in Florida Florida by camera with [26]. The The average average velocity velocity of of aa dart dart leader leader varied varied from from time resolution resolution O.51ls 0.5 ps [26]. aa time 6 lo6 to 2.5 2.5 Xx 10 lo77 mls m/s in in some some registrations; it was was (1.6-1.8) (1.6-1.8) x 10 lo77 mls m/s for for 55 x 10 to registrations; it three typical typical pictures in the the publication. publication. three pictures presented presented in It is is clear clear that that aa dart dart leader leader somehow somehow makes makes use use of of the the previous channel It previous channel with temperature, gas with aa different different temperature, gas density density and and composition. composition. There There are are several several tendency for the dart indications indications to to this. this. First, First, there there is is aa tendency for the dart leader leader velocity velocity to to decrease with increasing pause. This decrease with increasing duration duration of of the the interleader interleader pause. This is is because because the the trace trace channel to return return to the gas gas in in the channel is is gradually gradually cooled cooled to to the the original original condition. If If aa pause pause lasts lasts longer, longer, the subsequent component component may may take take its its condition. the subsequent own own way. way. Figure Figure 3.9 3.9 shows shows aa lightning lightning discharge discharge which which struck struck the the Ostankino Ostankino Television Tower Tower in in Moscow. Moscow. Some Some of of its its initial initial components components followed followed aa Television common but then the discharge Naturally, common channel channel but then the discharge trajectory trajectory changed. changed. Naturally, unperturbed air there there is is nothing nothing like like aa dart dart leader leader in in unperturbed air -- the the leader leader of of each each next component component develops develops in in aa step-wise step-wise manner. manner. next [ 15,271 Second, it it has has been been found found in in triggered triggered lightning lightning investigations investigations [15,27] Second, that aa dart dart leader leader requires requires aa current-free current-free pause for its its development. development. The The that pause for long-term which supports period long-term current, current, which supports the the channel channel conductivity conductivity in in the the period between two subsequent subsequent components, components, must entirely cease cease to to allow allow the the between two must entirely channel to lose its its conductivity. conductivity. Only Only then channel be ready channel to partly partly lose then will will the the channel be ready to to serve serve as as aa duct duct for for aa dart dart leader. leader. But But if if aa new new charged charged cloud cloud cell cell is is involved involved and raises raises the of the the channel channel with with current, current, an an M-component M-component is is and the potential potential of produced produced instead instead of of aa dart dart leader. leader. Its Its distinctive distinctive feature feature is is aa higher higher intensity intensity of the existing of the existing channel channel lacking lacking aa well well defined defined tip tip (figure (figure 3.10). 3.10). The The absence absence of of
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Available lightning lightning data
Figure 3.10. A streak photograph of well-branched lightning streak photograph of aa well-branched lightning with with M-components. M-components.
aa clear clear luminosity luminosity front front is is aa serious serious obstacle obstacle to to the the measurement measurement of of its its velocity. point to the velocity. Investigators Investigators often often point to aa nearly nearly simultaneous simultaneous increase increase of of the light light intensity intensity in in the the whole whole channel. channel. This This suggests suggests either either an an almost almost subsublight boundary of light velocity velocity of of the the leader leader or or an an exceptionally exceptionally smeared smeared boundary of its its front. In In contrast contrast to dart leader, leader, an an M-component M-component is is never never followed followed by front. to aa dart by aa distinct return stroke with high (l (10-100 kA) rapidly rapidly rising rising current current impulse. impulse. distinct return stroke with aa high 0-100 kA) Both the the external external view view and and photometric data obtained obtained from from streak streak Both photometric data made an photographs reveal photographs reveal clearly clearly aa dart dart leader leader tip. tip. Some Some authors authors [26] [26] made an attempt time variation attempt to to measure measure the the time variation of of the the leader leader light light intensity. intensity. Although Although the were performed the measurements measurements were performed near near the the time time resolution resolution limit, limit, they they indiindicate that that the the light light pulse front at at the the registration registration point rises for for 0.5-1I1S 0.5-1 ps cate pulse front point rises the dart and and then then is is stabilized stabilized for for 2-6I1S. 2-6 ps. Therefore, Therefore, with with the dart leader leader velocity velocity of of 7 lo7 m/s, the the extension extension of of the the front front rise rise is is 7.5-15m 7.5-15m with with the the full full pulse pulse 1.5 x 10 1.5 mis, m. It that the the front front length length of of 35-105 35-105m. It can can be be mentioned, mentioned, for for comparison, comparison, that M-component pulse front, kilometre length. M-component has has aa pulse front, if if any, any, of of aa kilometre length. It It is is important important for for the the theory theory of of dart dart leaders leaders that that they they always always move move from from aa cloud means that cloud down down to to the the earth. earth. This This means that the the voltage voltage source source that that excites excites the trace previous component them is them is 'connected' 'connected' to to the trace channel channel of of the the previous component right right in in the cloud. The The direction direction of of the leader does does not not matter matter much much because because the cloud. the previous previous leader their channels channels are are equally equally suitable suitable for for the the development development of of aa dart dart leader. leader. their
3.4
Lightning Lightning leader current current
We the values We can can only only guess guess about about the values of of descending descending leader leader currents currents or or estimate estimate them from them from indirect indirect data. data. We We shall shall make make such such estimations estimations in in section section 3.5, 3.5, using using leader leader charge charge data, data, also also obtained obtained indirectly. indirectly. Ascending Ascending leader leader currents currents are are Copyright © 2000 IOP Publishing Ltd.
Lightning leader current
101 101
1 - - - - - - 20 IDS - - - - I
o+-~
--+--=t.
200 400 600 Figure 3.11. A schematic schematic oscillogram current in oscillogram of of the the leader leader current in an an ascending ascending lightning. lightning.
not not difficult difficult to to measure, measure, and and there there have have been been many many measurements measurements of of this this kind. current detector is mounted mounted on on top of aa tower kind. Normally, Normally, aa current detector is top of tower dominating dominating the the locality locality [28-30]. [28-301. The The current current impulse impulse of of an an ascending ascending leader, leader, registered registered on on an an oscillogram, oscillogram, lasts lasts for for about about 0.1 0.1 s, s, corresponding corresponding to to the the time time of of ascending ascending leader development. development. The The current current nearly nearly always always rises rises in in time time (figure (figure 3.11). 3.11). The The leader current supplies supplies an an elongating elongating leader leader with with charges. charges. Physically, Physically, these these charges charges current are induced induced by by the the electric electric field field of of aa cloud. cloud. When When aa leader leader approaches approaches aa cloud, cloud, are going through through an an increasingly increasingly higher higher field, field, the the linear linear density density of induced going of induced charge TT increases. increases. Besides, Besides, the the leader leader goes goes up up with with an increasing velocity velocity charge an increasing V reducing the the time time for the charge charge supply. A combination combination of these factors factors V,,L , reducing for the supply. A of these raises the the current current i == TV At the the moment moment an leader starts raises 7VL. an ascending ascending leader starts its its L . At travel, current is 10A, end of travel, its its current is lower lower than than 10 A, whereas whereas at at the the end of the the travel, travel, it it may rise rise to to 200-600 200-600 A, with may with an an average average value value of of about about 100 100 A. A. Sometimes, Sometimes, just before just before the the leader leader begins begins its its continuous continuous elongation, elongation, impulses impulses with with an an amplitude of of several several amperes amperes may may arise arise against against the the background background of of aa millimilliamplitude ampere ampere corona corona current. current. Current triggered from thin wire Current oscillograms oscillograms of of an an ascending ascending leader leader triggered from aa thin wire picture. elevated by aa small rocket to to 100-300m elevated by small rocket 100-300m [13,31] [13,31] give give aa similar similar picture. They show the the same rising impulse impulse with with an an amplitude of 100-200 They show same slowly slowly rising amplitude of 100-200AA and duration duration 50-looms. 50-100 ms. It has no no overshoots overshoots at at the the front, front, even if the the and It has even if leader goes goes up up in in aa stepwise mode. leader stepwise mode. There are are no no reasons reasons to to suggest any principal principal difference difference between between average average There suggest any currents currents of of ascending ascending and and descending descending leaders. leaders. In In both both cases, cases, the the leader leader is is supplied by by charges charges induced induced by by the the electric electric field field of of aa storm storm cloud, cloud, and and the the supplied leader leader lifetimes lifetimes are are comparable comparable because because they they move move at at approximately approximately the the same same velocity. velocity. Qualitatively, the current variation of the first component component leader is Qualitatively, the current variation of the first leader is similar to to that that of of aa laboratory laboratory spark. When the the gap gap voltage voltage is is raised raised similar spark. When slowly, one one can can observe initial leader leader flashes flashes at at the the high-voltage high-voltage electrode, electrode, slowly, observe initial followed by by distinct distinct current current impulses impulses [32]. As for for long long spark spark steps, they followed [32]. As steps, they practically do do not not change change the the current current at the leader leader base. base. It has been been shown practically at the It has shown
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102 102
Available lightning data
[20] this is to be perturbation region [20] that that this is to be expected expected if if the the charge charge perturbation region is is separated separated from by an with high high from the the registration registration point point by an extended extended channel channel section section with resistivity resistivity and and distributed distributed capacitance capacitance (section (section 4.4). The The perturbation perturbation wave wave travelling the channel travelling along along the channel towards towards the the detector detector is is attenuated. attenuated. Of Of course, course, the the current current of of aa laboratory laboratory spark spark rarely rarely exceeds exceeds aa few few amperes, amperes, but but such such aa difference difference is is predictable. predictable. it it follows follows from from the the expression expression for for the the current current ii = higher velocity = TV T VL, cited cited above. above. A A lightning lightning leader leader has has an an order order higher velocity VL, and, and, at at least, least, an an order order larger larger linear linear charge charge TT (due (due to to the the voltage voltage being being 10-20 higher). All All in 10-20 times times higher). in all, all, this this increases increases the the current current to to within within the the anticipated two orders of of magnitude. magnitude. anticipated two orders no Now one judge about Now one can can judge about the the current current of of aa dart dart leader. leader. There There are are no direct was an direct registrations registrations of of this this current. current. One One exception exception was an attempt attempt at at its its measurement in triggered lightning just before before its with the the earth. measurement in aa triggered lightning just its contact contact with earth. This principally possible possible since point of This is is principally since the the point of contact contact is is known known exactly exactly -this is is the the point point of of wire wire fixation fixation to to the the earth. earth. The The wire wire evaporates evaporates completely, completely, this having passed the having passed the current current of of the the first first lightning lightning component. component. Using Using the the still still hot hot trace trace channel, channel, aa dart dart leader leader follows follows the the path path of of the the wire. wire. A A current current detector detector can can be be placed placed at at the the wire wire grounding grounding site. site. It is is much much more more difficult difficult to to interpret interpret the the recorded recorded oscillograms, oscillograms, because It because it time the it is is unclear unclear at at what what moment moment of of time the development development of of aa dart dart leader leader stops stops and the the return return stroke stroke with the high current begins. begins. Nevertheless, the and with the high current Nevertheless, the kA with published published current current measurements measurements vary vary from from 0.1 0.1 to to 66kA with the the average average value of of 1.7 1.7 kA kA [33]. [33]. The The lower lower limit limit of of the is more more typical for the value the range range is typical for the first component component (this (this may may be the next component, too, too, but but after after aa long long first be the next component, current-free pause, when previous trace totally current-free pause, when the the previous trace channel channel has has nearly nearly totally decayed). The The value value of of several several kiloamperes kiloamperes seems seems reasonable, reasonable, since since the the decayed). velocity of higher than velocity of aa dart dart leader leader is is 30-50 30-50 times times higher than that that of of the the first first component. component.
3.5
Field Field variation variation at the leader stage
The subdivision subdivision of of experimental experimental data data between between this and the the previous previous section section is is The this and somewhat somewhat arbitrary. arbitrary. Electric Electric field field measurements measurements provide provide information information about about leader while charge leader charge, charge, while charge and and current current are are related related by by leader leader velocity. velocity. On On the the whole, this is is aa general general problem. problem. If If the the observations observations were were arranged arranged properly whole, this properly was made measureand and the the data data analysis analysis was made carefully, carefully, relatively relatively simple simple field field measurements can can add add much much to to our our knowledge knowledge of of electrical electrical parameters parameters of of lightning. lightning. ments The knowledge knowledge of of the field itself itself is is rarely of importance, importance, probably, probably, except except in in the field rarely of The protection of some some applied applied problems problems of of lightning lightning protection of low low voltage voltage circuits. circuits. For For this reason, it is is not not the the measurements measurements but, rather, methodological methodological this reason, it but, rather, approaches to to their their treatment treatment which which are are significant. significant. So we we shall shall begin begin with with approaches the general principles underlying underlying aa treatment these treatment of of most most these approaches approaches and and the general principles lightning lightning stages. stages.
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Field variation at the leader stage Field
103 103
;;n;;)) ~;;7));';'
C
~
-
.Ue
Measurement of of fast fast variations variations in in the the electric electric field field during during the the lightning lightning Figure 3.12. Measurement development. development. Suppose the electric field field at at an an observation observation point earth’s surface surface Suppose the electric point near near the the earth's is Eo(O) at at the the moment moment of of the the lightning lightning start. start. When When the the discharge discharge is is comcomis Eo(0) pleted, the the field value Eo(tm). Eo(tm). The by the pleted, field takes takes aa new new value The field field has has changed changed by the value AEo = Eo(tm) over the the time time tt,.m. When When the measurement time time value I::!.Eo = Eo(tm) -- Eo(0) Eo(O) over the measurement is register the is relatively relatively short, short, it it is is more more convenient convenient to to register the field field change change rather rather than values. This usually done than to to measure measure its its values. This is is usually done with with electrostatic electrostatic antennas, antennas, i.e., through aa reservoir reservoir capai.e., metallic metallic conductors conductors (normally, (normally, flat) flat) grounded grounded through capacitor citor C (figure (figure 3.12). 3.12). If If the the capacitor capacitor and and the the measurement measurement circuits circuits connected connected to it it have have an an infinitely infinitely high leakage resistance the capacitor capacitor voltage, voltage, at at to high leakage resistance RI, R I , the any moment of time, is any moment of time, is U ( ) _ qc(t) _ EoEo(t)Sa c t - C C
(3.3) (3.3)
where Sa Sa is is the the area area of of aa flat flat antenna antenna and and qc is is the the charge charge induced induced on on it. it. If If RI where RI is resistance of is finite finite (which (which is is always always the the case case due due to to the the input input resistance of the the circuit circuit taking voltage voltage readings readings from the capacitor), requires the (3.3 ) requires the concontaking from the capacitor), the the use use of of (3.3) dition be easily the lightning dition RIC» RIC >> tt,,m , which which can can be easily met met for for the lightning duration", duration ~ 1100 2- ss* but becomes problematic between two but becomes problematic for for aa time time interval interval of of several several minutes minutes between two flashes. flashes. I::!.Eo(tm) requires An An accurate accurate measurement measurement of of the the field field change change AEo(t,) requires the the necessary necessary time time constant constant of of the the measurement measurement circuit circuit RIC RIC and and the the account account of of effects Eo, by effects of of external external field field variation variation in in the the atmosphere, atmosphere, Eo, by making making allowance allowance for the earth, mounted for local local effects. effects. (The (The antenna antenna may may be be raised raised above above the earth, say, say, mounted on building roof, that the be higher on aa building roof, so so that the field field there there will will be higher than than on on the the earth. earth. On On the hand, aa nearby nearby high the other other hand, high construction construction may may reduce reduce the the field, field, acting acting as as an an electrostatic value obtained electrostatic screen.) screen.) The The field field value obtained is is not not particularly particularly informative. informative. In we have In order order to to get get information information about about the the lightning lightning discharge, discharge, we have to to make make which have have certain certain assumptions assumptions concerning concerning the the distribution distribution of of charges charges which changed the the field. field. changed Let us begin begin with with aa simple simple illustration. illustration. Suppose Suppose aa lightning lightning leader leader Let us passing passing from from aa spherical spherical volume volume has has changed changed the the charge charge of of only only one one sign sign
Copyright © 2000 IOP Publishing Ltd.
104 104
A vailable lightning data Available
in in the the sphere sphere charge charge during during in aa storm storm cloud cloud cell. cell. If If there there are are other other changes changes in the leader travel, they are assumed to have been completely neutralized later, the leader travel, they are assumed to have been completely neutralized later, at the return stroke stage. If this assumption is correct, the measured value of at the return stroke stage. If this assumption is correct, the measured value of ~Eo(tm) can give an idea about the quantity of charge transported by the A E o ( t m )can give an idea about the quantity of charge transported by the leader leader from from the the cloud cloud to to the the earth: earth:
+
21fEo(H2 R2)3/2~Eo(tm) 2 7 r ~ o ( H+~R2)3/2AEo(tm) (3.4) H . (3.4) H Here, and R is is its its radial radial disdisHere, H is is the the altitude altitude of of the the charged charged storm storm centre centre and placement point. Both Both parameters parameters should should be be placement relative relative to to the the registration registration point. field registrations registrations at at measured measured by by an an independent independent method method or or simultaneous simultaneous field two from the the first first one. one. This This two more more points points should should be be made made at at given given distances distances from unknown values values of of H Hand R. will and R. will provide provide additional additional equations equations for for the the unknown Such from the the simple simple model model we we have have Such an an unambiguous unambiguous treatment treatment results results from parameter except except for for the the distance distance to to chosen, chosen, which which contains contains no no geometrical geometrical parameter the dipole model model deprives deprives the the the charge. charge. However, However, aa slightly slightly more more complicated, complicated, dipole electric field field measurements measurements measurement measurement treatment treatment of of this this advantage. advantage. Still, Still, electric have researchers owing owing to to their their have always always been been attractive attractive to to lightning lightning researchers of simplicity. with the the application application of simplicity. Interest Interest in in such such measurements measurements increased increased with lightning triggering triggering by by small small rockets rockets raising lightning raising aa grounded grounded wire wire to to 150-300 150-300 m m above the the earth's earth’s surface surface (triggered (triggered lightning). lightning). The The first first component component of of such such above lightning is is genuinely genuinely artificial, artificial, but but then then the the first first trace trace channel channel is is used used by by lightning practically natural natural dart dart leaders leaders travelling travelling to to the the earth. earth. Their Their point point of of contact contact practically so field field detectors detectors can with the the earth earth is is predetermined, predetermined, so can be be placed placed at at any any disdiswith tance from from the the leader. leader. This This registration registration system system is is quite quite sensitive sensitive and and capable capable tance of responding responding to to the the linear linear charge charge density density not far from of not far from the the leader leader tip tip when when it it approaches the earth. approaches the earth. To illustrate illustrate our our analysis, analysis, we we shall shall use use the field measurements To the field measurements described described in [34,35]. The authors of this work kindly made them available to to us us after after in [34,35]. The authors of this work kindly made them available their discussion at the IXth International Conference on Atmospheric Electheir discussion at the IXth International Conference on Atmospheric Elec1992. The The files files contained contained detailed tricity, held held in in St. St. Petersburg Petersburg in in 1992. of tricity, detailed records records of electric fields, taken during the flight of dart leaders, and of their electric fields, taken during the flight of dart leaders, and of their return return stroke currents. currents. Detectors Detectors were were placed placed at at the the distance distance of of R = = 500m 500 m and and stroke 30m from the contact point. Regretfully, the recordings at these 30 m from the contact point. Regretfully, the recordings at these distances distances were not not simultaneous simultaneous but but made made in in different different years. were years. Their Their comparison comparison is is still possible possible because because the the fields fields were were recorded recorded at still at the the same same time time as as the the return stroke stroke currents. currents. By By sorting sorting out out identical identical current return current oscillograms, oscillograms, one one can select select lightning lightning discharges discharges with with about about the can the same same leader leader tip tip potentials. potentials. This provides provides close close values values of of leader leader velocity velocity and and linear This linear charge charge density density in in the charge charge cover cover not not too too far far from from the the leader leader tip. Some representative the tip. Some representative oscillooscilloAE(t)/AEmax normalized normalized by grams of of ~E(t)/~Emax grams by their their amplitudes amplitudes are are shown shown in in figure 3.13. 3.13. They They correspond correspond to to discharges discharges with figure with really really close close currents currents in in the the (IM =6 6 kA kA at at point point R = = 500 500 m return strokes strokes (IM return = m and and 77 kA kA at at point point 30 30 m). m). The The amplitude values values of of field field variation variation ~Emax AE,,, over over the amplitude the time time of of the the dart dart leader leader ~QM AQM == =
Copyright © 2000 IOP Publishing Ltd.
105 105
Field variation at the leader stage
o 0
100
300 300
200
400
500
I
1.0 1.0-
0.8 0.8
~
0.6 ~ 0.6
w ~
-
0.4 0.4
0.2
20
10 10
Time, J.!S ps Time,
40 40
30
Oscillograms taken in Florida, Florida, USA [35], [35], from from the the vertical vertical field field comcomFigure 3.13. Oscillograms taken in ponent during ponent during the the development development of of the the dart dart leader leader in in the the subsequent subsequent component component of of lightning. The The detectors detectors were were positioned at 30 30 and and 500 500 m m from from the the point aa triggered triggered lightning. positioned at point of strike; strike; the the pulses are related related to to their their maximum maximum amplitudes. amplitudes. of pulses are 6.9 V/cm and 120 120 V V/cm, respectively. Note Note that that the the measuremeasuretravel were 6.9 travel were Vfcm and fcm, respectively. ments in in [35] [35] result result in in 6.E AEm,,/IM M const const at at every every point. There is is no ments point. There no max IIM ::::::: geometrical similarity similarity between between the AE( t)t)/AEmax at the the different different geometrical the pulses pulses 6.E( 16.Emax at points. On the contrary, there there is is aa sharp sharp difference difference in in the the rates of strength strength rates of points. On the contrary, rise, as was approaching the earth. rise, as aa dart dart leader leader was approaching the earth. The The field field increase increase in in the the range (0.5-1.0)6.E (0.5-1.0)AEm,, took 6.tl/2 Atl12 = 76J.!s 76ps for for point = 500m 500m and and only only range point R = max took ps for for point 30 m. m. 55 IlS point R == 30 These be treated treated in which aa dart These data data will will be in terms terms of of aa simple simple model, model, in in which dart leader with linear leader is is represented represented as as aa uniformly uniformly charged charged axis axis with linear charge charge density density rL. Naturally, the real real cover cover radius can be ignored in in the the field field calculation calculation TL' Naturally, the radius R, R c can be ignored R. We at at aa distance distance R. We shall shall show show below below that that field field calculations calculations can can only only take take into into account the charge distribution distribution along along aa relatively relatively short short length length behind behind the the tip, tip, account the charge comparable with R. R. This justify the the assumption This will will justify assumption of of TL rL being being constant, constant, comparable with near the because it refers to because it actually actually refers to aa short short length length of of about about R near the tip. tip. Therefore, Therefore, at the the earth, earth, with the the field field change change due due to the leader leader charge charge at at point point R at with the the to the allowance by the allowance for for its its mirror mirror reflection reflection by the earth, earth, is is described described as as 6.E _ ~ JH - 27l"co h (z2
Zdz
_
+ R 2)3/2 -
Copyright © 2000 IOP Publishing Ltd.
~[
27l"co (R2
I
+ h2)1/2
_
(R2
I
+ H2)1/2
]
(
3.5)
106 106
Available lightning data
where h is height of the earth where is the the height of the the leader leader tip tip from from the earth at at the the moment moment of of regisregisheight of base. The tration is the the height of the the leader leader base. The field field change change is is maximum maximum tration and and H is when the << H this this gives gives when the tip tip contacts contacts the the earth, earth, and and for for R « tlEmax
~ 2:~o (* - ~ ) ~ 2:C~R'
(3.6)
It It indeed indeed follows follows from from (3.6) (3.6) that that only only the the charge charge distribution distribution along along aa short short length length comparable comparable with with R is is important important for for field field evaluation. evaluation. (For (For example, example, at be less 5R, the the error error of of the the model model with with TL rL== const const will will be less than than 20% 20% at H > 5R,
for unless 1L toward for any any charge charge distribution, distribution, unless TL grows grows rapidly rapidly from from the the tip tip toward the base; but but there there is is no no reason reason for for this, this, because the channel channel field field E, is the base; because the E c is weak the cloud potential does vary much.) weak and and the cloud potential does not not vary much.) Formula Formula (3.6) (3.6) allows allows charge charge density density evaluation evaluation with with aa good good accuracy, accuracy, it reproduces the path of since the leader is is strictly strictly vertical vertical at at the the earth earth -- it since the leader reproduces the path of the the wire wire which the rocket rocket taking taking up up the which has has evaporated. evaporated. The The value value calculated calculated from measurements at m appears = 30 30m appears to to be be unexpectedly unexpectedly from the the measurements at point point R = 5 rL~ x 22 xX 10lop5 Cjm. Nearly as much much charge charge is is transported transported by long small: TL small: C/m. Nearly as by long laboratory sparks sparks (section (section 2.4). 2.4). The The potential of aa lightning lightning leader leader tip, tip, laboratory potential of U,, does not seem to be much larger than that of a laboratory spark. U does not seem to be much larger than that of a laboratory spark. , t According to to (2.8) (2.8) and and (2.35), (2.35), the the linear linear leader leader capacitance capacitance is is According 12 C C1 FZ ~21r€o/ln(H/Rd~(2-1O)x 2mo/ln ( H / R L ) x (2-10) x 1OF/m, F/m, even even with with indefinite indefinite leader leader j radius R LL.. From From this, we have have U U,t x rL/C1 2-10MV. radius this, we ~ TLlC ~ 2-IOMV. j x The velocity velocity of of aa dart dart leader leader proves proves to to be high. For For its its evaluation, evaluation, we we The be very very high. of A t l p , which is 5 ps for R = 30 m. Formula (3.5) shall use the measured value shall use the measured value of tltl/2' whichis51lsfor = 30m. Formula (3.5) gives tlE A E == tlE AE,,,/2 at h == V3R. J?;R. Hence, Hence, the average velocity velocity along along aa path path of of gives the average max /2 at 7 FZ 50 50m at the the earth's earth’s surface surface is is V VL FZ f i R / A t 1 / 2 = lo7 m/s, quite length h ~ length m at ~ V3R/ tltl/2 ~ 10 mis, quite L consistent with with direct direct measurements. measurements. It It should should be emphasized that that this this velocity velocity consistent be emphasized so it it is is the true velocity. velocity. refers to to the the perfectly path at at the the earth's earth’s surface, surface, so refers perfectly vertical vertical path the true Similar be made point Similar evaluations evaluations can can be made with with the the measurements measurements at at the the far far point R= = 500 500 m m but but with with aa lower lower reliability, reliability, since since the the parameter averaging is to be be parameter averaging is to 3 lo3 m with an unknown path. Neverthemade over a leader length of about made over a leader length of about 10 m with an unknown path. NevertheT~ ~ = 2.3 2.3 xX 10- 5 C/m Cjm and and V VL = 1.15 1.15 x 10 lo77 mls mjs are are found found to to less, the the values values of of TL less, L ~ be close to those above. It will be shown in the next section that an indefinite be close to those above. It will be shown in the next section that an indefinite trajectory may an error error much much larger larger than than the obtained difference difference in in trajectory may produce produce an the obtained TL and and V VL. So the dart leader of triggered lightning with the the values of TL the values of So the dart leader of triggered lightning with the . L definite path at the earth is is aa lucky lucky exception. exception. definite path at the earth powerAnother of tlE(t), A E ( t ) ,cited cited in in [35], [35], characterizes characterizes aa more more powerAnother illustration illustration of ful return stroke ful dart dart leader. leader. The The current current amplitude amplitude in in the the return stroke was was as as high high as as AE,,,max == 810 SlOVjcm, 40 kA. kA. The The maximum field change change was found to 40 maximum field was found to be be tlE VIcm, i.e., little more more than than aa value value proportional proportional to to current, current, while while the the characteristic characteristic i.e., aa little A t l p , decreased decreased to to 1.8Ils. 1.8 ps. Calculations Calculations similar similar to those time of of the the process, time process, tltl/2' to those 4 T~ x 1.35 X x 1010-4C/m, U,t ~ =20-30MV, and described above above give give TL described ~ 1.35 C/m, U 20-30 MV, and 7 VL 2.9 Xx 10 107m/s, thereby supporting supporting the the hypothesis of aa direct, direct, though though V ~ 2.9 mis, thereby hypothesis of L = not not very very strong, strong, dependence dependence of of the the leader leader velocity velocity on on the the tip tip potential. potential. For For
Copyright © 2000 IOP Publishing Ltd.
Perspectives of Perspectives of remote remote measurements measurements
107 107
the calculated calculated values values oflinear of linear charge charge and and velocity velocity at at the the earth, earth, the the leader leader curcurthe iL = T~ VL = 3.9 kA, only an order of magnitude lower rent is found to be only an order of magnitude lower rent is found to be i L = TL VL ::::;; 3.9 than the the current current amplitude amplitude in in the stroke. than the return return stroke.
3.6
Perspectives of remote remote measurements measurements Perspectives
What we we described described in in the the previous previous section section is is aa very very favourable favourable situation, situation, in in What which the of leader leader contact contact with the earth earth is is fixed fixed and and its its final final path which the point point of with the path is strictly strictly vertical, vertical, at at least, least, at at aa length length of of 150-300 150-300m above the earth. One One is m above the earth. should not not expect expect such such favourable favourable conditions conditions for for natural natural lightnings, lightnings, espeespeshould cially for for their their first first components. components. Still, Still, one one should should take take quietly quietly and and with with cially some scepticism scepticism the the idea idea of of indirect indirect remote remote measurements measurements of of lightning lightning some parameters. The to them them because, because, otherwise, his life parameters. The experimeter experimeter resorts resorts to otherwise, his life would turn out out too short to to bring bring his experiment to to aa conclusion. conclusion. ReconReconwould turn too short his experiment struction of of an an electromagnetic electromagnetic field field source source from from strength strength measurements measurements struction made at points is to aa fairly problem made at definite definite points is an an incorrect incorrect solution solution to fairly common common problem of electrodynamics electrodynamics in in various various areas areas of of science science and and technology. Lightning is is of technology. Lightning not an exception exception to to the We shall shall consider consider critically critically the the treatments treatments of of not an the rule. rule. We results obtained problems and results obtained from from solutions solutions to to such such problems and discuss discuss inverse inverse electroelectrostatic problems, as static problems, as applied applied to to the the lightning lightning leader. leader. Generally, the the density density of of space space charge charge p(x, p ( x , yy,. z) z ) between some boundary between some boundary Generally, surfaces can can be be found found if if the the electric electric field field in in the the whole whole confined confined volume volume is is surfaces known. Experimentally, this means known. Experimentally, this means simultaneous simultaneous field field measurements measurements at at an an infinitely points, which practically unfeasible. unfeasible. A well well infinitely large large number number of of points, which is is practically organized organized service service for for field field lightning lightning observation observation has, has, at at best, best, several several synchrosynchronized treatment of theoretical treatment of the the field field records records always always nized field field detectors. detectors. A theoretical suggests suggests an an a priori construction construction of of aa simplified simplified field field source source model. model. The The inverse be solved inverse problem problem can can be solved if if the the number number of of unknown unknown parameters parameters in in this model model does not exceed number of points. What this does not exceed the the number of registration registration points. What follows follows is the measurements is quite quite obvious. obvious. One One writes writes down down aa set set of of equations equations with with the measurements on the expression point (derived on the the right right and and the expression for for field field at at aa given given point (derived from from the the model with parameters) on model with yet yet unknown unknown charge charge parameters) on the the left. left. The The solution solution defines defines the the measurements permit. One the parameters parameters as as rigorously rigorously as as the measurements permit. One should should always always remember, remember, however, however, what what has has been been found found from from the the equations, equations, since since these these are are parameters phenomenon. How parameters of of aa speculative speculative model model rather rather than than aa real real phenomenon. How much much they they coincide coincide is is not not aa matter matter of of accuracy accuracy of of measurements measurements or or calculacalculations but that phenomenon under under study. ‘to the the phenomenon study. Most Most tions but that of of the the model model adequacy adequacy to often, possible errors often, it it is is here here that that possible errors originate. originate. shape 3.6.1 3.6.1 Effect of the leader shape Without but freWithout claiming claiming aa general general analysis, analysis, we we shall shall consider consider aa special special but frequently used model near-earth field variation at quently used model of of near-earth field variation at aa large large distance distance from from aa
Copyright © 2000 IOP Publishing Ltd.
Available lightning data
l08 108
r
H I
a
Q
b
0.02
-t:-' 0.00+-""'::;;;"'--...,...---.-~ ................--. <” 0.00UJ w U
L
z ::r:
1.0
-
(JJ°-0.02 WO -0.02
e
~
1
N N
I
R=H
-0.04 -0.04-
-0.06-0.06
.
Figure 3.14. Electric Electric field field variation variation at at the the earth, earth, evaluated evaluated for for aa large large distance distance from from the channel of of aa descending descending leader. leader. the vertical vertical channel
descending lightning lightning leader. leader. In In this this model, model, the the leader leader is is represented represented by by aa thin thin descending rL,and and the the vertical uniformly charged thread with aa linear linear charge charge density density TL' vertical uniformly charged thread with storm cell, cell, from from which which the the leader leader started, started, is is taken taken to to be be so so small small that that it it is is storm replaced charge Q at at height The value of Q may may be be unknown, replaced by by aa point point charge height H H.. The value of unknown, because the analysis analysis uses the time time variation variation of of the the field field rather rather than than its its because the uses the absolute absolute value value [1]. [l]. The field field at at point the earth earth (figure (figure 3. 3.14(a)) varies in in time time for for two two The point R near near the 14(a)) varies reasons. because of the charge reasons. The The absolute absolute field field decreases decreases because of the charge reduction reduction in in the the AQ = = TLL TLL carried carried away away by the leader leader on on its its way way storm cell cell by the charge charge ~Q storm by the by the to the earth. to the earth.tt The The second second component component is is due due to to the the charge charge accumulation accumulation on on the leader leader of of length length L; L ; as as the the leader leader moves moves on, on, this this charge charge goes goes down, down, the enhancing the field at at the the earth. earth. As aa result, result, for for the change of of the the vertical vertical enhancing the field the change field moment t with with L == VLt, where V is average leader field component component at at moment VLt, where VL is average leader L velocity, we we have have velocity,
~E(L) AE(L)= =_ -
+‘“I
AQH ( H -- xx) ) ddx x ~QH + ~JL (H 27rEo(H2 R2)3/2 27rEo x? R 2]3/2 .’ 2% 0o [[(H ( H-- x )’ + R2I3/’ 27rq,(H2+ R2)312
+
(3.7) (3.7)
This with the This expression expression takes takes into into account account the the doubled doubled field field associated associated with the earth-induced of the the integral integral gives gives the the known known earth-induced charge. charge. The The evaluation evaluation of expression expression
AE(L)= -
1
[ ( H- L)’
1
+ R2I1/’ - (H’ + R’)”’
-
(H‘
LH
}. (3.8)
+ R2)3/2
t This be ignored This component component may may be ignored in in field field measurements measurements at at aa small small distance distance from from the the leader, leader, as as 3.5. described in in section section 3.5. described
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Perspectives of of remote measurements
109 109
When will find find it it difficult difficult to to avoid avoid When analysing analysing this this expression, expression, an an experimenter experimenter will aa temptation. the function function AE(L) t1E(L) has has an an temptation. In In the the range range of of R/H R / H < 1.4, 1.4, the extremum (figure 3.14(b)) which can be easily recorded by an oscillogram. extremum (figure 3.14(b)) which can be easily recorded by an oscillogram. which the the lightning lightning started started What What remains remains to to be be done done is is to to find find the the height height H H aatt which be equal to the average height of the storm front), to (it may be taken to (it may be taken to be equal to the average height of the storm front), to measure point and and the the leader leader (e.g., (e.g., measure the the distance distance between between the the observation observation point is known known and and from the sound velocity is from the the thunder thunder peal peal delay delay time, time, since since the sound velocity of sound wave excitation is near the earth's surface), and to find the point the point of sound wave excitation is near the earth's surface), and to find tip descended to the the height height H H - L, Lm the the moment moment of of time time when when the the leader leader tip descended to by calculation, from (3.8), the leader length L corresponding to the by calculation, from the leader length L,m corresponding to the time is is registered registered exactly exactly by by extremal extrema1 point. point. At At least least one one more more moment moment of of time the contact with with the the earth, earth, giving giving the oscillogram oscillogram -- the the moment moment of of the the leader leader contact rise indicates this this moment moment by by aa rise to to the the return return stroke. stroke. The The oscillogram oscillogram indicates field in figure figure 3.14(b) 3.14(b) defines defines the the field strength strength overshoot. overshoot. The The time time interval interval t1t At in path length H - L, L m at at the the earth: earth: leader leader average average velocity velocity along along the the path length H the boundaries boundaries of of the the measured measured V VL M (H ( H -- LL,)/At. Note that that one one of of the L :::::: m )/ t1t. Note of sign sign reversal reversal of of the the field field length length might might also also be be found found from from the the moment moment of being of (3.8) that that the the curve curve A t1E(L) being registered. registered. It It follows follows from from the the analysis analysis of E(L) intercepts the the abscissa abscissa if if the the registration registration point intercepts point lies lies at at aa distance distance R Rz :::::: (0.8(0.81.4)H from from the the vertical vertical path path axis. axis. Technically, Technically, the 1.4)H the reference reference point point is is easier easier to to find find than than the the extremum. extremum. It with eating. eating. If one substitutes substitutes the the It is is known known that that the the appetite appetite comes comes with If one geometrical values of of A t1E(L) into (3.8), E ( L ) into geometrical parameters parameters used used and and the the measured measured values Together with the the velocity, velocity, one one can can find find the the average average linear linear charge charge density density TL' T ~Together . with this provides provides the the average average current current in in the of time time this the leader leader for for the the final final period period of At: iiL M TL T ~VFL '. ~The The . calculation of of charge charge density density was t1t: calculation was replaced replaced in in [36] [36] by by L :::::: graphical differentiation differentiation of of the the oscillogram oscillogram E( t ) at E(t) at the the point point corresponding corresponding graphical to the the moment moment of of leader leader contact contact with with the the earth; earth; this, this, however, however, gave gave aa to low accuracy. accuracy. Therefore, Therefore, the the electric electric field field registration low registration only only at at one one point point on on the earth's earth's surface surface seemed seemed to to be be sufficient sufficient to to evaluate evaluate one one of of the the least least the accessible parameters parameters -- the the leader leader current current in accessible in aa descending descending lightning lightning discharge. discharge. Let us us now now try try to to assess assess this this situation situation without Let without considering considering the the measuremeasurement errors. errors. Obviously, Obviously, the the main main error error is is associated associated with with finding finding the the starting starting ment point of of aa lightning lightning spark spark and and the the distance distance to it. A 10% error to it. A common common 10% error in in point measurements gives gives much much larger larger errors errors in in evaluations measurements evaluations of of leader leader velocity velocity and current. current. This This always always happens happens when when one one deals of and deals with with the the difference difference of two comparable comparable parameters. parameters. We We shall shall focus focus on on errors two errors of of the the model model itself. itself. The problem problem of of leader leader branching branching effects effects will will be be ignored. ignored. After After all, all, one one can can The always consider a dart leader which has no branches. The representation always consider a dart leader which has no branches. The representation of aa real real leader leader as as aa vertical vertical axis axis is is quite quite another another matter. of matter. Any Any photograph photograph so a shows numerous bendings of a lightning trajectory, straight vertical vertical shows numerous bendings of a lightning trajectory, so a straight leader is is nothing nothing more more than than aa speculative speculative mathematical mathematical concept. concept. To To assess assess leader its implications, implications, let let us us make make another another step step and its and consider consider aa tilted tilted straight straight
Copyright © 2000 IOP Publishing Ltd.
110 110
Available lightning data data
0.25
I1'f
+20
H~ /
0.20
1:a>o t~E
0.15
~
...
....l
ffi
0
<:l
~
C"l
0.10 0.05 0.054 0.00 0.0
0.2
0.4
0.6
L/H L/H
0.8
1.0
Figure 3.15. Electric Electric field field variation at the the earth earth for for aa leader leader deviating deviating from from its its vertical vertical variation at path; = 0.7. 0.7. path; R R/I H =
leader. moves in vertical plane plane passing leader. Suppose Suppose aa leader leader moves in aa vertical passing through through aa field field detector and and is is tilted towards it it by by the the angle angle 0: Q relative relative to to the the vertical line detector tilted towards vertical line (figure we have have 3.15). Then, Then, instead instead of of (3.8), (3.8), we (figure 3.15).
.6.E(L) = _
TL
{
27fco sin 0:
+ JL o [(H - x)2
LH (H 2 + a2 )3/2 (H - x) dx } + (a 2 + xtano:?]3/2
(3.9)
tan 0:. a. The The result of numerical integration of of (3.9) (3.9) is is where a == R where R -- H tan result of numerical integration presented in in figure figure 3.15 3.15 for for R Even aa slight slight tilt tilt from from the vertical presented R == 0.7H. Even the vertical ( a z~z 20°) 20") entails entails aa nearly nearly three-fold three-fold (two-fold (two-fold for for R / H == 0.7) 0.7) change change line (0: line R/H in the the pulse amplitude .6.E(L) A E ( L ) and and in in the the parameters parameters usually derived from from in pulse amplitude usually derived field measurements at the earth's surface. surface. The The value value of of L corresponding corresponding to field measurements at the earth's to the the extremum extremum .6.E(L) A E ( L ) depends depends only only slightly slightly on on the the tilt, tilt, but but the the signal signal ampliamplitude make one parameters tude variation variation is is sufficient sufficient to to make one treat treat the the derived derived leader leader parameters only matter if the only as as estimations estimations of of orders orders of of magnitude. magnitude. It It is is quite quite another another matter if the leader shape shape is is recorded recorded simultaneously simultaneously with electric field field registration at leader with electric registration at two The leader leader trajectory trajectory can can then in space space quite quite two points. points. The then be be reconstructed reconstructed in records can accurately, processing of accurately, and and aa computer computer processing of records can completely completely eliminate eliminate this type this type of of error. error.
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Perspectiws of remote measurements measurements Perspectives of
111 III
Effect of linear charge charge distribution distribution 3.6.2 3.6.2 Effect The with aa uniform uniform charge The model model of of aa leader leader with charge distribution distribution and and the the concept concept of of aa storm cell cell as as an an electrode electrode with capacitor battery battery supplying supplying conduction conduction storm with aa capacitor current to to the the leader leader are are extremely extremely far far from from reality. reality. A storm storm cloud cloud does does current not look plate, to to which which aa lightning not look like like aa giant giant capacitor capacitor plate, lightning leader leader is is connected connected galvanically motion to galvanically during during its its motion to the the 'plate' ‘plate’ of of opposite opposite sign, sign, i.e., i.e., to to the the earth. earth. In reality, the the cloud In actual actual reality, cloud charge charge is is concentrated concentrated on on hydrometers hydrometers which which do do not possess the not contact contact one one another another and and their their assemblage assemblage does does not not possess the properties properties of aa metallic metallic electrode. electrode. A better better analogy analogy would that of of an an electrode-free electrode-free of would be be that voltage electrode spark, than of spark, rather rather than of aa spark spark starting starting from from aa high high voltage electrode of of aa laboratory laboratory generator. generator. To metallic rod To illustrate illustrate this, this, consider consider aa small small metallic rod suspended suspended along along the the by aa field field vector vector in in an an inter-electrode inter-electrode gap, gap, where where the the field field is is supported supported by high-voltage has no no contact high-voltage generator. generator. The The rod rod has contact with with the the generator generator poles. poles. Two rod ends, Two sparks sparks of of opposite opposite sign sign are are excited excited simultaneously simultaneously at at the the rod ends, i.e., the region region of i.e., in in the of enhanced enhanced local local field field (figure (figure 3.16). 3.16). The The charges charges appearing appearing on the sparks must be polarization charges. on the sparks must be regarded regarded as as polarization charges. This This is is the the way way aa
Figure 3.16. Streak photographs of Streak photographs of aa simultaneous simultaneous development development of of aa positive positive and and aa of 50 50 cm cm in in length length in in aa uniform electric negative leader leader from from the the ends ends of of aa metallic rod of negative metallic rod uniform electric 3) channel positive leader; 5) the field: (1) rod; rod; (2, (2,3) channel and and streamer streamer zone zone ofa of a positive leader; (4, (4,5) the same same for for aa field: (1) negative negative leader. leader.
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112 112
A Available vailable lightning lightning data
metallic metallic conductor conductor is is polarized polarized when when itit is is introduced introduced into into an an electric electric field. field. Similarly, Similarly, aa charged charged storm storm cloud cloud possessing possessing no no conductivity conductivity is is only only aa field field source source in in the the space space extending extending to to the the earth. earth. A plasma plasma conductor conductor arising arising in in this way way or or other other is is polarized polarized in in the the field field and and grows, grows, being being supplied supplied by by polarpolarthis ization ization current. current. This This system system is is definitely definitely not not aa perpetuum mobile. mobile. Its Its energy energy source source is is the the electric electric energy energy of of the the cloud cloud field. field. As the the leader leader develops, develops, this this energy energy decreases, decreases, in in accordance accordance with with the the conservation conservation law. law. If If the the external external field field and and the the conductor conductor are are homogeneous, homogeneous, the the linear linear density density of of polarization polarization charge charge is is equal equal to to zero zero exactly exactly at at the the conductor conductor centre centre and and its its absolute absolute value value rises rises towards towards the the ends ends of of different different polarities. polarities. As long long as as the the conductor conductor has has no no contact contact with with the the high-voltage high-voltage generator generator terminals, terminals, its its total total charge, charge, naturally, naturally, remains remains equal equal to to zero. zero. The The latter latter is is also also valid valid when when the the field field and and conductor conductor are are inhomogeneous. inhomogeneous. Using Using numerical numerical methods, methods, one one can can find find the the polarization polarization charge charge distribution distribution for for any any electric electric field. field. The The distributions distributions presented presented in in figure 3.17 3.17 have have been been found found by by the the equivalent equivalent charge charge method method [37]. [37]. This This figure method is is simple simple and and convenient convenient for for long long conductors, conductors, like like those those used used to to method simulate lightning lightning leaders. leaders. simulate Numerical computations computations show show that that aa uniform uniform field field in in aa perfectly perfectly conconNumerical ducting rod rod creates creates aa polarization polarization charge charge T7 rising rising almost almost strictly strictly linearly linearly ducting from the the rod rod centre centre towards towards its its ends. ends. The The ends ends are are an an exception, exception, because because from
t
4
R.=O.Skm
~15C
6km~
..@ 2
~
/l.r
~
J
3km! > } > > > >
1
Ian 0 10
...s:: U
-2
-4 Figure 3.17. Polarization Polarization charge charge distribution distribution along along aa straight straight conductor conductor (a (a leader leader Figure system) in in the the cloud cloud dipole dipole field, field, with with allowance allowance for for aa dipole dipole reflection reflection in in the the conductconductsystem) ing earth. earth. ing
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Perspectives remote measurements measurements Perspectives of of remote
113 113
the the rod rod radius) radius) rises rises rapidly. rapidly. The The the charge charge density density here here (along (along aa length length of of the = ax, charge be approximated as Tr(x) charge distribution distribution at at the the rod rod base base can can be approximated as (X)= where The reader reader will will soon soon see see where the the coordinate coordinate origin origin x is is at at the the rod rod centre. centre. The that small at at larger larger distances, distances, if if that the the contribution contribution of of the the end end charges charges ±q fq is is small radius r. r. the the rod rod length length 2d 2d is is much much greater greater than than its its radius For is concentrated concentrated on on the the outer outer For simplicity, simplicity, let let us us assume assume that that the the charge charge is and that that the the potential potential rod rod surface, surface, as as is is the the case case when when it it has has aa conductivity, conductivity, and will The rod rod centre centre will will be be taken taken as as will be be calculated calculated along along the the longitudinal longitudinal axis. axis. The the zero zero point point of of the the external external field field Eo Eo potential. potential. The The potential potential p 'P at at the the point point x the is -Eox, the the end end charges, charges, pq, 'P q, is aa sum sum of of potentials potentials created created by by the the external external field, field, -Eox, and and the the charges charges distributed distributed along along the the rod, rod, 'PT: pr:
'PT(X) = _a_Jd ydy 47f€0 -d [(y - x)2 + r 2]1/2 = _a_ {[(d _ x)2
+ r 2]1/2
47f€0
+ + + + +
1
2 '12 (d - x ) [(a- x12 + r2] _- [(d [(d + x)2 x12 + r2]1/2 r2]1/2 + xln x In (d - x) + [(d - x)2 + r ]1/2 } -(d [(d + X)* x)2 + rr22]1/2 -(d + xx)) + [(d p2
+ +
';::j
~ [In 4(d
47f€0
+
2
2
r
2
x
) _
(3.10) (3.10)
2].
Here, to the the rod rod sites sites lying lying far far from from its its Here, the the last last approximate approximate expression expression refers refers to ends, Id ± fxl X I »>> r.r. Here, Here, the the term term 'Pq pqcan can be be neglected, neglected, and and we we shall shall approxiapproxiends, mately have have 'PT pT -- Eox Eox ';::j = O.0 . With With the the actual actual charge ( xand ) the charge distribution distribution ~r(x) and the mately end charges charges providing providing 'Pq' pq, the the rigorous rigorous equality equality 'P(x) p(x)= = 0 0 should should be be valid valid end along the the whole whole rod rod length. length. By By relating relating the approximate equality the approximate equality to to the the along centres of of the the semi-axes semi-axes x = = ±d/2, f d / 2 , we we find find centres a';::d
27f€oEo 27f€oEo = In (V3d/r) - 1 In V3d/er
(3.11) (3.11)
The potential potential at at the the rod rod ends ends must must be calculated with The be calculated with the the account account of of their higher higher charges. charges. Assuming Assuming this this charge charge to their to be be concentrated concentrated along along the the end circumference, circumference, the the potential potential at at the the centre centre of of the the end end plane plane (at (at the the end points x = = ±d f d on on the the axis) axis) can can be be described described as as points 'P(±d) = 'Pq
+ 'PT(±d) - (±Eod)
';::j
0,
q
'Pq = ±-4-' 7f€or
The potential potential 'PT pr must must now now be be calculated calculated from from the The the unsimplified unsimplified (3.12) that the end charge is (3.10). It follows from (3.10). It follows from (3.12) that the end charge is approximately approximately q x 27f€orE 27r~orEod and by by aa factor factor of of q';::j od and ad22 ad d K=-NN ';::j 2rln (V3d/er) K = 24 2q 2r In (&/er)
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(3.12) (3.12) formula formula equal equal to to (3.13) (3.13)
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A Available vailable lightning lightning data
smaller smaller than than the the charge charge distributed distributed over over each each half half (it (it is is an an order order of of magnitude magnitude smaller d/r~ z 100). 100). Therefore, Therefore, the the account account oflocalized of localized tip tip charge charge may may be be smaller for for d/r necessary to the the tip, tip, necessary only only for for the the calculation calculation of of electric electric field field in in the the region region close close to at aa distance distance less less than than lOr 10r from from it. it, In In aa remote remote region, region, where where measurements measurements at it is is sufficient sufficient to to consider consider are usually usually made, made, such such aa subtlety subtlety is is unnecessary unnecessary -- it are only only the the charge charge distribution distribution along along the the leader leader channel. channel. Clearly, Clearly, this this is is not not aa uniform uniform distribution, distribution, taken taken for for granted granted by by some some researchers. researchers. It It is is time time to to look look at at the the shape shape of of aa field field strength strength pulse pulse at at the the earth, earth, deterdetermined mined by by the the charge charge of of aa linearly linearly polarized polarized vertical vertical axis axis with with charge charge ~ ( x==) ±ax f a x per per unit unit length. length. It It is is defined defined by by the the algebraic algebraic sum sum of of terms terms T(X) from the the positively positively and and negatively negatively charged charged semi-axes semi-axes and and is is equal equal to to from
=ra
AE(L) _ !J.E(L) -
JL
_a_
-L - L 27TEO 2rEo
x ( H -- x) X) dx dx x(H 2]3/2 [(H [ ( H-- x)2 x)’ + R R2I3/’
where where LL are are the the lengths lengths of of leader leader sections sections which which have have moved moved away away from from the the starting point point to to the the earth earth and and upwards. upwards. Integration Integration with with (3.11) (3.1 1) gives gives starting
L L !J.E(L) Eo [ L _ L A E ( L ) _= 2 -In(V3L/er) [(H_L)2+ ]1/2 - [(H+L)2+ 2]1/2 In (fiL/er) [ ( H - L)’ R R2]‘/’ [ ( H L)’ + R R2I1/’ Eo 2)1/2f [H + (H (H’2 + R R2)”’]2 [H ] -- In In -----------;;-'-----:--';--;,::-----'-------'-------;;--~.,._;;;_ 2 2 {H L + [(H L)2 + R ]1/2}{H + L + [(H + L)2 + R ]1/2} {H [ ( H- L)* R2I1/’}{H [ ( H L)’ R2]’/’J
[
+
+
+
+
+
+
+ +
+
+
(3.14) (3.14)
Here, H is is the the height height of of the the leader leader start, start, r IS is ItS its radius, radius, and and R is is the the Here, distance between between the the leader leader axis axis and and the the observation observation point. point. It It can can be be distance A E ( L ) of of (3.14) (3.14) rises rises smoothly smoothly with with L and and has has shown that that the the function function !J.E(L) shown no extrema. extrema. no The linear linear charge charge distribution distribution assumed assumed in in the the above above illustration illustration is, is, of of The course, another another speculation speculation (section (section 4.3). 4.3). Moreover, Moreover, aa leader leader goes goes up up and and course, down non-uniformly, non-uniformly, and and the the field field in in the the earth-cloud earth-cloud gap gap is is far far from from down being uniform: uniform: its its strength strength decreases decreases towards towards the the earth. earth. This This limits limits the the being linear linear charge charge growth growth from from the the start start downward. downward. The The finite finite channel channel conductivconductivis impossible impossible to to ity exhibits exhibits similar similar behaviour, behaviour, reducing reducing the the tip tip potential. potential. So itit is ity find the the actual actual charge charge distribution distribution exactly exactly without without knowing knowing these these paramparamfind eters. eters. Thus, Thus, aa processing processing of of field field oscillograms oscillograms can can give give nothing nothing more more than than what they they actually actually show. show. The The field field at at aa point point is is an an integral integral effect effect of of the the what whole combination combination of of charges charges created created or or transported transported by by aa given given moment moment whole of time. time. It It is is probably probably worth worth speculating speculating about about registrations registrations but but one one of should assess assess the the results results soberly, soberly, considering considering all all possible possible variants variants and and should is, of of course, course, to to insuring oneself oneself whenever whenever possible. possible. The The best best insurance insurance is, insuring increase the the number number of of registration registration points points and and parameters parameters determined determined by by increase independent independent methods. methods.
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Lightning return stroke
3.7
115 115
Lightning Lightning return return stroke
All lightning lightning hazards hazards are are associated associated with with the the return return stroke, and this this accounts All stroke, and accounts for the of investigators as much investigators to to learn learn as much as as possible possible about about this this for the great great effort effort of discharge stage. stage. It It has has been been established established that that the the contact contact of of aa descending descending discharge or aa grounding grounding electrode electrode produces produces aa return return lightning leader leader with with the the earth earth or lightning wave wave of of current current and and voltage. voltage. It It travels travels up up along along the the leader leader channel, channel, partially partially neutralizing neutralizing and and redistributing redistributing the the charge charge accumulated accumulated during during the the leader leader development development (figure (figure 3.18). 3.18). The The travel travel is is accompanied accompanied by by an an increased increased light light the channel, wave front. intensity intensity of of the channel, especially especially at at the the wave front. At At the the earth, earth, the the wave front front intensity intensity acquires its maximum maximum over over 3-41ls As the the wave wave wave acquires its 3-4ps [31]. [31]. As goes up up to to the the cloud, cloud, the the wave wave intensity intensity steepness steepness and amplitude decrease decrease goes and amplitude many-fold, indicating indicating aa considerable decay. Judging Judging by by streak pictures, the the many-fold, considerable decay. streak pictures, region of of aa high high light light intensity intensity at at the the wave wave front extends to to 25-110 m. The The region front extends 25-1 10m. whole wave wave travel travel takes takes 30-50 30-5Ops. This time time is is especially especially convenient convenient for for IlS. This whole electron-optical electron-optical methods methods of of streak streak photography. photography. However, However, available available attempts attempts to to use use such such methods methods can can hardly hardly be be considered considered successful. successful. A A serious serious obstacle obstacle is is the the exact exact synchronization synchronization of of aa streak streak camera camera and and lightning lightning contact contact with with the the earth. earth. Although Although there there are are many many synchronization synchronization methods, methods, they they have have no simple no simple technical technical solutions solutions and and are are seldom seldom used used in in lightning lightning experiments. experiments. Continuous (e.g. (e.g. sinusoidal) electron streak photography has has not not justified justified Continuous sinusoidal) electron streak photography hopes. Basic Basic results results on on return return stroke velocities have have been been obtained obtained using using hopes. stroke velocities cameras with with aa mechanical mechanical image image processing, processing, which which do do not not need need synchronisynchronicameras zation (Boyce (Boyce camera). camera). zation
Eo
Eo
1
1 ...+... +
:t 1
M
3.18. Scheme of the the return return stroke propagation after after the the contact contact of descendFigure 3.18. Scheme of stroke propagation of aa descend= 0). A leader leader brings brings potential potential U < 0; 0; 1ZM is ing leader leader with with the the earth earth (at moment t = ing (at moment M is return stroke current. return stroke current.
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116
A vailable lightning data Available
3.7.1 Neutralization Neutralization wave velocity velocity The measurements measurements made made half half aa century century ago ago [25,39] [25,39] and and those those performed performed The recently recently [40] [40] indicate indicate aa high high velocity velocity of of aa return return current-voltage current-voltage wave. wave. The The 7 minimum measured measured values values are are close close to to (1.5-2) (1.5-2) x 10 minimum lo7 mls mjs and and the the maximum maximum ones an order 0.5-0.8 of speed c. A ones are are an order of of magnitude magnitude higher, higher, reaching reaching 0.5-0.8 of light light speed velocity speed does does not deal with velocity comparable comparable with with light light speed not mean mean that that we we deal with relativistic relativistic particles particles or or purely purely electromagnetic electromagnetic perturbations. perturbations. The The wave wave velocity is is the the phase phase velocity velocity of of the the process. process. velocity There are are not not so There so many many successful successful optical optical registrations registrations of of the the return return stroke, stroke, the number number of of really really good good ones ones being being about about 100. 100. Most Most of of the the available available data data the concern subsequent subsequent lightning lightning components. components. This This is is natural natural because because every every concern successfully registered registered discharge discharge includes includes the the return return strokes strokes of of several several compocomposuccessfully nents. The The wave wave velocities velocities of of subsequent subsequent components components are are somewhat somewhat higher higher nents. than those those of of the the first first ones. ones. According According to to [40], [40], the the first first component component has has an an averaverthan 7 age velocity velocity V V,r :::::; x 9.6 9.6 Xx 10 lo7 mjs while while the the subsequent subsequent ones ones are are aa factor factor of of 1.25 1.25 age mls higher. Similar Similar data data are are cited cited by by other other authors authors for for subsequent subsequent components components of of higher. lightning discharges discharges triggered triggered from from aa grounded grounded wire wire elevated elevated by by aa rocket. rocket. lightning To illustrate illustrate the the statistical statistical velocity velocity spread spread in in individual individual measurements measurements To and in in those those made made by by different different researchers, researchers, figure figure 3.19 3.19 shows shows integral integral and distribution curves curves for for the the data data of of [25] [25] and and [40]. [40]. The The first first and and subsequent subsequent distribution
1.0 0.8 0.8
g.--O
3 :E cd ro D .L:J
e&
Q.,
0.6 0.6 -
0.4 0.4 0.2
-
0.01 0.0
- .
0.05 0.05
*
.
1
I
0.1 0.1
0.2 0.2 vv,ic ,Ie
0.5 0.5
1
3.19. Velocity Velocity distribution distribution of of the the lightning lightning return return stroke: stroke: (1) (1) averaged averaged over over the the Figure 3.19. [25];(2) (2) averaged averaged over over 1.3 1.3km km above above the the earth earth [40]. [40]. visible channel channel length length [25]; visible
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Lightning return stroke
117 117
components were were not separated. Within Within aa 50% probability, there is is aa 2-fold 2-fold components not separated. probability, there difference between between the the velocities. velocities. Earlier Earlier measurements measurements generally generally give give lower lower difference return stroke stroke velocities. velocities. The The point is that that most most measurements measurements performed performed return point is during the the 1980s 1980s were were two-dimensional, two-dimensional, usually usually providing providing higher velocities, higher velocities, during whereas the earlier data data had had allowed allowed conclusions conclusions only only about about the the vertical vertical whereas the earlier component of of velocity. velocity. Moreover, Moreover, the the application application of of improved improved optics optics and and component photographic materials, materials, as as well well as as higher higher relative motion rates of the the photographic relative motion rates of image and and film, film, improved improved the time resolution of streak streak photographs. As aa image the time resolution of photographs. As result, the value obtained obtained in in the the 1980s 1980s was was more accurate and and result, the velocity velocity value more accurate higher because the measurements were averaged over the initial stroke higher because the measurements were averaged over the initial stroke length of of about about 1km 1 km at at the the earth's earth’s surface, surface, where the wave 1.5-2 length where the wave moves moves 1.5-2 times times faster, faster, rather rather than than over over the the whole whole stroke stroke length. length. All return stroke All measurements measurements show show that that the the return stroke velocity velocity gradually gradually decreases the wave wave front decreases and and that that the the velocity velocity V V,r drops drops abruptly abruptly when when the front passes through the point branching. The passes through the point ofleader of leader branching. The latter latter fact fact suggest suggest aa certain certain relation between stroke velocity velocity and and the the current current transported by the the wave: relation between the the stroke transported by wave: so the the at the branching point, the current is divided among the branches, at the branching point, the current is divided among the branches, so velocity becomes lower. The knowledge of this relation could improve the velocity becomes lower. The knowledge of this relation could improve the calculation calculation accuracy accuracy of of overvoltages overvoltages in in electrical electrical circuits circuits during during lightning lightning disdischarges. Unfortunately, the charges. Unfortunately, the available available data data are are insufficient insufficient to to allow allow finding finding this relation relation reliably. this reliably. Simultaneous Simultaneous registrations registrations of of current current and and velocity velocity have been made made only return strokes have been only for for return strokes of of subsequent subsequent components components of of triggered lightnings but they triggered lightnings but they cannot cannot provide provide aa representative representative statistics. statistics. With With reference reference to to [12,41], [12,41], there there is is note note in in [1] [l] about about aa satisfactory satisfactory agreement agreement between these registrations registrations and between these and Lundholm's Lundholm’s semi-empirical semi-empirical formula formula Vr/c V,/c == (1 (1 + 40/I 40/1M)-1’2, where 1lM is aa return return stroke stroke current current amplitude amplitude M )-1/2, where M is expressed 3.7.2). The The lack lack of of factual factual data data is is sometimes sometimes expressed in in kA kA (see (see section section 3.7.2). compensated compensated by by aa superposition superposition of of distribution distribution statistics. statistics. It It is is assumed assumed that values of probability that the the values of current current and and velocity velocity characterized characterized by by an an equal equal probability correspond to each this correspond to each other. other. There There are are no no serious serious arguments arguments in in favour favour of of this operation but it the lack better method. method. operation but it is is used used for for the lack of of aa better 3.7.2 3.7.2 Current amplitude amplitude
The parameter. Most hazards of The current current amplitude amplitude is is an an important important lightning lightning parameter. Most hazards of lightning whose lightning are are associated, associated, directly directly or or indirectly, indirectly, with with stroke stroke currents, currents, whose registration has much time time and Very few were made registration has taken taken much and effort. effort. Very few of of them them were made by direct using aa shunt belt [28,29,42-46]. by direct methods, methods, using shunt and and aa Rogovski Rogovski belt [28,29,42-461. Still Still fewer by equipment fewer direct direct measurements measurements were were made made by equipment with with aa wide wide dynamic dynamic range, range, which which can can register register both both powerful powerful impulses impulses with with an an amplitude amplitude to to 200 which are 200 kA kA and and low low currents currents of of aa few few hundreds hundreds of of amperes, amperes, which are equally equally physics. important important for for the the understanding understanding of of the the lightning lightning physics. have been been made A of measurements measurements have made by by magnetic magnetic detecdetecA large large number number of tors. tors. Such Such aa detector detector represents represents aa rod rod several several centimetres centimetres in in length, length, made made
Copyright © 2000 IOP Publishing Ltd.
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Available A vailable lightning data
from rod detectors from magnetically magnetically hard hard steel. steel. Preliminarily Preliminarily demagnetized demagnetized rod detectors were were placed at placed at aa fixed fixed distance distance from from aa conductor conductor aimed aimed at at leading leading lightning lightning current current to to the the earth. earth. This This could could be be aa grounding grounding lead lead of of aa lightning lightning conductor conductor or or aa metallic tower metallic tower of of aa power power transmission transmission line. line. With With the the appearance appearance of of lightning lightning current, the detector be within within the range of current, the detector proves proves to to be the range of its its magnetic magnetic action action and and becomes magnetized. magnetized. One the residual residual steel becomes One measures measures the steel magnetization magnetization and and calculates the current problem. The calculates the current by by solving solving the the inverse inverse problem. The advantages advantages of of this method magnetic detectors this method are are its its simplicity simplicity and and low low cost. cost. Usually, Usually, magnetic detectors are are installed by the to obtain the necessary necessary statistics. they installed by the thousand thousand to obtain the statistics. However, However, they can yield nothing but aa current can yield nothing else else but current impulse impulse amplitude. amplitude. Of Of course, course, by by markmarking the ends the direction ing the ends of of the the detector, detector, one one can can also also determine determine the direction of of current current and to the negative). The and attribute attribute it it to the lightning lightning type type (positive (positive or or negative). The accuracy accuracy of of current very low reasons. current measurements measurements is is very low for for several several reasons. First, with aa simple First, there there are are few few objects objects with simple system system of of current current spread spread over over metallic constructions. metallic constructions. A single single conductor conductor would would be be ideal ideal in in this this respect, respect, because because it it excludes excludes current current branching. branching. In In reality, reality, lightning lightning current current is is distribudistributed among among many many conductors, conductors, the the distribution distribution pattern being unpredictable ted pattern being unpredictable since it it varies varies with with temporal temporal parameters of the the impulse. impulse. We We shall shall illustrate illustrate since parameters of this situation situation with with reference reference to to aa simple simple system system consisting consisting of of two two parallel inducthis parallel inductively connected branches with their own inductances L1 and L2, mutual tively connected branches with their own inductances L 1 and L 2 , mutual inductance M and resistances resistances R1j and and R2. Suppose aa rectangular rectangular current current inductance M,, and R 2 . Suppose short risetime risetime is is applied applied to to the system. The The current current distribudistribuimpulse I with impulse with aa short the system. tion between the two two branches is described described as as tion between the branches is . di2 di2 di2 di j . Rjl j +L j - +M- = R 2 12 +L 2 -d +M -d . dt dt t t
Initial currents currents iilo and ii20 at the the stage stage when when current current II(t) ( t ) is is stabilized stabilized to to ZI Initial lO and 20 at are generated generated over over aa very very short short time time equal equal to to the the I risetime. The branch are risetime. The branch currents, therefore, therefore, rise from zero zero very very quickly. quickly. The The reactive reactive components components of of currents, rise from voltage drop drop rv -di/dt produced by them are are much much larger larger than than the the ohmic ohmic voltage di/dt produced by them ones "'-'i -i that can be neglected for for the the time time being. Hence, we we have have ones that can be neglected being. Hence, ii1o/i20 /i = (L M)/(L M), and the initial current, say, in the first branch lO 20 = (L2 2 - M ) / ( L11- M ) , and the initial current, say, in the first branch = Z(L2 ) / ( Lj 1+ L2 2 M ) . When When the transitional process, whose is iilo is I(L 2 -- M M)/(L L 2 -- 2M). the transitional process, whose lO = duration is is defined defined by the time time constant constant ti.t At = = (L] (Li+ L2 2 M ) / ( R I+ R 22),) , duration by the L 2 -- 2M)/(R] is over, over, currents currents iilx = IR2/(R1 and ii2x = ijocR ilxR1/R2 are established established is IR 2 /(R 1 + R2) R 2 ) and 1cc = 20c = 1 / R 2 are in the circuits. The The durations durations of of lightning lightning currents currents are are usually usually comparable comparable in the circuits. with the constant ti.t. At. Therefore, Therefore, aa magnetic magnetic detector detector placed in one one of of with the time time constant placed in the branches branches will will register register aa current current intermediate intermediate between the initial initial and and estabestabthe between the lished values values having having aa maximum maximum amplitude, amplitude, since since the residual magnetization lished the residual magnetization of the rod contains contains information information only only about about the the maximum maximum magnetic field of of of the rod magnetic field current. For this reason, one can can calibrate calibrate aa magnetodetector for deriving deriving aa current. For this reason, one magnetodetector for full current current amplitude amplitude only only if if the impulse shape shape is is known. known. This This cannot cannot be full the impulse be done in in aa real experiment, so so one one has has to to resort to aa rough rough estimation estimation of of current current done real experiment, resort to distribution over over metallic constructions and and use it in in data data processing. distribution metallic constructions use it processing.
+
+
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Second, the the operating operating range range of of the magnetization curve curve is is not not large, large, Second, the rod rod magnetization a linear to saturation region may produce additional and the transition from and the transition from a linear to saturation region may produce additional errors in in data data processing. processing. To To avoid avoid saturation, saturation, the the magnetodetector magnetodetector is is placed errors placed far from from the the conductor, conductor, which which creates creates difficulties difficulties in in data data processing of lightlightprocessing of far nings with low current current and and low low magnetic field. Besides, Besides, when the distance distance nings with low magnetic field. when the between conductor and and aa detector detector is is large, large, the the magnetic magnetic field field effects effects of of between aa conductor other metallic metallic elements elements with current are are hard hard to to take into account. account. So So aa other with current take into 100% error error does does not seem too too high high for for magnetodetectors, magnetodetectors, even even when several 100% not seem when several detectors are are placed at different different distances distances from from aa current current conductor. conductor. Their Their detectors placed at records sufficient material for engineering engineering estimations estimations or or for for aa records provide provide sufficient material for qualitative comparison comparison of of storm storm intensity intensity in in different different regions, regions, but they are are qualitative but they insufficient for for theory. theory. Organization Organization of of direct direct registrations takes much much time insufficient registrations takes time and effort. effort. There There are are no no more more than than aa hundred successful registrations and hundred successful registrations made over aa decade. decade. Let Let us see what what information information can can be be derived derived from from them. them. made over us see Current impulse impulse amplitudes amplitudes vary vary widely, widely, from from 2-3 2-3 to to 200-250 200-250 kA. Current kA. Some magnetodetector magnetodetector measurements give even even 300-400 300-400 kA, kA, but but these these Some measurements give amplitudes seem seem doubtful. doubtful. According According to to [42,46], [42,46], the the integral integral amplitude amplitude disdisamplitudes tributions for for the the first first and and subsequent subsequent lightning lightning components components obey obey the the sosotributions called lognormal lognormal law, law, in in which it is is current current logarithms, logarithms, rather rather than currents called which it than currents themselves, probability of themselves, that that meet meet the the normal normal distribution distribution criterion. criterion. The The probability of lightning lightning with with aa current current larger larger than than 1ZM, is defined defined as as M , is I [kA] (3.16) (3.16) is where where (lg (lg I)av I)av is an an average average decimal decimal logarithm logarithm of of the the currents currents measured measured and and (JIg olg is is the the mean mean square square deviation deviation of of their their logarithms. logarithms. This This approximation approximation cannot be considered cannot be considered accurate. accurate. The The relative relative deviation deviation of of the the value value of of (3.16) (3.16) from be several from the the real real one one may may be several tens tens percent; percent; it it may may be be even even more more for for pracpractically ranges. Nevertheless, Nevertheless, lognormal tically important important current current ranges. lognormal distributions distributions allow allow measurement measurement comparison comparison and and serve serve as as aa guide guide to to engineering engineering estimations. estimations. For example, example, about about 200 200 current current oscillograms oscillograms for for lightnings lightnings that that struck struck the the For 70m tower on on the the San San Salvatore Salvatore Mount Mount in in Switzerland Switzerland [42] [42] satisfactorily satisfactorily 70 m tower obey the the lognormal lognormal law law with with (lg (lgZ)av 1.475 and and (JIg olg= = 0.265 0.265 for for the the first first obey I)av == 1.475 means that component component currents currents of of aa negative negative lightning lightning discharge. discharge. This This means that the the 50% current value is be 30 current value is estimated estimated to to be 30 kA; kA; 95% 95% of of lightnings lightnings must must have have kA and kA. The probability of currents 5% of of lightnings lightnings 80 80kA. The probability of currents exceeding exceeding 44kA and 5% 100 kA kA is is expected expected in in 2% 2% of of cases cases and and higher currents currents rapidly rapidly decreases: decreases: 100 higher 200 kA in % of 200kA in less less than than 0.1 0.1% of cases cases (figure (figure 3.20). 3.20). It It should should be be emphasized emphasized the distribution be treated with caution. again again that that the distribution boundaries boundaries must must be treated with caution. The The curve shape shape in in the low current current range range strongly strongly depends depends on on the the sensitivity sensitivity of of curve the low the measuring instruments used used (its (its left-hand left-hand limit limit is is usually usually taken taken to to be be the measuring instruments 1-3 measurements in 1-3 kA kA in in distribution distribution plots). plots). There There are are few few measurements in the the high high current current
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11
0.01 0.01
I
50
'
I
100 100
.
I
.
I
150 150 200 Lightning Lightning current, current, kA
.
1
250
Lognormal distributions distributions of of return stroke currents: currents: A, the the first first comcomFigure 3.20. Lognormal return stroke ponent of ponent of aa negative negative lightning lightning with with (lg (lg I).v I)," == 1.475 1.475 and and O"lg clg= = 0.265; 0.265; B, B, subsequent subsequent components with with (lg (lgZ),v 1.1 and and O"lg olg= = 0.3; 0.3; C, C, positive positive lightnings lightnings with components I).v == 1.1 with (lgZ)," == 1.54 1.54 and and O"lg clg== 0.7. 0.7. (lgI).v range: it it is is considered considered as as good good luck luck if if they provide aa reliable order of of magnimagnirange: they provide reliable order tude. Note that negative lightning been tude. Note that negative lightning currents currents above above 200 200 kA kA have have never never been registered registered reliably. reliably. The The approximation approximation of of data data on on subsequent subsequent lightning lightning components components in in [42] [42] gives high currents. A lognormal much lower lower integral integral probability probability for for high currents. A lognormal distridistrigives aa much = 1.1 1.1 and and (/lg clg= = 0.3. 0.3. The The bution can be satisfactorily described described by by (lg (lg& bution can be satisfactorily I)av = kA, 5% kA, and calculated calculated 50% 50% current current is is 12.5 12.5kA, 5% current current is is only only 39 39kA, and the the kA is % chance chance for for aa subsequent subsequent component component to to exceed exceed 100 lOOkA is close close to to 0.1 0.1% (figure (figure 3.20). 3.20). The positive lightnings, whose number number is 10% of of the the The statistics statistics for for positive lightnings, whose is about about 10% total registrations, is All descending positive lightnings total registrations, is less less representative. representative. All descending positive lightnings are are one-component. one-component. The The integral integral current current distribution distribution for for them them has has aa large large spread. probabilities of spread. The The probabilities of both both low low and and high high currents currents are are larger larger without without to 35 an an essential essential change change of of the the 50% 50% value. value. The The 50% 50% value value is is close close to 35 kA, kA, i.e., the same negative lightning. i.e., it it is is nearly nearly the same as as for for the the first first component component of of negative lightning. An the lognormal positive curAn approximate approximate description description of of the lognormal distribution distribution of of positive currents in be made made with with (lgI)av 1.54 and and (/lg clg= = 0.7 0.7 (figure (figure 3.20). 3.20). rents in [42] [42] can can be (lgI)av== 1.54 Positive high current Positive high current lightnings lightnings are are more more frequent frequent than than negative negative ones. ones. A A kA can with aa 5% 100 kA can be be expected expected with 5% probability probability corresponds corresponds to to 250 250 kA, kA, and and 100
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20% probability. 26 successful successful registrations registrations of of positive positive lightnings lightnings in in 20% probability. Among Among 26 [43], only only one one showed showed 300 300 kA current. It It seems seems likely likely that positive lightlight[43], kA current. that many many positive nings were, under the the observation observation conditions conditions [43], [43], ascending ascending ones, ones, which nings were, under which may account account for for the the large large spread. spread. Such Such lightnings lightnings have practically no no may have practically return stroke, stroke, and and the the equipment equipment seems seems to to have registered the the relatively relatively return have registered low leader leader current current of of the the final final development development stage. stage. These These data data were low were used used to to derive the integral distribution distribution extending extending to the low low current current region. region. derive the integral to the The great great importance importance of of lightning lightning current current statistics statistics to to applied applied lightning lightning The protection necessitated necessitated aa unification of theoretical theoretical distribution distribution curves. curves. OtherOtherprotection unification of wise, engineers engineers would would have unable to to compare compare the the frequency frequency of of harmful harmful wise, have been been unable lightning protection efficiency. being done within the lightning effects effects and and protection efficiency. This This work work is is being done within the frame of of the the CIGRE CIGRE (Conference (Conference Internationale Internationale des des Grands Grands Reseaux Reseaux frame Electrique haute tension) Electrique 11a haute tension) -- an an operating operating international international conference conference on on high-voltage networks. Data high-voltage networks. Data on on current current from from all all over over the the globe globe are are collected collected and analysed. analysed. However, However, there there is is no no unified unified approach approach to to these these data: data: different different and data are are discarded discarded for for different different reasons, reasons, so so that that the the distributions distributions obtained obtained data differ markedly. markedly. For For example, example, aa report report submitted submitted to to [47] [47] compares compares two two loglogdiffer normal laws laws with with (lgI)av (lg&" == 1.4 1.4 and and 1.477 1.477al, = 0.39 0.39 and and 0.32. 0.32. The The latter latter is is normal O"lg = preferable for power transmission lines, lines, since since the measurements for for objects objects preferable for power transmission the measurements higher than were excluded higher than 60 60 m m were excluded from from this this derivation derivation (power (power transmission transmission lines his book, book, Uman Urn an [1] table of lines are are usually usually lower). lower). In In his [l] gives gives aa table of lightning lightning currents currents mostly mostly based based on on the the measurements measurements of of [42]. [42]. Attention to Attention to details details is is inevitable, inevitable, since since slight slight corrections corrections in in parameter parameter distributions may cause probabilities of distributions may cause manifold manifold changes changes in in the the calculated calculated probabilities of currents above above 100 100 kA, especially important important in in lightning lightning protection of currents kA, especially protection of important Both theory important objects. objects. Both theory and and applications applications suffer suffer from from aa lack lack of of lightlightning measurements. We We shall here some key issues be discussed ning current current measurements. shall list list here some key issues to to be discussed in below. in more more detail detail below. We have mentioned height. It We have mentioned the the importance importance of of an an object's object's height. It has has been been known since that h high known since Benjamin Benjamin Franklin's Franklin's experiments experiments that g h constructions constructions attract more lightnings. the process process of attract more lightnings. It It seems seems likely likely that that the of attraction attraction depends depends on the statistics on the the potential potential of of aa descending descending leader. leader. If If this this is is so, so, the statistics of of descenddescending various height may prove prove different: there ing leader leader currents currents for for objects objects of of various height may different: there will be be aa kind this case, reliable will kind of of lightning lightning separation. separation, In In this case, aa comparison comparison of of reliable current various objects resolve the the much current statistics statistics for for various objects could could help help resolve much debated debated problem of problem of lightning-object lightning-object interaction interaction mechanism. mechanism. Of this connection Of interest interest in in this connection is is the the following following fact. fact. In In the the case case of of aa very very high construction, of the the ascending ascending type type having having high construction, many many first first components components are are of no return the first no return stroke. stroke. But But the first component component is is followed followed by by subsequent subsequent descenddescending ing components, components, whose whose average average stroke stroke currents currents are are lower lower than than in in subsequent subsequent high components buildings. This components affecting affecting low low buildings. This suggests suggests an an involvement involvement of of high that an ground the formation ground constructions constructions in in the formation of of storm storm clouds. clouds. It It appears appears that an ascending high construction before the has ascending leader leader starts starts from from aa high construction before the cloud cloud has matured. Its potential are, therefore, lower. lower. This matured. Its charge charge and and potential are, therefore, This accounts accounts for for
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the the fact fact that that subsequent subsequent components components discharging discharging an an immature immature cloud cloud will will transport lower lower stroke stroke currents currents than than aa mature mature cloud. cloud. transport Finally, Finally, an an important important issue issue is is the the effect effect of of grounding grounding resistance resistance of of objects objects on on lightning lightning current. current. This This may may provide provide information information on on the the resistance resistance of of lightning itself. itself. This This resistance resistance is is to to be be introduced introduced in in equivalent equivalent circuits, circuits, lightning when calculating calculating overvoltages overvoltages affecting affecting various various electrical electrical circuits. circuits. This This when problem is is still still much much debated: debated: some some investigators investigators suggest suggest the the substitution substitution of of problem lightning channel channel by by aa current current source source with with an an 'infinite' ‘infinite’ resistance, resistance, others others aa lightning ascribe to to the the channel channel the the wave wave resistance resistance of of aa common common wire wire (about (about 3000). 300 0). ascribe It would would not not be be hard hard to to solve solve this this problem problem if if we we had had at at our our disposal disposal reliable reliable curcurIt rent rent statistics statistics for for objects objects of of various various height height but but different different grounding grounding resistances. resistances. No such such statistics statistics exist exist yet. yet. To To speed speed up up the the work work in in this this area area and and to to reduce reduce its its No cost, various various remote remote registration registration techniques techniques are are being being employed. employed. They They register register cost, electromagneticfields fields and and coordinates coordinates ofpoints of points where where the the lightning lightning strikes strikes (ide(ideelectromagnetic ally, ally, the the lightning lightning trajectory), trajectory), followed followed by by the the solution solution of of the the inverse inverse problem problem for the the field field source, source, i.e., i.e., lightning lightning current current (see (see section section 3.7.4). 3.7.4). for There is is also also an an increasing increasing number number of of direct direct current current registrations registrations from from There lightnings triggered triggered from from aa wire wire lifted lifted by by aa rocket rocket to to the the height height of of 150150lightnings 250m. The The first first component component of of aa triggered triggered lightning lightning (ascending (ascending leader) leader) has has 250m. no return return stroke; stroke; therefore, therefore, one one deals deals only only with with subsequent subsequent components. components. A A no comparison of of such such registrations registrations with with natura11ightning natural lightning currents currents was was made made comparison in Alabama, Alabama, USA USA [15]. [15]. The The statistics statistics were were not not particularly particularly representative representative in (45 measurements), measurements), so so no no principal principal differences differences were were revealed. revealed. The The lognormal lognormal (45 distribution of of currents currents corresponded corresponded to to the the parameters parameters (lg (lg& = 1.08 1.08 and and I)av = distribution olg== 0.28, 0.28, nearly nearly the the same same as as those those obtained obtained in in Switzerland Switzerland for for subsequent subsequent O"lg components of of natural natural lightnings lightnings [42]. [42]. We We should should like like to to warn warn the the reader reader components against aa possible possible overestimation overestimation of of this this coincidence. coincidence. The The comparison comparison against involved measurements measurements from from geographical geographical points points separated separated by by large large disdisinvolved tances, whereas whereas the the global global variation variation of of lightning lightning parameters parameters still still remains remains tances, unclear. A more more important important thing thing is is that that the the lightnings lightnings studied studied in in [42] [42] unclear. cannot be be regarded regarded as as totally totally natural. natural. They They struck struck aa 70-m 70-m tower tower on on aa cannot mountain elevated elevated at at 600 600m above the the earth's earth’s surface surface close close to to aa lake. lake. The The mountain m above conditions here here are are more more similar similar to to those those of of lightning lightning triggering triggering than than to to its its conditions natural development development in in aa flat flat country. country. Lightning Lightning parameters parameters are are known known to to natural differ with with altitude: altitude: currents currents registered registered by by magnetodetectors magnetodetectors at at an an altitude altitude differ of 1-2 1-2km were two two times times lower lower than than in in aa flat flat country, country, for for less less than than 50% 50% of km were probabilities [48]. [48]. probabilities
3.7.3 Current impulse impulse shape shape and and time characteristics characteristics 3.7.3
Records of of lightning lightning current current impulses impulses look look more more like like abstractionists' abstractionists’ Records pictures -- they they are are so so diverse diverse and and fanciful. fanciful. The The conventional conventional approximation approximation pictures Z(t) = =1 Io[exp (-at) -- exp exp (-!3t)], which which is is of aa impulse impulse by by two two exponents exponents I(t) of 0 [exp (-at) suggested in in various various guides guides to to equipment equipment testing testing lightning lightning resistance, resistance, is is suggested
,I)@-(
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Lightning return stroke Lightning
10
t, f.iS
o
20
40
60
80
100
t, f.iS
Figure 3 .21. A schematic 3.21. schematic oscillogram oscillogram of of aa current current impulse impulse in in the the first first component component of of aa negative lightning. lightning. negative
intended intended for for currents currents of of laboratory laboratory sources sources simulating simulating lightning, lightning, rather rather than than for natural natural lightning. lightning. Let Let us us try try to to identify identify the the main main features features of of the the time time for of current, current, essential essential for for the the understanding understanding of of the the return return stroke stroke variation of variation mechanism and and applications. applications. mechanism The most most reliable reliable data have been been obtained obtained for for first component currents The data have first component currents of aa negative negative lightning. lightning. This current is is easy easy to to register, register, since the impulse impulse front front of This current since the takes several several microseconds microseconds and an oscillographic oscillographic record record reproduces reproduces it in takes and an it in detail. A A sketch sketch of of aa current current impulse impulse averaged averaged over over many many oscillograms oscillograms is is detail. shown in in figure figure 3.21 3.21 in in two two time time scales. Note the the concave concave shape shape of of the the shown scales. Note front. An An expression expression of of the the type type 11 -- exp looks least for its its front. exp (-{3t) (-pt) looks least suitable suitable for description. The The first current peak peak is is often often followed followed by by aa higher higher one, and description. first current one, and evaluation of the impulse impulse risetime risetime tf is is associated with some some reservations. reservations. evaluation of the associated with For example, measured the the time time of of current rise from 2 kA, aa value value For example, [42] [42] measured current rise from 2kA, close to to the the resolution resolution threshold, threshold, to to the the first maximum ., 1 In this this case, case, close first maximum ZM , In of negative lightnings had the risetime of the first component about 50% about 50% of negative lightnings had the risetime of the first component over over 5.5Ils, 5.5 ps, 5% 5% exceeded exceeded 181ls, 18 ys, and and another another 5% 5% less less than than 1.81ls. 1.8 1s. The The knowlknowledge of the risetime allows calculation of the average impulse edge of the risetime allows calculation of the average impulse slope slope AI AI av == 1Z,/tf, However, the the calculation calculation of of electromagnetic electromagnetic fields fields of of lightning lightning M /tf' However, and and the the evaluation evaluation of of possible possible hazards hazards require require aa maximum maximum slope slope AI,,,max == (dl/ (dZ/dt),,,, rather than than an an average average one. one. The The error error in in evaluations evaluations AI dt)max, rather of of this this parameter parameter from from current current oscillograms oscillograms may may be be very very large, large, because because one has has to to replace replace the the tangent tangent to to the the I(t) one Z ( t ) curve curve by by aa secant. secant. Nevertheless, Nevertheless, this this operation operation has has aa sense sense for for aa fairly fairly long long impulse impulse of of the the first first component. component. The integral integral distribution distribution of of the the values, values, like like the the current current itself, itself, is is described described The by the the lognorma11aw lognormal law with with the the parameters parameters (lg (lgZ)," = 1.1 1.1 and and CTlg glg= 0.255, 0.255, if if by I)av = the slope slope is is expressed expressed in in kAIIlS. kA/ps. It It results results in in 12 12kAIIlS kA/ps for for 50% current, current, and and the the the slope slope exceeds exceeds 33 33 kAIIlS kA/ys with with aa 5% 5% probability. probability. To To describe describe the the electromagnetic electromagnetic effect effect of of lightning lightning current, current, let let us us find find 2 placed in aa frame frame of of area area SS = 1m 1 m2 placed at at distance distance D = = 1m 1m the induced induced emf emf UM, in the from from the the channel channel or or aa grounding grounding conductor, conductor, when when the the first first component component current current flows flows through through itit (the (the frame frame is is in in aa plane plane normal normal to to the the current current AI max == 33 33 kAIlls, kA/ps, we we have have magnetic field). field). Even Even for for aa moderate moderate steepness steepness AI magnetic U M, = = = /-LaAImaxS(211"D)-1 p0AImaxS(2~D)-' = 6.6kV, 6.6 kV, where where /-La p0 = = 411" 47r x 10lo-' 7 Him Hjm is is vacuum vacuum
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magnetic permeability. permeability. The The role role of of aa frame frame can can be be performed performed by by any any metallic metallic magnetic structure within within the the construction construction affected affected by by lightning-wires, lightning-wires, wall wall fittings, fittings, rails, rails, structure metallic stripes stripes touching touching each each other, other, etc. etc. At At the the site site of of aa poor poor contact, contact, induced induced metallic emf will will produce produce aa spark, spark, much much more more effective effective than than that that in in an an electric electric lighter. lighter. emf This is is dangerous dangerous because because the the spark spark may may come come in in contact contact with with an an explosive explosive gas gas This mixture. mixture. Sometimes, lightning lightning current current behaves behaves in in aa dual dual way, way, creating creating the the inducinducSometimes, at the the resistance resistance of of the the grounding grounding electrode electrode UR, = = IR. /R. It It tion emf emf and and voltage voltage at tion is important, important, therefore, therefore, to to have have knowledge knowledge about about the the relation relation between between the the is current amplitude amplitude and and maximum maximum slope. slope. Although Although both both parameters parameters obey obey current the same same lognormal lognormal law, law, no no correlation correlation has has been been found found between between them. them. the is bad bad for for engineering engineering applications, applications, for for one one has has to to calculate calculate the the This is This probabilities of each current with a whole set of possible slopes. probabilities of each current with a whole set of possible slopes. There have have been been attempts attempts at at aa more more detailed detailed description description of of the the current current There impulse front. front. They They were were initiated initiated by by the the CIGRE CIGRE mentioned mentioned above above to to impulse handle hazardous hazardous effects effects of of lightning on power power transmission transmission lines. lines. A A set of handle lightning on set of additional parameters parameters has has been been suggested suggested to to reduce reduce errors errors in in current oscilloadditional current oscillogram processing processing and and some regions of of the the impulse impulse front have been been described gram some regions front have described quantitatively. This This is illustrated in in figure figure 3.22 requires no no comment. quantitatively. is illustrated 3.22 and and requires comment. [47]. The processing technique used did Some of the results were cited in did Some of the results were cited in 1471. The processing technique used not lead to considerable data refinement, since the 50% maximum slope not lead to considerable data refinement, since the 50% maximum slope AI max = kA/llS is only two two times times larger than the the 50% 50% average AI,,, = 12 12kA/ps is only larger than average slope slope A ~o% = 5 o y O / t f 5 o= 30/5.5 = 5.5kA/llS. 5.5 kA/ps. But of 22 is is essential A5O'io = Z/50'/0/00% =~ 30/5.5::::;: But the the factor factor of essential to to of ultrahigh ultrahigh voltage voltage insulation. the electrical electrical strength the strength of insulation.
-I
Eo
-
t
~O%
~O%
------\-
\ . .\ \ '
1·moxl
.\
\'.\
\'.\,
Figure 3.22. 3.22. The The distribution distribution of of current current impulse impulse parameters parameters in in the the return return stroke, stroke, Figure based based on on oscillograms. oscillograms.
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Current have aa shorter shorter risetime. risetime. In In Current impulses impulses of of subsequent subsequent components components have the probability and and 0.2 0.2).!s for aa the work work cited cited above above [42], [42], tf 9 < 1.1 1.1 ).!S ps for for aa 50% 50% probability ps for 5% with caution caution because because this this value value 5% probability. probability. The The latter latter should should be be treated treated with is equipment. The The impulse impulse front front is close close to to the the resolution resolution limit limit of of the the measuring measuring equipment. in is mentioned mentioned in in [l] [1] with with in subsequent subsequent components components is is likely likely to to rise rise faster. faster. It It is reference many digital digital registrations registrations the the reference to to other other publications publications that that in in many current the first detector reading reading (for (for 0.2 0.2 ps). ).!s). to aa maximum maximum during during the first detector current could could rise rise to The in subsequent subsequent components components The maximum maximum slope slope of of aa impulse impulse front front in obeys, law: (lg (lg AI,,),,,max)av = = 1.6 1.6 obeys, in in the the first first approximation, approximation, the the lognormal lognormal law: and 40 kA/ps kA/).!s with with aa 50% 50% probability probability and O'lg glg= = 0.35. 0.35. The The value value of of AI AI max exceeds exceeds 40 and probability. When When affected affected by by such such aa 5% probability. and is is larger larger than than 120 120 kA/).!s kA/ps with with aa 5% would exceed exceed 25 25 kV kV in in the the steep steep impulse, impulse, the the amplitude amplitude of of induced induced voltage voltage would above above example example of of aa frame. frame. Current In 55% of cases, cases, the the front front Current of of positive positive lightnings lightnings rises rises slowly. slowly. In % of duration ).!s. With of air air duration was was tf 9 > 200 200ps. With these these impulses, impulses, the the electric electric strength strength of gaps 2.6, formula formula gaps of of several several metres metres in in length length is is close close to to aa minimum minimum (section (section 2.6, (2.51)). than aa (2.51)). The The voltage voltage with with tf::::: tr 200).!s 200ps is is much much more more dangerous dangerous than 'common' micro‘common’ lightning lightning overvoltage overvoltage impulse impulse with with aa risetime risetime of of several several microseconds. nonseconds. Minimum Minimum breakdown breakdown voltage voltage in in air air gaps gaps with with aa sharply sharply nonuniform than in in aa standard standard uniform field field (see (see formula formula (2.52)) (2.52)) is is about about 1.5 1.5 times times lower lower than lightning with the the conconlightning overvoltage overvoltage impulse impulse of of 1.2/50).!s 1.2/50 ps (in (in accordance accordance with is the the ventional ventional way way of of presenting presenting time time characteristics characteristics of of aa impulse, impulse, 1.2 1.2 is risetime ).!s). risetime and and 50 50 is is the the impulse impulse duration duration at at 0.5 0.5 amplitude, amplitude, all all in in ps). The protecThe duration duration of of aa current current impulse impulse is is as as important important for for lightning lightning protection practice practice as as the the risetime. risetime. Impulse Impulse duration duration is is usually usually characterized characterized as tion as aa time span between its beginning and the moment its amplitude decreases time span between its beginning and the moment its amplitude decreases by half. half, Since Since current current is is related related to to the the neutralization neutralization wave by wave travelling travelling along along the the time time of of the the the channel, channel, the the impulse impulse duration duration tt,p is is comparable comparable with with the 8 wave mls and length wave travel. travel. If If its its velocity velocity is is V V,r ::::: E 10 108m/s and the the average average channel channel length the value value of of ttpp will will be be several several tens tens of of microseconds. microseconds. A isis 33km, km, the A similar similar value isis derived derived from from experimental experimental data. data. The The impulse impulse duration duration in value in the the first first component of of aa negative negative lightning lightning is is above above 30, 30, 75 75 and and 200).!s 200 ps for component for the the probprobabilities 95, 95, 50 50 and and 5%, 5%, respectively. respectively. For For subsequent subsequent components, components, the the abilities 6, 32 32 and and 140 140 IlS ps for for the the same same probabilities. impulse is is much much shorter: shorter: 6, impulse probabilities. Positive Positive lightnings must must be be longer longer because because most most of of the the positive positive charge charge of of aa storm storm lightnings cloud is located 2-3km higher than the negative charge. Indeed, is cloud is located 2-3 km higher than the negative charge. Indeed, ttpp is above 230).!s 230ps with with aa 50% 50% probability. probability. The The shortest shortest durations durations for for positive above positive lightnings are are the the same same as as for for the the first first component component of of aa negative negative one. one. 'Anom‘Anomlightnings alously’long long impulses impulses stand stand out out against against this this background background -- about about 5% 5% ofposiof posialously' tive currents decreased to half the amplitude for 2000 ps. tive currents decreased to half the amplitude for 2000 ).!s. Today, we we know know nothing nothing about about the the nature nature of of superlong superlong positive Today, positive impulses. One One thing thing is is clear: clear: they they are are unrelated unrelated to to the the wave wave processes processes in in impulses. the lightning lightning channel. channel. One One may may suggest suggest that that hydrometeor hydrometeor charge charge is is accumuaccumuthe lated and and descends descends to to the the earth earth due due to to an an ionization ionization process process in in the the positively positively lated
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charged region of nature of this charged region of aa cloud. cloud. But But we we can can only only speculate speculate about about the the nature of this process producing producing final process final current current of of 100 100 kA kA and and ask ask why why it it is is manifested manifested only only in in positive lightnings. positive lightnings. 3.7.4 3.7.4 Electromagnetic Electromagnetic field field
Electromagnetic to those Electromagnetic field field of of lightning lightning is is familiar familiar to those leaving leaving aa TV TV or or radio radio set set on on during during aa thunderstorm. thunderstorm. Sound Sound and and video video noises noises inform inform about about aa storm begins. Lightning storm long long before before it it actually actually begins. Lightning was was the the first first natural natural radio radio station by the radio engineering of radio engineering for for testing testing their their receivers. receivers. station used used by the founders founders of The by A S Popov in S Popov in 1885 1885 is is still still Russia's Russia’s The lightning lightning detector detector designed designed by national pride. pride. For meteorologists surveyed national For many many years years meteorologists surveyed approaching approaching storm by registering pulses of of electroelectrostorm fronts fronts by registering so-called so-called atmospherics atmospherics -- pulses magnetic of magnetic radiation radiation from from lightning lightning discharges discharges occurring occurring hundreds hundreds of kilometres away. the late kilometres away. In In the late 1950s, 1950s, much much interest interest in in atmospherics atmospherics was was due due to the nuclear to the nuclear weapon weapon race: race: suspiciously suspiciously similar similar to to radiation radiation pulses pulses from from nuclear explosions, explosions, they they interfered interfered with the diagnostics diagnostics of of the latter. nuclear with the the latter. It is is clear clear from from the the foregoing foregoing that that in in aa return stroke the charge accumuaccumuIt return stroke the charge lated by aa leader varies and redistributed rapidly rapidly along lated by leader cover cover varies and is is redistributed along the the channel, channel, producing variation producing variation of of the the static static component component of of the the electric electric field. field. Charge Charge variation occurs simultaneously with the propagation of a current wave variation occurs simultaneously with the propagation of a current wave along along the the channel, channel, inducing inducing aa magnetic magnetic field. field. The The induction induction emf emf varying varying in in time time gives gives rise rise to to an an induction induction component component of of the the electric electric field. field. Finally, Finally, variation in the the current current dipole dipole moment moment (a (a leader leader channel channel can can be variation in be regarded regarded as mirror reflection by the of its its mirror reflection by the earth) earth) gives gives rise rise as aa dipole, dipole, with with the the account account of to radiation component to an an electromagnetic electromagnetic wave wave producing producing aa radiation component of of the the electric electric field with with aa concurrent concurrent magnetic radiation component. component. There There is is another another magmagfield magnetic radiation magnetostatic one one proportional proportional directly directly to to current. current. netic component component -- aa magnetostatic netic It is is common common practice to distinguish distinguish between between the near and and far far regions regions of of It practice to the near electromagnetic radiation. radiation. In In the the near near region, static field field components components may may be be electromagnetic region, static dominant: the the electric electric component, component, damped damped in in proportion proportion to to the the cubic cubic dominant: distance, to the magnetic component, with the dipole dipole centre, centre, and and the the magnetic component, varying varying with distance Y to distance be neglected F2.These These can can be neglected for for the the far far region, region, because because they they are are distance as as ,-2. much than the E,, H ::::; Now, after much smaller smaller than the radiation radiation components components E cz ,-1. Y - ’ . Now, after these these preliminary how preliminary remarks, remarks, we we shall shall turn turn to to experimental experimental data data showing showing how much the shape of of aa registered registered pulse pulse varies varies with with distance distance between lightning much the shape between aa lightning discharge and and aa field field detector. detector. discharge The shapes shapes of of return return stroke stroke radiation radiation pulses pulses are are shown shown schematically schematically in in The 3.23 for for the the near near and and far far regions. regions. At At large large distances, distances, where where the the static static figure 3.23 figure components of of magnetic and electric electric fields fields are are nearly completely damped, damped, components magnetic and nearly completely E(t) H(t) the pulses E( t ) and and H ( t ) become become geometrically geometrically similar. similar. Both Both are are bipolar bipolar the pulses and high front well defined and have have aa high front slope, slope, aa well defined initial initial maximum maximum and and several several smaller pulse slope, producing the smaller ones ones along along the the slowly slowly falling falling pulse slope, producing the effect effect of of damping oscillations. oscillations. Note that the oscillation period period is is smaller smaller than than the the damping Note that the oscillation
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Lightning return stroke
E, Vim
127 127
H,Alm
r =2km
2000 1500
2.0 "-
1000 500
,, "" ,
1.0
" t, f.JS
0
6
50
100
E,Vlm
150 H,A/m
r =100 km
0.01
4 2 0
0.005
t, JLs 50
Figure pulses of Figure 3.23. Schematic Schematic oscillograms oscillograms of of electromagnetic electromagnetic pulses of lightning lightning in in the the near near (top) (top) and and far far (bottom) (bottom) zones zones at at the the distances distances 2km 2 km (top) (top) and and lOOkm 100 km (bottom). (bottom).
double time of the the wave wave run run along along the channel. After After passing the zero zero point, double time of the channel. passing the point, the part opposite nearly the the pulse pulse part opposite in in sign sign rises rises and and then then decreases decreases with with nearly the same same rate; its its amplitude amplitude is is 2-3 2-3 times times smaller smaller that that the the first first 'half ‘half period’. rate; period'. The proportionality of The inverse inverse proportionality of radiation radiation components components to to the the distance distance from was the reason why why measurements presented from the the radiation radiation source source was the reason measurements are are presented Ybas = = 100 100 km km in the above form: form: they they are are normalized normalized to to the distance rbas in the above the basic basic distance 5 as E:,, = 1010-5Em,,r in metres. metres. For For the the first first lightning lightning component, component, as E:nax = Emaxr with with r in the average average values values of of the the initial initial pulse peak of of the vertical component, component, pulse peak the vertical the EA,,, lie within within 5-l0Y/cm 5-10 V/cm [49-54] [49-541 (compare: (compare: radio radio receivers receivers detect detect well well E:nax, lie signals of of 1mY/m lmV/m in in the the medium bandrange). The The electric electric component component of of signals medium bandrange). subsequent is 1.5-2 1.5-2 times times smaller. smaller. The The spread spread of of subsequent lightning lightning components components is measurements measurements is is as as large large as as that that oflightning of lightning currents. currents. The The standard standard deviation deviation (JE range 35-70% oE is is in in the the range 35-70% for for the the first first lightning lightning component component and and 30-80% 30-80% for for
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Available lightning data
subsequent subsequent ones. ones. The The horizontal horizontal component component of of magnetic magnetic field field strength strength 12 (po/~O)-'/2E~ax varies respectively. respectively. Magnetic Magnetic induction induction H':nax == (/Lo/cor varies / E':nax Bmax = /LoHmax is km from B,,, = poHmax is about about 10lo-*8 T T at at aa distance distance of of 100 100 km from the the lightning. lightning. The pulse of The radiation radiation pulse of the the first first lightning lightning component component rises rises to to the the initial initial peak processing, the risetime is peak with with an an increasing increasing rate. rate. In In oscillogram oscillogram processing, the risetime is arbitrarily the initial arbitrarily subdivided subdivided into into two two components: components: the initial slow slow one one of of 3-51ls 3-5 ps duration the final duration and and the final fast fast one one taking taking 1-0.llls. 1-0.1 ps. The The standard standard deviation deviation is is also also large large here: here: 30-40% 30-40% of of the the average average value value for for the the slow slow front front and and about the fast about 50% 50% for for the fast front. front. In In the the final final stage, stage, the the signal signal rises rises for for about about 0.5-1.0E~,,. With With some some reservations, subdivision into into aa slow slow and and fast fast 0.5-1.0E':nax. reservations, aa subdivision component component can can be be also also made made for for radiation radiation pulse pulse of of the the return return stroke stroke of of subsequent be more subsequent lightning lightning components. components. But But it it would would be more correct correct to to consider consider that the the rise to the initial peak occurs quickly quickly there, there, for for 0.15-0.6Ils. 0.15-0.6 ps. Note that that rise to the initial peak occurs Note that the risetimes for the first and and subsequent subsequent components components are are close close to to those those of of the risetimes for the first their their current current impulses impulses in in aa return return stroke. stroke. The moment of sign sign reversal reversal for for radiation radiation pulses of the the first first components components The moment of pulses of is delayed, relative to the onset of a return stroke, by 50ps in temperate is delayed, relative to the onset of a return stroke, by 50llS in temperate latitudes the tropics reversal for latitudes [54] [54] and and by by 90 90 IlS ps in in the tropics [52]. [52]. The The sign sign reversal for subsequent subsequent components occurs occurs by factor of of 1.3-1.5 1.3- 1.5 earlier. earlier. The The time time for for maximum maximum field field components by aa factor to be established after the sign reversal is of the same order of magnitude as to be established after the sign reversal is of the same order of magnitude as that prior to the reversal. that prior to the reversal. The radiation components near The radiation components E and and H are, are, naturally, naturally, present present in in the the near region, too, but they than the region, too, but they are are much much smaller smaller than the static static component. component. One One exception is is the the initial initial moments of time. The initial initial peaks peaks in in oscillograms oscillograms exception moments of time. The E(t) H(t) be attributed to radiation, radiation, since E ( t ) and and H ( t ) should should be attributed to since the the static static field field components did not have enough time to reveal themselves. The monotonic components did not have enough time to reveal themselves. The monotonic rise of of electric electric field field over over 20-50IlS, 20-50ps, the the time time long long enough enough for for the the radiation radiation rise component to to be damped, is is nearly due to to electrostatic electrostatic effect. effect. The The component be damped, nearly totally totally due induced powerful, because because the induced electrostatic electrostatic field field is is quite quite powerful, the charge charge accumulated accumulated by the the stepwise by stepwise leader leader of of the the first first component component or or by by the the dart dart leader leader of of subsequent components components is is neutralized neutralized during during the the return return stroke. stroke. For For subsequent example, the the electrostatic electrostatic field field changes changes by by several several kV/m kV/m at at the the distance distance of of example, 1km IlS (for 1 km from from the the channel channel lightning lightning during during the the first first 50 50ps (for the the subsequent subsequent times lower than for component, component, the the signal signal is is 2-3 2-3 times lower than for the the first first one one ); aa slower slower field rise may field rise may continue continue for for about about 100 100 Ils. ps. All All in in all, all, the the field field of of the the first first lightlightradiation ning component ning component is is an an order order of of magnitude magnitude higher higher than than the the initial initial radiation rise. With km, the to 15-20 15-20km, the radiation radiation component component rise. With increasing increasing distance distance r to becomes dominant over over the the others, others, and and the the initial initial radiation radiation peak becomes dominant peak becomes becomes an an absolute absolute maximum maximum of of the the registered registered signal. signal. The magnetostatic component the near region is not so The magnetostatic component in in the near region is not so important. important. Still, Still, at at aa distance distance of of 1km, 1 km, it it contributes contributes as as much much to to the the signal signal as as the the radiation component here is high radiation component (figure (figure 3.23). 3.23). The The magnetic magnetic induction induction here is as as high 5 as T. maximum is lop5 T. The The absolute absolute magnetic magnetic field field maximum is achieved achieved later later than than the the as 10because stroke stroke current current peak peak registered registered at at the the earth's earth's surface. surface. This This is is clear clear because
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Total Total lightning flash duration duration and processes in the intercomponent pauses
129 129
the the magnetostatic magnetostatic component component is is proportional proportional not not only only to to the the current current but but to to the conductor length. the conductor length. The The length length increases increases as as aa neutralization neutralization wave wave travels travels from the earth up same reason, for the from the earth up to to the the cloud. cloud. For For the the same reason, the the times times for the first first and subsequent subsequent components components do and do not not differ differ much. much. The The duration duration of of pulse pulse B(t) in in the the near near region region is is comparable comparable with with that that of of current current inducing inducing aa B(t) magnetic field. field. magnetic
3.8
Total lightning lightning flash flash duration duration and and processes processes in in the Total intercomponent pauses pauses intercomponent
A descending descending negative negative lightning lightning flash flash has has on on average average two two or or three three compocompoA nents, each terminated by by aa more more or less powerful powerful current the nents, each terminated or less current impulse impulse of of the return stroke. stroke. The number of of components components in in an an ascending lightning return The average average number ascending lightning is four. The The maximum lightning flash flash may may be be as as is four. maximum number number of of components components in in aa lightning large large as as 30. 30. The The pauses pauses between between the the components components ~tcom At,,, vary vary from from several several milliseconds milliseconds to to hundreds hundreds of of milliseconds. milliseconds. With With aa 50% 50% probability, probability, their their duration duration exceeds exceeds 33 33 ms; ms; the the integral integral distribution distribution curve curve is is described described by by the the lognormal law law with with the the parameters parameters (lg (lg ~tcom)av At,,,)av == 1.52 1.52 and and (jIg olg== 0.4, 0.4, at at lognormal ~tcom [ms]. total flash with the the number At,,, [ms]. The The total flash duration duration varies varies with number of of components. components. Negative one-component one-component flashes the shortest ones, since their current current Negative flashes are are the shortest ones, since their often ceases right after the return return stroke, less than than aa millisecond. millisecond. An An often ceases right after the stroke, for for less ascending one-component one-component positive positive flash pass current ascending flash can can pass current for for aa longer longer time, time, 0.5 s, in spite of the the absence absence of of aa return return stroke. course, this this is low current, current, 0.5 s, in spite of stroke. Of Of course, is aa low than 1kA. The average average flash duration is is close close to to 0.1-0.2s 0.1-0.2 s and and the the less than 1 kA. The flash duration less maximum s. These maximum is is 1.5 1.5s. These large large times times are are discernible discernible by by the the naked naked eye, eye, so so lightning lightning flickering flickering is is not not aa physiological physiological by-product by-product of of vision vision but but aa physical physical reality. reality. Intercomponent pauses pauses take take most most of of the the flash flash time. time. They They cannot cannot be be said said Intercomponent to be A lightning by current nearly all the to be current-free. current-free. A lightning leader leader is is supplied supplied by current nearly all the time, and and this this current current is is high high enough enough to to support plasma in in aa state close to to time, support plasma state close that of steady-state arc. arc. Current of an an intercomponent intercomponent pause pause is is referred referred that of aa steady-state Current of to as continuous to as continuous current, current, which which is is aa fairly fairly ambiguous ambiguous term. term. Average Average continuous current varies varies between between 100 and 200A. 200 A. Nearly Nearly as high current current continuous current 100 and as high supplies an an arc arc in in aa conventional welding set set used used for for cutting cutting metal metal sheets supplies conventional welding sheets or or for for welding welding thick thick pipes. pipes. Most Most thermal thermal effects effects of of lightning lightning are are associated associated with with its its continuous continuous current, current, rather rather than than with with return return stroke stroke impulses impulses which which are are more more powerful powerful but but shorter. shorter. The The highest hghest continuous continuous current current measured measured [55] A. Continuous [55] was was 580 580A. Continuous current current usually usually slowly slowly decreases decreases with with time. time. In In aa one-component having no return stroke, one-component ascending ascending lightning lightning having no return stroke, the the contact contact of of the leader leader with with the the cloud cloud is is terminated terminated by by charge overflow from the cloud to the charge overflow from the cloud to the same value. the earth as aa decreasing continuous current the earth as decreasing continuous current of of about about the same value. Cloud discharging by continuous current can can be be easily registered by by an elecCloud discharging by continuous current easily registered an electric field Field varies varies monotonically, monotonically, as as long flows through through tric field detector. detector. Field long as as current current flows
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A vailable lightning data Available data
the current of of 100 A extracts, extracts, the channel. channel. These These are are appreciable appreciable changes, changes, since since current lOOA from the time time 0.1 0.1 s. s. The The field field on on the the earth earth AQ ~ x lOC 1OC over over the from aa cloud, cloud, charge charge DoQ DoE = AQ/(27qH2) DoQ/(27rc oH 2) M~ 200V/cm 200V/cm if if right right under under aa cloud cloud changes changes by by the the value value A E= the km; at at distance distance rr = = 10 10 km km from from the height height of of the the charged charged cell cell centre centre is is H H= = 3 km; the r2)3/2] zz ~ 5V/cm. 5V/cm. Similar Similar values values the lightning lightning axis, axis, DoE AE = = DoQH/[27rco(H2 A Q H / [ 2 q , ( H 2+ Y~)~'~] were were registered registered during during observations. observations. by slowly slowly rising rising and and as as slowly slowly Continuous Continuous current current flow flow is is accompanied accompanied by decreasing up to to 11 kA. kA. These These are are MMdecreasing current current impulses impulses with with an an amplitude amplitude up components typical M-component M-component is is about about components of of lightning. lightning. The The risetime risetime of of aa typical 0.5ms, the level level 0.5) 0.5) is is twice twice as as much, much, an an 0.5ms, an an average average impulse impulse duration duration (on (on the average M-components with with current current up up A, although although M-components average amplitude amplitude is is 100-200 100-200 A, to Pulsed current current rise rise is is always always to 750 750 A A have have also also been been registered registered [56,57]. [56,57]. Pulsed accompanied intensity of of the the whole whole channel, channel, accompanied by by an an increase increase in in light light emission emission intensity from photographs (even (even taken taken slowly) slowly) do do from the the cloud cloud down down to to the the earth. earth. Streak Streak photographs not wave front front similar, similar, say, say, not show show the the propagation propagation of of aa well well defined defined emission emission wave most of of the the channel channel flares flares up up to to the the tip tip of of aa dart dart leader. leader. It It seems seems as as if if most simultaneously, propagates down down from from aa simultaneously, although although excitation, excitation, no no doubt, doubt, propagates cloud measurements of of [58]). [58]). cloud with with aa high high velocity, velocity, (2.7-4)x (2.7-4)x 10 lo77 mls mjs (from (from measurements Two as ascending ascending ones. ones. In In later later Two M-components M-components were were identified identified in in [58] [58] as measurements, the the existence existence of of ascending ascending processes processes were were questioned, questioned, because because measurements, there the appearance of an an inducing inducing there were were no no clear clear physical physical reasons reasons for for the appearance of perturbation perturbation at at the the earth's earth's surface. surface. Variations in in current current and and electric electric field field ofM-components of M-components were Variations were registered registered in triggered triggered lightning lightning flashes flashes at at aa short short distance distance from in from the the channel channel (r = = 30 30 m) m) [57].The The field field variation variation of of aa vertical vertical component component at [57]. at the the earth earth is is shown shown in in figure figure 3.24. The The pulse pulse DoE A E rises rises to to its its maximum maximum 70/is 70ps earlier 3.24. earlier than than the the current current impulse. The The field field rises rises and and decreases decreases at at nearly impulse. nearly the the same same rate. rate. The The pulse pulse component of of field field perturbation perturbation is is nearly nearly completely completely damped component damped while while the the current still still has has aa high high amplitude. amplitude. current
+
I- --
o 0.5
2?0 200
I
400
600 m
b
f.ls t, PJ-
,e-
1.0
loO1 1.5
E, kV/m E,kV/m Figure 3.24. 3.24. Superimposed Superimposed schematic schematic oscillograms oscillograms of M-component electric Figure ofM-component electric field field and and [57]. current at at the the earth earth [57]. current
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Flash charge and normalized normalized energy
131 131
The number number ofM-components of M-components in in aa flash flash may may even even be be larger larger than than that that of of The subsequent components, components, but they are are of of little little interest interest to to lightning lightning protection protection subsequent but they practice their charge charge and and current current are are too too low. low. Theoretically, Theoretically, however, however, these these practice -- their components are are of of great great interest, interest, because because they they seem seem to to contain contain information information components on unobservable occurring in in storm storm clouds. clouds. It It is is quite quite likely likely that that on unobservable processes processes occurring these give rise to aa dart dart leader leader with with aa return stroke or or to to aa these processes processes give rise to return stroke stroke-free M-component. M-component. Some Some authors authors [27] [27] believe that an an M-component M-component stroke-free believe that is always always formed formed against against the the background background of of continuous continuous current, current, whereas is whereas aa necessary prerequisite for aa dart dart leader leader is is aa current-free current-free pause, pause, during during necessary prerequisite for which the grounded lightning lightning channel channel partly loses its its conductivity. conductivity. This This is is which the grounded partly loses very important important detail detail shedding shedding light light on on processes processes occurring occurring in in aa storm storm aa very cloud after after aa grounded grounded plasma plasma channel channel of of the the first first lightning lightning component component cloud has penetrated it. it. The The transport transport of of the earth’s zero zero potential potential to cloud by by has penetrated the earth's to aa cloud conducting channel, channel, resulting in aa rapid rapid increase increase in in the the cloud cloud electric electric aa conducting resulting in field in in the the vicinity vicinity of of the the channel channel top, top, is is aa powerful powerful stimulus stimulus for for gas gas discharge discharge field processes there there (for (for details, details, see see sections sections 4.7 4.7 and and 4.8). 4.8). processes
3.9
Flash Flash charge charge and and normalized normalized energy
During During intercomponent intercomponent pauses, pauses, charge charge is is transported transported from from aa cloud cloud to to the the earth earth by by both both powerful powerful return return stroke stroke impulses impulses and and continuous continuous current, current, the the latter being being much much lower lower but but longer-living. longer-living. The The contributions contributions of of these these currents currents latter to to the the total total charge charge effect effect are are comparable. comparable. With With aa 50% 50% probability, probability, the the stroke stroke charge transported transported by by the the first first component component of of aa negative negative flash flash is is over over 4.5C, 4.5C, charge while while 5% 5% of of flashes flashes transport transport over over 20C 20C and and another another 5% 5% less less than than l.1C 1.1C [42]. [42]. The The lognormal lognormal law law described described above above is is suitable suitable for for an an approximate approximate reprerepre0.653 sentation of of the the integral integral distribution distribution curve curve with with the the values values (lg (lg Q)av == 0.653 sentation and (Jlg olg== 0.4. 0.4. The The return return strokes strokes of of subsequent subsequent components components have, have, for for the the and same probabilities, probabilities, five five times times smaller smaller charges charges due due to to their their shorter shorter duration duration same and lower lower currents. currents. The The largest largest spread spread of of charge charge measurements measurements is is charactercharacterand istic of positive lightning, in agreement with the diversity of their shape and and istic of positive lightning, in agreement with the diversity of their shape duration. Positive pulse charges exceed 16C with a 50% probability, 150C duration. Positive pulse charges exceed l6C with a 50% probability, l50C with aa 5% 5% probability, probability, and and are are less less than than 2C with with aa 5% 5% probability. probability. These These with seem to be positive lightning with no return stroke. For the description of seem to be positive lightning with no return stroke. For the description of integral integral charge charge distribution distribution for for positive positive pulses, pulses, the the lognormal lognormal parameters parameters may be be taken taken to to be be (lg (lg Q)av Q)av== l.2 1.2 and and (JIg rlg== 0.6. 0.6. may We have have already already mentioned mentioned that that the the charge charge of of aa lightning lightning flash flash is is always always We larger than than the the sum sum of of charges charges transported transported by by the the return return strokes strokes of of the the first first larger and and subsequent subsequent components, components, since since aa substantial substantial contribution contribution to to the the total total charge is is made made by by continuous continuous current. current. The The total total negative negative flash flash charge charge exceeds exceeds charge 7.5C with with aa 50% 50% probability, probability, 40C 40C with with aa 5% 5% probability, probability, and and is is nearly nearly the the 7.5C same same as as the the first first negative negative pulse pulse charge charge in in the the least least powerful powerful flashes. flashes. The The total with 95%, 50% 50% and and 5% 5% total positive positive charge charge is is appreciably appreciably larger larger -- with
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Available lightning data data
probabilities, probabilities, it it exceeds, exceeds, respectively, respectively, 20, 20, 80 80 and and 350C. 350C. One One cannot cannot say say that that the the charge charge transported transported by by aa flash flash is is very very large. large. For For comparison, comparison, even even aa very very large large lightning lightning charge charge of of 350C 350C flows flows through through the the arc arc of of aa conventional conventional welding welding unit unit for for 3-5 3-5 s.s. Charge Charge transport transport is is accompanied accompanied by by energy energy release. release. An An average average negative negative flash flash with with aa charge charge Q Q == 10C 1OC and and gap gap voltage voltage 50 50 MV MV dissipates dissipates about about QU == 8 55 Xx 10 J, which 10sJ, which is is equal equal to to the the energy energy released released by by aa 100 100 kg kg trinitrotoluene trinitrotoluene explosion. explosion. While While most most energy energy is is released released within within the the lightning lightning trace, trace, the the problem problem of of energy energy release release and and heating heating of of metal metal constructions constructions is is of of much much interinterest. est. Normally, Normally, the the resistance resistance of of metallic metallic conductors conductors and and that that of of aa grounding grounding electrode much less electrode are are much less than than the the equivalent equivalent resistance resistance of of aa lightning lightning channel channel R[ RI ~ = U / 1IMM (1M (IM is is the the impulse impulse amplitude amplitude of of aa return return stroke); stroke); R[ RI ~ = I1 kn kR if if U ~ = 50 kA. Therefore, lightning can be regarded as a current x 50 50 MV MV and and 1ZM = 50 kA. Therefore, lightning can be regarded as a current M the object's source, is independent independent of of the object’s resistance. resistance. Any Any source, assuming assuming that that current current 1IM M is conductor with lightning conductor with lightning current current flow flow releases releases the the energy energy
1;
2 dt ( K / R ) R . K= = R J~ ii2 dt == (K/R)R,
KIR =
1;
i2 dt
proportional practical calculations, proportional to to the the conductor conductor resistance resistance R. For For practical calculations, data data on ‘normalized’ energies K / R characterizing lightning only are published. on 'normalized' energies / R characterizing lightning only are published. According the According to to [42], [42], 95%, 95%, 50% 50% and and 5% 5% probabilities probabilities correspond correspond to to the 4 5 7 2 measured values exceeding 2.5 x lo4, 6.5 x IO5 and 1.5 x 107A2s for measured values exceeding 2.5 10 , 6.5 X 10 and 1.5 10 A s for 3 5 2 positive flashes , 5.5 A s for positive flashes and and 6.0 6.0 x 10 lo3, 5.5 Xx 10 IO44 and and 5.5 5.5 x 10 105A2s for negative negative flashes, flashes, respectively. respectively. For For subsequent subsequent components components of of negative negative flashes, flashes, the the respective values magnitude smaller respective values are are an an order order of of magnitude smaller and and do do not not contribute contribute much to to the total energy potency of much the total energy release. release. To To get get an an idea idea about about thermal thermal potency of lightning, evaluate the heat of a steel conductor with a cross section of lightning, evaluate the heat of a steel conductor with a cross section of 5 S == 1cm nR em, 1 cm’.2 . With With resistivity resistivity p == 10lop5 cm, the the energy energy density density released released by by aa 7 2 3 powerful positive positive flash flash (K/R ( K / R == 1.5 1.5 x 10 lo7 A’s)s) is is (K/R)(p/S2) ( K / R ) ( p / S 2== ) 150J/cm 150 J/cm3, A powerful , with with the the conductor conductor temperature temperature increasing increasing by by 40°C. 40°C. Owing Owing to to Joule Joule heat, heat, aa lightning lightning flash flash is is capable capable of of burning burning down down only only aa very very thin thin conductor conductor with with aa cross cross section section less less than than 0.1 0.1 cm’. cm2 . In In many many cases, cases, however, however, heating heating just just by by several several hundred degrees degrees may may become become hazardous. hazardous. hundred
3.10
Lightning temperature and radius radius Lightning
Plasma temperature temperature is is usually usually measured measured by by spectroscopic spectroscopic methods. methods. LightLightPlasma ning spectroscopy spectroscopy is is aa hundred hundred years years old, old, and and it it was was used used even even before before ning photography photography and and field-current field-current measurements. measurements. Reviews Reviews of of spectroscopic spectroscopic 591 together results results can can be be found found in in Uman’s Uman's books books [l, [1,59] together with with extensive extensive data on on lightning lightning plasma plasma are are still still very very scarce. scarce. references. However, However, direct direct data references. Lightning Lightning spectra, spectra, naturally, naturally, contain contain lines lines of of molecular molecular and and atomic atomic oxygen oxygen and and nitrogen, nitrogen, as as well well as as singly singly charged charged ions ions N2, N 2 , argon, argon, cyane cyane and and some some
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Lightning temperature and radius
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other impurities. impurities. No doubly charged charged ions ions have have been detected, indicating indicating that that other No doubly been detected, the does not not exceed exceed 30000 30 000 K. K. Measurements Measurements of of time time resolved resolved the temperature temperature does NI1 (N') line line intensities intensities show show that that the the return stroke temperature temperature reaches reaches N II (N+) return stroke 30 000 K for for the the first first lOllS 10 ps [59,62] [59,62] and and drops drops to to 20 20 000 000 K K in in 20IlS. 20 ps. Average 30000K Average temperatures are are estimated estimated to be about about 25000 25 000 K. K. These These results results are are obtained obtained temperatures to be assuming that that aa plasma channel is is optically optically transparent transparent and and that that the the assuming plasma channel excitation of of atoms atoms in in the the plasma plasma is is equilibrium equilibrium (of (of the the Boltzmann Boltzmann type). type). excitation The estimations estimations justify this assumption. assumption. The justify this Electron densities densities found found from from the the Stark Stark broadening of the lines Electron broadening of the Ha He> lines are 10 1 018' cm* ~ m3 -for for ~ the the first first 51ls 5ps of of the the stroke stroke life. life. Under Under thermodynamic thermodynamic are equilibrium conditions conditions at at T == 30000 30 000 K, K, this this value value of of nnee corresponds corresponds to equilibrium to the pressure pressure of atm [63]. About 10 IlS later, lops later, nnee decreases decreases to to 10 l O17' 'cm~ m -3 ,~ , the of 88atm [63]. About corresponding to to the drop down down to to the atmospheric pressure. pressure. corresponding the pressure pressure drop the atmospheric Then value of the N N IIII line Then the the value of n nee remains remains unchanged unchanged over over the the time time of of the line registration. This This does does not not seem seem strange. strange. Equilibrium Equilibrium electron electron density density in in registration. air wide temperature air at at p == const const == 1atm 1 atm changes changes only only slightly slightly in in aa wide temperature 17 range 15 cmrange 15000-30 000-30 000 000 K, K, remaining remaining about about 10 1017 ~ m3-. As As ~ . the the channel channel cools cools down, the ionization x = when the down, the ionization degree degree x = nee/ /NN certainly certainly decreases, decreases, but but when the pressure reaches reaches the the gas pressure the atmospheric atmospheric value, value, the gas density density N rises rises simul· simulN does does not not change change much. much. High High intensity intensity taneously. For this this reason, reason, nnee == xxN taneously. For radiation is IlS (from to 1000 Ils). The is observed observed for for about about 100 loops (from 40 40 to lOOOps). The first first radiation peak is by another hundreds of microseconds peak is often often followed followed by another one one several several hundreds of microseconds later. later. Spectroscopic measurements were mostly mostly made during aa return return stroke, stroke, Spectroscopic measurements were made during but some register the but some authors authors (64] [64] managed managed to to register the spectrum spectrum of of aa 2-m 2-m portion portion of of aa stepwise N II stepwise leader. leader. The The leader leader tip tip temperature temperature calculated calculated from from the the N 11 lines lines lies lies within 20000-35000 the radiation within 20 000-35 000 K. K. The The diameter diameter of of the radiation region region is is less less than than 35 unavailable. It unlikely that 35 cm. cm. More More accurate accurate evaluations evaluations are are unavailable. It seems seems unlikely that this temperature temperature is is characteristic characteristic of of the whole leader leader channel. channel. Rather, Rather, the the this the whole experiment temperature rise rise during experiment registered registered aa short short temperature during aa powerful powerful step step which was akin akin to to aa miniature stroke (section (section 2.7). 2.7). The The step-induced step-induced which was miniature return return stroke perturbation involving involving part of the the channel channel region region is is most most likely likely to to be be perturbation part of damped damped rapidly rapidly along along the the leader leader length. length. It is is not only the the plasma dynamics but the channel channel radius, radius, too, too, which which It not only plasma dynamics but the still the radius, still remains remains enigmatic. enigmatic. In In making making evaluations evaluations of of the radius, one one usually usually relies photographs. But But in this case, to agree the relies on on photographs. in this case, it it is is very very important important to agree on on the kind be the kind of of radius radius being being evaluated. evaluated. This This may may be the radius radius of of the the channel, channel, through which current flows flows during during the the leader leader and and stroke stroke stages. stages. Clearly, Clearly, through which current such aa radius radius will will include include the the best best conducting conducting and, and, hence, the hottest hottest core core such hence, the of the the plasma channel. Or, Or, one one can can follow follow another another approach. approach. When When solving solving of plasma channel. the problem problem of of electric electric field field variation variation during during the the lightning lightning development, development, one one the of the the space space charge charge has to to deal deal with with the the radius radius of of the the leader leader cover cover where where most most of has is concentrated. concentrated. This This is is the the charge charge radius radius of of lightning. lightning. Therefore, Therefore, each each time time is we radius, we we speak speak of of radius, we must must define define exactly exactly what what we we mean. mean.
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Available A vailable lightning data
Here, we we shall shall use use the the concept concept of of channel channel radius radius as applied to to the the region region Here, as applied where the the lightning lightning current current is is accumulated accumulated and and the the concept concept of where of cover cover radius radius to to the region region where where most most of of the the space space charge charge is is concentrated. concentrated. The The former former can.be can.be the determined, to to some some extent, extent, by by using using optical optical methods, methods, although although this this is is aa comcomdetermined, plicated plicated task. task. With With reference reference to to the the optical optical measurements measurements [65], [65], one one usually usually is accessible to deals with radii radii of of several centimetres. This resolution is deals with several centimetres. This resolution accessible to modern cameras cameras at distance of kilometre, but but the the cameras cameras must must modern at aa distance of about about aa kilometre, have the highest highest resolution resolution possible. possible. Anyway, Anyway, we have the we have heard about about have never never heard the application application of of such such perfect perfect optical optical equipment equipment in in lightning lightning research. research. the In addition addition to to using using special-purpose special-purpose optics, optics, the the experimentalist experimentalist must must In match perfectly perfectly the the sensitivity sensitivity of of photographic photographic materials materials and and exposures. exposures. match A longer longer exposure exposure produces produces aa halo, halo, increasing increasing the the actual actual radius. radius. Unless Unless A special special measures measures are are taken, taken, the the error error may may be be very very large, large, especially especially for for flashes with with aa high high light light intensity. intensity. For For some some reasons, reasons, the the optical optical radius radius of of flashes lightning channel channel may may exceed exceed manifold manifold the the thermal thermal radius. radius. Such Such an an aa lightning effect effect was was observed observed in in studies studies of of spark spark leaders leaders in in laboratory laboratory conditions conditions [66]. [66]. Registration Registration of of the the thermal thermal radius radius appears appears problematic problematic even even for for trigtriggered lightning, lightning, with with aa fixed fixed point point of of contact contact with with the the earth. earth. For For natural natural gered is much much more more complicated. complicated. As As for for the the cover cover radius, radius, lightning, this this task task is lightning, there is is no no reliable reliable technique technique for for its its registration registration at at all. all. So So lightning lightning radius radius there measurements cannot cannot provide provide unquestionable unquestionable data, data, and and the the researcher researcher is is measurements to rely rely on on theoretical theoretical evaluations evaluations only. only. to
3.1 1 What can can one one gain gain from from lightning lightning measurements? measurements? 3.11 It was was not not our our task task to to review review all all experimental experimental studies studies on on lightning: lightning: this this has has It [l, 591. We We believe believe that that the the latest latest experimental experimental data data will will been well well done done in in [1,59]. been
be presented presented in in aa new new Uman Uman book book now now in in preparation. preparation. But But the the basic basic facts facts be have been been discussed discussed here, here, and and we we can can now now ask ask ourselves ourselves whether whether the the have available data data are are sufficient sufficient to to build build lightning lightning theory theory and and to to check check itit by by available experiment. experiment. The situation situation with with lightning lightning is is somewhat somewhat similar similar to to that that for for aa long long The laboratory spark, spark, i.e., i.e., experiments experiments give give mainly mainly external external parameters parameters of of aa disdislaboratory charge. In In the the laboratory, laboratory, these these are are velocities velocities of of the the major major structural structural elements elements charge. (streamers and and leaders), leaders), their their initiating initiating voltages, voltages, currents, currents, transported transported (streamers charges, and, and, possibly, possibly, some some other other characteristics characteristics Sometimes, Sometimes, we we have have charges, some information information on on channel channel radii, radii, or or on on the the time time variation variation of of radii, radii, or or some scarce data data on on plasma plasma parameters. parameters. But But that that is is all. all. scarce The arsenal arsenal oflightning of lightning researchers researchers is is much much smaller. smaller. First, First, they they have have no no The information about about the the voltage voltage in in the the cloud-earth cloud-earth gap gap at at the the lightning lightning start, start, information and there there are are no no data data on on the the initial initial distribution distribution of of electric electric field. field. Both Both literally literally and and figuratively, figuratively, the the bulk bulk of of aa storm storm cloud, cloud, where where aa descending descending leader leader and originates, isis obscure. obscure. Measurements Measurements made made at at the the earth's earth’s surface surface cannot cannot originates,
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help much, much, because because the the number of registration registration points points is is too too small, small, so so it it is is help number of impossible to to reconstruct the initial initial field field distribution distribution along along the the whole whole impossible reconstruct the lightning path. path. lightning The fine structure of of aa lightning lightning flash flash is is not clear either. either. Observations Observations The fine structure not clear give no information about the size of the streamer zone in lightning give no information about the size of the streamer zone in aa lightning leader, and even the existence of such a zone is largely speculative. do leader, and even the existence of such a zone is largely speculative. Nor Nor do we know the origin and structure of volume leaders, which are responsible we know the origin and structure of volume leaders, which are responsible for the stepwise pattern of aa negative leader, at at least, least, observable observable in in laboratory laboratory for the stepwise pattern of negative leader, conditions. There is no information the track track of conditions. There is no information on on the the gas gas state state in in the of aa preceding component, preceding component, when when aa dart dart leader leader travels travels along along it. it. The The only only dart dart leader parameter is its its velocity. leader parameter that that has has been been measured measured is velocity. What What has has just just been been that space listed refers primarily primarily to to the the return return stroke. listed refers stroke. It It appears appears that space charge charge neutralization the basic basic process occurring in in it it -- is is related to the the fast fast neutralization -- the process occurring related to radial propagation propagation of of streamers streamers away away from from the the channel. channel. This This is is the the way way radial the there are the cover cover charge charge is is supposed supposed to to change. change. But But there are no no experimental experimental data data on this process, nor nor can can we we hope obtain any any in in the near future. future. on this process, hope to to obtain the near Most Most available available findings findings concern concern lightning lightning currents currents and and transported transported As in usually registered charges. charges. As in aa laboratory laboratory spark, spark, lightning lightning currents currents are are usually registered at the the earth's earth’s surface, surface, so so we we have have data data on on leader leader currents currents for for ascending ascending disdisat charges only. only. There There are are no no direct direct measurements measurements of of currents currents for for descending descending or or charges dart dart leaders, leaders, the the latter latter fact fact being being especially especially disappointing. disappointing. There There are are more more or or less return strokes, but the less detailed detailed descriptions descriptions of of currents currents for for return strokes, but the measurements measurements made with the restrict the the possibilities possibilities of made at at one one point point (that (that of of contact contact with the earth) earth) restrict of both theoretical physicist physicist and practical engineer. the current both aa theoretical and aa practical engineer. Data Data on on the current wave the leader because then wave damping damping along along the leader are are important important for for the the former former because then he may try try to reconstruct the the plasma plasma conductivity he may to reconstruct conductivity variation. variation. The The latter latter needs needs them them to to be be able able to to calculate calculate the the lightning lightning electric electric field field at at the the earth earth and to both both ground and in in the the troposphere, troposphere, because because it it is is hazardous hazardous to ground objects objects and and aircraft. aircraft. Lightning Normally, they Lightning current current statistics statistics deserves deserves special special attention. attention. Normally, they the occurrence probability of are used in are used in calculations calculations of of the occurrence probability of lightning lightning with with the impulse hazardous parameters, e.g., hazardous parameters, e.g., aa critically critically fast fast rise rise of of the impulse front front and/or and/or amplitude. requirements on amplitude. The The practical practical requirements on the the calculation calculation reliability reliability are are provide the extremely extremely high. high. Indeed, Indeed, it it is is impossible impossible to to provide the necessary necessary accuracy, accuracy, using lognormal parameter distributions. Any approximation using lognormal parameter distributions. Any approximation of of an an actual actual distribution would be be approximate, the range distribution lognormally lognormally would approximate, especially especially in in the range of of large protection. The may be be as high large values values important important for for lightning lightning protection. The error error may as high as mind when as 100%. 100%. One One should should keep keep this this in in mind when comparing comparing calculations calculations of of hazardous lightning the available protection. hazardous lightning effects effects and and the available experience experience in in object object protection. by aa theorist This This is is the the reality reality not not to to be be ignored ignored either either by theorist attempting attempting to to protection. create create aa lightning lightning model model or or by by an an engineer engineer working working on on lightning lightning protection. No matter theorist may be, he No matter how how ingenious ingenious aa theorist may be, he will will not not be be able able to to check check his his model, filling by general model, filling the the gaps gaps by by laboratory laboratory spark spark data data or or by general physical physical considerations. practical lightning usually gained one usually gained considerations. As As for for practical lightning protection, protection, one
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Available lightning data Available
the unfortunate unfortunate experience experience from from analyses of emergencies emergencies that that resulted resulted from the analyses of from the lack lack of of knowledge knowledge of of atmospheric atmospheric electricity. electricity. the
References References [1] Uman M M 1987 (New York: York: Academic Academic Press) Press) p p 377 377 [l] Uman 1987 The Lightning Discharge (New (London, New York: Academic 1, 2 [2] Golde Golde R RH (ed) 1977 1977 Lightning (London, [2] H (ed) New York: Academic Press) Press) vols vols 1,2 [3] Clouds and Precipitation (Lenin(Lenin[3] Imyanitov Imyanitov II M M 1970 1970 Aircraft Electrization in Clouds grad: Gigrometeoizdat) Gigrometeoizdat) pp 210 210 grad: [4] [4] Gunn Gunn R R 1948 1948 J. J . Appl. Phys. 19481 19 481 [5] Gunn Gunn R R 1965 1965 J. J . Atmos. Sci 22 498 498 [5] [6] Evans Evans W WH H 1969 1969 J. J . Geophys. Geophys. Res. Res. 74 939 939 [6] J . Geophys. Geophys. Res. Res. 79 1761 1761 [7] Winn Winn W W P, P, Schwede Schwede G G Wand W and Moore Moore C CB B 1974 1974 J. [7] B and and Holmes Holmes C CR R 1981 1981 J. J. Geophys. Geophys. Res. 861187 86 1187 [8] Winn Winn W W P, P, Moore Moore C CB [8] [9] Kazemir Kazemir H H Wand W and Perkins Perkins F F 1978 1978 Final Report, Kennedy Space Center Contract Contract [9] CC 69694A [lo] Newman Newman M M M, M, Stahmann Stahmann JJ R, R, Robb Robb JJ D D et al1967 all967 J. J . Geophys. Geophys. Res. Res. 72 4761 4761 [10] [ l l ] Kito Kito Y, Y, Horii Horii K, K, Higashiyama Higashiyama Y Y and and Nakamura Nakamura K K 1985 1985 J. J . Geophys. Geophys. Res. 90 90 [11] 6147 6147 [12] Hubert Hubert P P and and Mouget Mouget G G 1981 1981 J. J . Geophys. Geophys. Res. Res. 865253 86 5253 [12] [13] Hubert Hubert P, P, Laroche Laroche P, P, Eybert-Berard Eybert-Berard A A and and Barret Barret L L 1984 1984 J. J . Geophys. Geophys. Res. Res. 89 [13] 251 1 2511 [14] Idone Idone V V PP and and Orville Orville R RE E 1984 1984 J. J . Geophys. Geophys. Res. Res. 89 7311 7311 [14] a1 1993 J. J . Geophys. Geophys. Res. Res. 9822887 98 22887 [15] Fisher Fisher R R G, G, Schnetzer Schnetzer G G H, H, Thottappillil Thottappillil Ret R et al1993 [15] [16] Wang Wang D, [16] D, Rakov Rakov V V A, A, Uman Uman M MA A et al1999 a1 1999 J. J . Geophys. Geophys. Res. Res. 1044213 104 4213 [17] Malan Malan D D JJ and and Schonland Schonland F FG G 1951 1951 Proc. R. R. Soc. Soc. London Ser. Ser. A 209158 209 158 [17] [18] Malan Malan D D JJ 1963 1963 Physics of of Lightning (London: (London: English English Univ. Univ. Press) Press) pp 176 176 [19] Malan Malan D D JJ 1963 1963 J. J . Franklin Inst. Inst. 283 526 526 [19] (Boca Raton: Raton: CRC CRC Press) Press) [20] Bazelyan Bazelyan E EM M and and Raizer Raizer Yu Yu PP 1997 1997 Spark Discharge (Boca [20] 294 pp 294 (2nd edn) edn) (Oxford: (Oxford: Pergamon) Pergamon) pp 418 418 [21] Chalmers Chalmers JJ A A 1967 1967 Atmospheric Electricity (2nd [21] [22] Berger Berger K [22] K and and Fogelsanger Fogelsanger E E 1966 1966 Bull. Bull. SEV S E V 5713 57 13 11 [23] Schonland Schonland B B 1956 1956 The The Lightning Discharge. Handbuch der Physik 22 (Berlin: (Berlin: [23] Springer) Springer) pp 576 576 [24] B, Malan Malan D D and and Collens Collens H H 1938 1938 Proc. Proc. Roy. Roy. Soc. Soc. London Ser. Ser. A 168 168 [24] Schonland Schonland B, 455 455 D and and Collens Collens H H 1935 1935 Proc. Proc. Roy. Roy. Soc. Soc. London Ser. Ser. A 152 152 [25] Schonland Schonland B, B, Malan Malan D [25] 595 595 D M, M, Rakov Rakov V V A, A, Beasley Beasley W W Hand H and Uman Uman M MA A 1997 1997 J. J . Geophys. Geophys. Res. Res. [26] Jordan Jordan D [26] 102 22.025 10222.025 G, Schnetzer Schnetzer G G H, H, Thottappilli1 Thottappillil Ret R et al1992 a1 1992 Proc. Proc. 9th Intern. Intern. Con! Conf. on on [27] Fisher Fisher R R G, [27] Atmosph. Electricity 3 (St (St Peterburg: Peterburg: A A II Voeikov Voeikov Main Main Geophys. Geophys. Observ.) Observ.) pp 873 873 [28] McCann McCann G G 1944 1944 Trans. Trans. AlEE A I E E 63 63 1157 1157 [28] S E V 56 56 No No 11 22 [29] Berger Berger K K and and Vogrlsanger Vogrlsanger E E 1965 1965 Bull SEV [29] [30] Baze1yan Bazelyan E E M, M, Gorin Gorin B B Nand N and Levitov Levitov V V 11978 I 1978 Physical and Engineering Funda[30] of Lightning Protection Protection (Leningrad: (Leningrad: Gidrometeoizdat) Gidrometeoizdat) pp 223 223 (in (in Russian) Russian) mentals ofLightning
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[31] Extrait de de la 1a Revue Revue Generale Genera1e de de [31] Saint Privat dAllier d’Allier Research Group 1982 1982 Extrait
l'E1ectricite, I’Electricite, Paris, Paris, September September Gorin B Nand Elektrichestvo 2 29 29 Shkilev A AV V 1974 1974 Elektrichestvo Gorin B N and Shkilev Idone 6159 Idone V V P, P, Orville Orville R RE E 1985 1985 J. J . Geophys. Geophys. Res. 90 6159 Rubinstein P 1992 Froc. 9th Intern. Intern. Conf. Con! on Rubinstein M, M, Uman Uman M M A A and and Thomson Thomson P 1992 Proc. Atmosph. Voeikov Main Main Geophys. Geophys. Observ.) Observ.) pp 276 276 Atmosph. Electricity 1 (St (St Peterburg: Peterburg: A A II Voeikov [35] J. Geophys. Res. 100 8863 8863 M, Rachidi Rachidi F, F, Uman Uman M M A et al1995 a1 1995 J. [35] Rubinstein Rubinstein M, [36] [36] Thomson Thomson E EM M 1985 1985 J. J . Geophys. Geophys. Res. 90 8125 8125 [37J Equipment [37] Kolechizky Kolechizky E E C C 1983 1983 Electric Field Calculation for High-Voltage Equipment (Moscow: Russian) (Moscow: Energoatomizdat) Energoatomizdat) pp 167 167 (in (in Russian) [38] 6555 [38] Jordan Jordan D DM M and and Uman Uman M MA A 1983 1983 J. J . Geophys. Geophys. Res. 88 6555 [39] Froc. Roy. Roy. Soc. Soc. London Ser. A 143 143654 H 1934 1934 Proc. 654 [39] Schonland Schonland Band B and Collens Collens H [40] 879703 [40] Idone Idone V V PP and and Orville Orville R RE E 1982 1982 JJ.. Geophys. Geophys. Res. 87 9703 [41] 1385 R E, E, Hubert Hubert P P et al1984 a1 1984 J. J . Geophys. Res. 89 1385 [41] Idone Idone V.P, V.P, Orville Orville R [42] 4123 [42] Berger Berger K, K, Anderson Anderson R R Band B and Kroninger Kroninger H H 1975 1975 Electra 41 23 [43] Schweiz. Elekrtotech. Elekrtotech. Ver.63 Ver. 63 1403 1403 [43] Berger Berger K K 1972 1972 Bull. Bull. Schweiz. [44J and Lightning Protection [44] Gorin Gorin B B Nand N and Shkilev Shkilev A AV V 1979 1979 in in Lightning Physics andLightning (Moscow: Inst.) pp 99 (Moscow: Krzhizhanovsky Krzhizhanovsky Power Power Engineering Engineering Inst.) [45] 29 [45] Gorin Gorin B B Nand N and Shkilev Shkilev A V V 1974 1974 Elektrichestvo 2 29 [46] 8) 238 238 Trans. South Afr. lEE ZEE 69 (Pt (Pt 8) [46] Eriksson Eriksson A A JJ 1978 1978 Trans. [47] 65 [47] Anderson Anderson R RB B and and Eriksson Eriksson A A JJ 1980 1980 Electra 69 65 [48] in Lightning Lightning Physics and Lightning Lightning a1 1974 1974 in [48] Alizade Alizade A A A, A, Muslimov Muslimov M M M M et al (Moscow: Krzhizhanovsky Krzhizhanovsky Power Power Engineering Engineering Inst.) Inst.) pp 10 10 Protection (Moscow: [49] Darveniza M M 1984 IEEE Trans. Trans. P PAS [49] Master Master M M J, J, Uman Uman M M A, A, Beasley Beasley W W Hand H and Darveniza 1984 ZEEE AS Pas-103 2519 Pas-l032519 [50] 8471 [50] Krider Krider E E PP and and Guo Guo C C 1983 1983 J. J . Geophys. Geophys. Res. 88 8471 [51] [51] Cooray Cooray V V and and Lundquist Lundquist S S 1982 1982 J. J . Geophys. Geophys. Res. 87 11203 11203 V and and Lundquist Lundquist S S 1985 1985 JJ.. Geophys. Geophys. Res. 90 6099 [52] Cooray Cooray V [52] 6099 [53] McDonald McDonald T T B, B, Uman Uman M M A, A, Tiller Tiller JJ A A and and Beasley 1979 JJ.. Geophys. [53] Beasley W WH H 1979 Geophys. Res. 84 1727 841727 [54] Lin Lin Y T, T, Uman Uman M MA A et al1979 a1 1979 JJ.. Geophys. Geophys. Res. 84 6307 [54] 846307 [55] J. Geophys. Res. 84 842432 [55] Krehbiel Krehbiel PP R, R, Brook Brook M M and and McCrogy McCrogy R R A 1979 1979 J. 2432 [56] Thottappilli1 Thottappillil R, R, Goldberg Goldberg JJ D, D, Rakov Rakov V A, Uman 1995 JJ.. Geophys. [56] V A, Uman M MA A et a1 al1995 Geophys. Res. 100 100 25711 25711 Res. [57] Rakov Rakov V V A, A, Thottappillil Thottappillil R, R, Uman Uman M MA and Barker 1995 JJ.. Geophys. Res. [57] A and Barker P PP P 1995 100 25701 10025701 [58] Malan Malan D D JJ and and Collens Collens H H 1937 1937 Proc. 175 [58] Froc. R. R. Soc. Soc. London A 162 175 [59] Uman Uman M M A 1969 1969 Lightning (New (New York: York: McGraw-Hill) [59] McGraw-Hill) [60] Orvill Orvill R RE E 1968 1968 J. J . Atmos. Sci. Sci. 25 827 827 [60] [61] Orvill Orvill R RE E 1968 1968 J. J. Atmos. Sci. Sci. 25 839 839 [61] [62] Orvill Orvill R RE E 1968 1968 J. J . Atmos. Atmos. Sci. Sci. 25 852 852 [62] [63] Kuznetsov Kuznetsov N NM M 1965 1965 Thermodynamic Functions and Shock Adiabata [63] Adiabata for High (Moscow: Mashinostroenie) Mashinostroenie) (in Temperature Air (Moscow: (in Russian) Russian) [64] Orvill Orvill R RE E 1968 1968 J. J. Geophys. Geophys. Res. Res. 73 73 6999 6999 [64] E 1977 1977 in in Lightning, Lightning, vol vol 1, 1, R R Golde Golde (ed) (ed) (New [65] Orvill Orvill R RE [65] (New York: York: Academic Academic Press) Press) 281 pp 281 [66] Positive Discharges in Air Gaps at Les Renardieres 197s 1977 1977 Electra 53 31 [66J - 1975 31 [32] [32] [33] [33] [34] [34]
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Chapter 4
Physical Physical processes processes in in a lightning Iight ning discharge discharge Here Here we we shall shall discuss discuss the the basic basic phenomena phenomena occurring occurring in in aa lightning lightning discharge: discharge: descending negative negative leader, leader, an an ascending ascending positive positive leader, leader, the the return return strokes strokes aa descending of the the first first and and subsequent subsequent components, components, aa dart dart leader, leader, and and some some others. others. of Lightning may may travel travel not not only only from from aa cloud cloud towards towards the the earth, earth, or or from from aa Lightning grounded object object towards towards aa cloud, cloud, but but it it may may also also start start from from aa body body isolated isolated grounded plane, aa rocket, rocket, etc. etc. About About 90% 90% of of all all descending descending from the the earth earth -- aa plane, from discharges are are negative negative and and about about as as many many ascending ascending discharges discharges are are positive. positive. discharges For this this reason, reason, an an ascending ascending leader leader is is said said to to be be positive. positive. Available Available experiexperiFor mental data data on on lightning lightning as as such such are are oflittle of little use use in in our our attempts attempts to to explain explain the the mental mechanisms underlying underlying the the above above processes. processes. There There are are very very few few observations observations mechanisms So, one one has has to to resort resort to to specspecthat might might shed shed light light on on their their physical physical nature. nature. So, that ulations, invoking invoking both both theory theory and and experimental experimental data data on on aa long long laboratory laboratory ulations, spark, which which relate relate primarily primarily to to aa positive positive leader. leader. Since Since this this process process is is most most spark, simple (to (to the the extent extent aa lightning lightning process process may may be be considered considered simple), simple), we we simple of an an ascending ascending positive positive leader. leader. shall begin begin with with the the discussion discussion of shall
4.1 4.1
An ascending ascending positive positive leader
4.1.1 The The origin origin 4.1.1
The lightnings lightnings people people observe observe most most frequently frequently are are descending descending discharges, discharges, The which originate originate among among storm storm clouds clouds and and strike strike the the earth earth or or objects objects located located which on its its surface. surface. However, However, constructions constructions over over 200 200m high and and those those built built in in m high on of mountainous regions regions suffer suffer mostly mostly from from ascending ascending lightnings. lightnings. These These are are of mountainous nearly as as much much interest interest to to the the physicist physicist as as the the seemingly seemingly common, common, descending descending nearly discharges. An An ascending ascending leader leader is is initiated initiated by by aa charge charge induced induced by by the the elecelecdischarges. tric field field of of aa storm storm cloud cloud in in aa conducting conducting vertically vertically extending extending grounded grounded tric If aa metal metal conductor conductor of of height height h with with aa characteristic characteristic radius radius of of the the object. If object. r << h is is fixed fixed on on the the earth earth and and then then affected affected by by aa vertical vertical rounded top top r« rounded 138 138 Copyright © 2000 IOP Publishing Ltd.
Ann ascending positive leader A
139 139
external field field Eo, field El x Eoh/r >> Eo is created created by by the the induced induced charge charge at at external Eo, aa field E) ~ Eoh/r » Eo is the conductor conductor top top (see (see section section 2.2.7). 2.2.7). This This field field rapidly rapidly decreases decreases in in air air (for (for aa the distance of of several several r values), values), creating creating aa potential potential difference difference between between the the distance conductor end end and and the the adjacent adjacent space, space, 6AU When the the cloud cloud bottom bottom conductor Ux ~ Eoh. Eoh. When is charged charged negatively negatively and and the the vector vector Eo is directed directed from from the the earth earth up to is Eo is up to the cloud, cloud, the the grounded grounded conductor conductor becomes charged, since since the the the becomes positively positively charged, field makes makes some some of of the negative charges charges leave leave the the metal metal to to go go down down to to field the negative the earth. earth. the No stringent conditions conditions are are necessary for the the field field E) E , to to initiate initiate the the air air No stringent necessary for ionization (at (at sea sea level level E) El ~ xE E,i ~ x 30 30 kV/cm) kV/cm) or or for for aa corona corona discharge discharge to to ionization arise at at the the pointed parts of of aa high high structure structure (it (it is is necessary necessary to to have have arise pointed parts El ~ x40-31kV/cm for r = = l-10cm). 1-10cm). The The conditions conditions for for aa leader leader to to be E) 40-31 kV/cm for be initiated in the streamer corona stem are much more rigorous. The energy initiated in the streamer corona stem are much more rigorous. The energy estimations made in section section 2.6 2.6 show show that that there there is is no no chance chance for for aa leader leader to to estimations made in arise if if the the leader leader tip U,, or, more exactly, its excess over the external arise tip potential potential U or, more exactly, its excess over the external , I potential at at the the tip, AU = =U U,I -- U U,, is less less than than !:lUI. AVrm,,~ x 300-400kV. 300-400 kV. This This potential tip, !:lU o, is estimate is is supported supported by experiments with with leaders, leaders, ';hose whose streamer streamer zones zones estimate by experiments have contact with with the the electrode electrode of of opposite opposite sign sign at at the the initial initial moment moment of of have no no contact time. Therefore, for the desired potential difference A Ut,, to be produced, time. Therefore, for the desired potential difference !:l U tmin to be produced, x !:lU AUrm,,/EO x 20-30m 20-30m if if the the average average the structure structure must must have, have, at at least, least, h ~ ) Eo ~ the lmi field of of the the storm storm cloud cloud at at the the site site of of the the grounded grounded object object is is "" -150 V/cm. 150 V/cm. field On the the other other hand, hand, even even if if aa leader leader is is produced produced at at such such aa low low potential, potential, On AVfm,,, it can can hardly hardly travel travel for for aa large large distance. distance. The The leader leader current current will will be be too too !:l Ulmin , it low to to heat the channel channel to to aa sufficiently sufficiently high high temperature. temperature. As result, the the low heat the As aa result, channel resistance resistance will will be be too too high high so so that that aa very very strong strong field field will will be be required required channel to E e is, to support support the the current current in in the the channel. channel. The The channel channel field field E, is, however, however, limited by by the the external external field field Eo. Indeed, aa grounded grounded body of height height h, h, from from Eo. Indeed, body of limited which leader has started, possesses zero potential. potential. Having Having covered covered which aa positive positive leader has started, possesses zero the leader leader tip tip acquires acquires the the potential = -EeL. -E,L. Here, Here, the the the distance distance L, the L, the potential UI, = potential of the the unperturbed external fields fields is is U U.o = = -Eo(L + h), h), and and we we have have potential of unperturbed external
+
+
AUt = =U UtI -- U U0o = = !:lUi AU, + (Eo (Eo -- E,)L. !:lUI Ee)L,
AU, = = Eoh. !:lUi Eoh.
(4.1) (4.1)
For aa leader leader to to develop develop from from the the initial initial threshold threshold conditions, conditions, the the potential For potential difference UI, should not decrease relative to Ui . For AU should not decrease relative to the the initial initial value value of of !:l AU,. For difference !:l this, the the average average channel channel field field E, be lower lower than than the the external external field field Eo. this, Ee must must be Eo. However, aa mature mature channel channel possesses falling current-voltage current-voltage characteristic characteristic However, possesses aa falling E, (i). A A decrease decrease in in E, to "" 100 100 V/cm V/cm requires requires aa channel channel current current higher than Ee(i). Ee to higher than We discussed discussed this this issue issue in in sections sections 2.5.2 2.5.2 and and 2.6. 2.6. With With the the approximation approximation A. We 11A. accepted there there (E (E,e ~ x b/i b / i and and b =300 =300 VAjcm), leader current current is is to exceed accepted VA/cm), the the leader to exceed ,,iimin = = b/E b/Eo x 2 A at Eo x 150 V/cm. o ~ 2A at Eo ~ l50V/cm. see how large the the potential difference !:l AU u It should should be to make make the the Let us how large potential difference be to Let us see current exceed exceed imino i,. In chapter chapter 2, 2, we we derived derived formula formula (2.35) (2.35) relating relating the the current In channel current current behind leader tip to the tip potential U,I and and the the leader leader channel behind the the leader tip to the tip potential U vL. That That formula formula was was applicable applicable to the laboratory laboratory conditions conditions velocity VL' velocity to the N
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140 140
Physical processes in a lightning discharge
considered in in that that chapter, chapter, when when aa leader leader travelled travelled through through the considered the rapidly rapidly decreasing field field of of aa high-voltage high-voltage electrode. electrode. Having Having covered covered aa distance distance of of decreasing only aa few few radii radii of of the electrode curvature, curvature, usually usually very small, the the leader leader only the electrode very small, tip potential, U tip found found itself itself in in aa space space with with aa nearly nearly zero zero potential, U,o « << U U,. The neglect neglect t . The of the the external external field field was was justifiable in that that case. case. An An ascending ascending leader leader is is quite quite of justifiable in another matter: the the potential DoU another matter: potential difference difference A U t, == Ut, -- U U,o continuously continuously increases the leader increases with with the leader velocity, velocity, as as its its tip tip approaches approaches aa charged charged cloud, cloud, since the tip region of since the tip enters enters aa region of an an ever ever increasing increasing external external field. field. Hence, Hence, we we have IUtl I Uti « << IUol I Uoi and and the the value value of of A U t, largely largely determined determined by U,.o. Therefore, Therefore, have DoU by U the approximate must be rewritten in the approximate formula formula of(2.35) of (2.35) must be rewritten in its its general general form, form, with with U t, replaced Ut : replaced by by Do AU,:
.
.
’
1= =
~TEOAU~VL 271'E ODoUt VL In(L/R) ln(L/R) ,’
aut = U, - U,
(4.2)
where R R is where is the the effective effective radius radius of of the the leader leader charge charge cover. cover. The available available leader leader theory theory fails fails to to provide clear and and convincing convincing The provide aa clear physical expression to describe the the relationship relationship between between vL and AU,. VL and DoUt . So physical expression to describe we shall we shall further further use use the the empirical empirical relation relation suggested suggested in in section section 2.6: 2.6: I 2 aa == 15m/sV 1 5 m / ~ V/ ’. / ~ .
(4.3) (4.3) This relation relation was derived from from experimental experimental data data on on rather rather short short gaps, gaps, in in This was derived which the the tip potential could be taken taken to to that that of which tip potential could be to be be identical identical to of aa high high voltage voltage electrode.t e1ectrode.t In In accordance accordance with with (4.2), (4.2), expression expression (4.3) (4.3) corresponds corresponds to to the the l 3 relation / also VL :::::: Now, = iill3 also supported supported by by some some laboratory laboratory experiments. experiments. Now, relation vL using the the value value of of iimin, we shall shall find find AUi which provides provides the the leader leader viability: viability: using DoUi which min , we VL t )I/2, wL == a(DoU a(AU,)”2,
Do UI
_ [b In(L/R)]2/3 ---,-'--'-'--'Eo 271'Eoa
mm
(4.4)
Assuming L:::::: L = 10 10m and R:::::: R = I1 m m for for aa still-short still-short initial initial leader leader at at Assuming m and Eo 150V/cm, we obtain AVjmln = 3.1 3.1 MV MV and and hmin 210m. The The result result Eo == 150V/cm, we obtain DoUimin :::::: hmin == 210m. of that of of this this simple simple estimation estimation agrees agrees with with that of lightning lightning observations. observations. In In aa flat make up flat country, country, ascending ascending lightnings lightnings make up an an appreciable appreciable fraction fraction of of the the total number number of of strikes strikes affecting affecting grounded grounded objects objects of of about about that that height. height. total The The continuous continuous ascending ascending leader leader of of aa triggered triggered lightning lightning (initiated (initiated from from aa grounded wire rocket above above the the earth) earth) is is also also excited excited at at about about grounded wire raised raised by by aa rocket 200m. that the the value value of of Eo in the the calculations calculations is is somewhat somewhat 200 m. Note Note that Eo used used in larger than than those measured at at the the earth earth surface. surface. The The storm storm cloud cloud field field larger those measured near near the the earth earth is is always always attenuated attenuated by by the the space space charge charge introduced introduced in in the the air air by by corona corona discharges discharges from from thin thin conductors conductors of of small small height, height, such such as as tree branches, shrubs, tree branches, shrubs, grass, grass, constructions, constructions, etc. etc. Some Some measurements measurements show show that that at height of m. that the the field field at at the the earth earth is is half half that at aa height of 10-20 10-20m. The streamer streamer theory theory has has been advanced further. further. Note, for comparison, comparison, that the streamer streamer tt The been advanced Note, for that the is V V,s "" RZ AU, from formulae formulae (2.6) (2.6) and and (2.8) (2.8) with with tJ.U, AU, instead instead of of U, U, and and E, const. velocity ~U, from Em == const. velocity is
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An ascending positive leader
141 141
development and current current 4.1.2 Leader development Two main leader parameters are accessible accessible to to measurement: measurement: its its velocity velocity and and Two main leader parameters are the current current through through the the channel channel base contacting aa grounded grounded object. object. The The the base contacting current is is due due to to the charge pumped by an an electric electric field field into into the the growing growing current the charge pumped by leader. If If the the field field in in the the leader leader channel channel is is lower lower than than the the external external field field leader. (otherwise the the leader leader is is non-viable), non-viable), the the difference difference between between the the tip tip potential (otherwise potential and the the unperturbed unperturbed potential potential at at its its site site becomes becomes larger larger as as the the leader leader becomes and becomes longer (see (see expression expression (4.1)). (4.1)). According According to to (4.2) (4.2) and and (4.3), (4.3), the the leader leader current, current, longer proportional to to ii '" AU:”, also rises. The current current rise rise becomes more rapid as flU;/2, also rises. The becomes more rapid as proportional the leader leader becomes longer, especially especially when when the leader reaches reaches the the region of aa the becomes longer, the leader region of very high high cloud cloud field. field. The The rising rising current current heats the channel channel more, so that that its its very heats the more, so linear resistance resistance and and field field drop. drop. With With time, time, the the channel channel becomes linear becomes aa nearly nearly perfect conductor. Grounded Grounded at at its its base, base, the the channel channel possesses same perfect conductor. possesses the the same potential U(t, U ( t ,x) everywhere along along its its length, length, including including the the tip, tip, which which is is potential x) everywhere low relative relative to to the the absolute absolute external external potential potential IUo(x)l. IUo(x)I. In this this case, case, the the low In value of of A U == U(t, U ( t ,x)x )-- Uo(x) Vo(x)i=::j z --Uo(x) varies only only slightly slightly with with time time at at flU Uo(x) varies value every point point x along along the the channel. channel. every The linear linear leader leader capacitance capacitance C C1, given by by formula formula (2.8) (2.8) with with length length LL The 1, given and cover cover radius radius R R instead instead of of I1 and and r,Y, also also varies varies very very little. little. Indeed, Indeed, the the and cover radius radius behind behind the the tip tip is is about about the the same same as as the the streamer streamer zone zone radius radius cover which, according according to to (2.39), (2.39), is is R R == AUt/2E,, where E E,s i=::j M5 5 kV/cm kV/cm is is the the which, flU t /2E s , where streamer streamer zone zone field field under under normal normal conditions. conditions. The The height height of of the the charged charged region region centre centre in in the the cloud, cloud, H i=::j PZ 3 km, km, is is much much greater greater than than that that of of the the leader starting starting point, point, h i=::j PZ 200 200m. Suppose the the leader leader length length is is greater greater than than leader m. Suppose but smaller smaller than than H by by such such aa value value that that the the cloud cloud field field non-uniformity non-uniformity h but along the the channel channel can can be be neglected. neglected. We We then then have have AU, M lUo(L) IUo(L)II i=::j M IEoLI, IEoLI, along flUt i=::j and the the value value of of L/ LIR M 2E 2E,/Eo under the logarithm in C1 of (4.2) and / and R i=::j Eo under the logarithm in C of (4.2) and 1 s (4.4) is is independent independent of of time. time, If If this this relation relation does does change, change, which which happens happens (4.4) when aa leader leader rises rises so so high high that that it it enters enters the the region region of of aa rapidly rapidly increasing increasing when external field, field, the the logarithm logarithm changes changes much much more more slowly. slowly. Thus, Thus, the the linear linear external charge T(X) ~ ( x i=::j M ) C 1, flU A U remains remains nearly nearly constant constant in in time time at at every every leader leader point. point. charge But But if if there there is is no no charge charge redistribution redistribution along along the the channel, channel, the the current current in in it, it, i(t, x), x), does does not not change change along along its its length length but but changes changes only only in in time. time. Entering Entering the the channel channel through through its its grounded grounded base, base, the the current current supplies supplies charge charge only only to to the front front leader leader portion. portion. The The current current in in the the base base is is the the same same as as in in the the channel channel the right behind behind the the tip. tip. It It is is defined defined by by formula formula (4.2) (4.2) with with Aut M IUo(L)I, I Uo(L)I,close close flU t i=::j right to the the unperturbed unperturbed potential potential of of the the cloud cloud charge charge at at the the tip tip site. site. Similarly, Similarly, the the to leader leader velocity velocity can can be be found found from from (4.3). (4.3). Therefore, Therefore, the the velocity velocity and and current current of of aa fairly fairly long long leader leader (long (long relative relative to the the start start height), height), which which develops develops in in the the average average field field Eo, Eo, are are described described as as to N
1.‘ = 27fEoa(flUt)3/2 2TEoa(AUt)3’2 AU , = EoL, e=2.72 e = 2.72.. WL = a(ALJ,)1/2, /= V3 ,flUt=EoL, ....
In(2 3EsleEo) ’ ln(2&E,/eEo)
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(4.5)
142 142
Physical processes in a lightning discharge
The numerical numerical factor factor ../3/ f i / e e in in the the second second expression expression of of (4.5) (4.5) has has resulted resulted The from from aa more more rigorous rigorous calculation. calculation. This This result result is is obtained obtained if if the the linear linear charge charge T(X) is is calculated calculated directly directly from from external external field field Eo Eo with with r(x) r(x) == const const x, x, as as in in r(x) section C1 and and A f:i.U(x), U ( x ), section 3.6.2, 3.6.2, rather rather than than from from average average linear linear capacitance capacitance C 1 as as was was done done in in the the derivation derivation of of (4.2). (4.2). The The current current is is found found from from ii == dQ/dt, dQ/dt, where where Q Q is is the the net net charge charge of of the the conductor conductor (the (the integral integral of of r(x) ~(x) in x). x). in It It follows follows from from (4.5) (4.5) that that the the current current rises rises rapidly rapidly with with time time as as the the leader develops, develops, whereas whereas the the velocity velocity increases increases much much more more slowly: slowly: leader 12 3 2 3 U, = = dL/dt dL/dt :::::: xL L1I2 M tt and and i:::::: iM L L3I2 M tt3 at at Eo Eo == const. const. In In stronger stronger external external / :::::: VL / :::::: EOmuch much faster faster than than the the velocity. velocity. fields, the the leader leader current current also also rises rises with with Eo fields, 4 lo4 cmjs Numerically, cm/s 500 m has has i == 4.5A 4.5 A and and VL == 4 x 10 Numerically, aa leader leader with with LL == 500m in Eo == 150V/cm, 150V/cm, i.e., i.e., about about the the same same values values as as for for an an in an an average average field field Eo extremely m and to 2000 2000m and Eo Eo to to extremely long long laboratory laboratory spark. spark. An An increase increase of of LL to 300V/cm gives the the typical typical lightning lightning parameters: parameters: ii == 120 120A and 300 V/cm gives A and 5 U, = = 12 12 Xx 10 105cm/s. A rapid rapid rise rise of of the the leader leader current current and and aa much much slower slower VL cm/s. A increase increase of of its its velocity velocity were were inevitably inevitably registered registered in in observations observations of of both both [ 1,2]. These These estimations estimations reasonably reasonably agree agree natural and and triggered triggered lightnings lightnings [1,2]. natural with with measurements. measurements. In In contrast contrast to to (4.2) (4.2) and and (4.3), (4.3), expression expression (4.5) (4.5) ignores ignores the the voltage voltage drop drop U,t « << 1lUo(L)l. It is is easy easy to to see see the the validity validity across the the leader leader channel channel because because U across Uo(L) I. It of this this assumption assumption in in the the next next approximation approximation using using the the derived derived formulae. formulae. of With M [ ]ilip',1, the the voltage voltage drop drop across across With the the voltage-current voltage-current characteristic characteristic E :::::: 12 the channel channel decreases decreases as as UL, :::::: E EeL:::::: EcL M L L'I2 M C t-' l with with the the leader leader developdevelop/ :::::: the lEoiL grows grows even even faster faster ment. At At the the tip tip site, site, on on the the contrary, contrary, 1U0(L) IUo(L) == IEolL ment. L tt22 if if one one takes takes into into account account the the increase increase of of the the average average external external than L:::::: than E,e and and field along along the the channel channel during during its its travel travel up up to to the the cloud. cloud. Note Note that that E field U,L do do not not drop drop to to zero zero in in reality reality but but only only decrease decrease to to aa certain certain limit, limit, because because U the field field Ee(i) Ec(i) in aa very very heated heated channel channel with with high high current current is is stabilized stabilized due due to to the in the greater greater plasma plasma energy energy loss loss for for radiation radiation (the (the current-voltage current-voltage charactercharacterthe E,e == c + b/ b / ii rather rather than than as as E E,e == b/ b/i). This issue, issue, istic should should be be expressed expressed as as E istic i). This however, is is of of no no importance importance to to an an ascending ascending leader, leader, since since its its current current however, U o ( L )» />> U U,.L . becomes very very high high only only when when the the tip tip reaches reaches the the region region with with 1 Uo(L)1 becomes If If one one desires desires to to refine refine these these simple simple results results by by taking taking account account of of the the voltage drop, drop, charge charge redistribution, redistribution, and and current current variation variation along along the the channel, channel, voltage one one should should regard regard itit as as aa long long line, line, as as was was done done with with the the streamer streamer in in section section 2.2.3. U ( t . x ) ,charge charge per per unit unit length length r(x, r(x,t), t), 2.2.3. The The distributions distributions of of potential potential U(t,x), and current current i(x, i ( x ,t)t ) along along the the line line can can be be described described by by equations equations similar similar to to and (2.13) (2.13) and and (2.14): (2.14):
+
1
d r ai di _ 0 fh at ax at + dx -= 0,,
+
dU
dX = E,(i),
--
i(L) i(L) == r(L)vL r(L)vL
(4.6) (4.6)
where E E,e is is the the longitudinal longitudinal channel channel field field expressed expressed through through current current i(x, i ( x , t) t) where from the the current-voltage current-voltage characteristic characteristic (the (the field field in in (2.13) (2.13) was was expressed expressed from
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An ascending positive leader
143 143
through current current and and linear linear resistance, E,e = = iR,). iR1). The The leader leader velocity velocity VL uL is is through resistance, E uL. given, for for example, example, by by formula formula (4.3): (4.3): dL/dt dL/dt == VL' given, Equations (4.6) (4.6) must involve an an electrostatic electrostatic relation charges Equations must involve relation between between charges and potentials. In aa simple simple approximation, approximation, expressions expressions (2.13) (2.13) and and (4.2) (4.2) and potentials. In were allowed allowed to to contain contain aa local local relation, ~ ( x= = ) C, C1[U(x) [ U ( x )-- Uo(x)], U o ( x ) ] through through , were relation, T(X) linear capacitance capacitance C,. C1.This This approximation approximation was was shown shown by by many many calculations calculations linear to be quite the case to be quite acceptable acceptable to to the case of of aa uniform uniform or or weakly weakly non-uniform non-uniform external field, field, but it appears appears insufficiently insufficiently rigorous rigorous for for aa strongly strongly nonnonexternal but it uniform field, which which aa leader leader crosses crosses on on entering entering aa storm storm cloud. cloud. In In fact, fact, uniform field, the potential at every every point along the the channel channel length length is is also also created created by the potential at point along by charges located located at at adjacent adjacent channel channel sites. sites. To To simplify simplify the the non-local relation, charges non-local relation, the leader leader charge charge can can be assumed to to be be concentrated concentrated on on aa cylindrical cylindrical surface surface the be assumed with an effective effective cover cover radius radius R; R; then desired relation relation takes takes the the form form with an then the the desired 1
JL
7rco
0
A U ( x , t)t) = = U(x, U ( X ,t)t) -- Uo(x) U ~ ( X= =)-46.U(x,
r ( z ,t) t) dz dz T(Z, 2 '/2 . [(z -- x) [(z x ) +~ R2] R2I1l2 *
(4.7) (4.7)
The boundary boundary conditions conditions for for the set of of integral integral differential differential equations equations (4.6) (4.6) The the set and (4.7) (4.7) are are described described by the third third equality equality in in (4.6) (4.6) and and U(O, U ( 0 ,t) t) = = 0, 0, since since and by the the leader leader base base is is grounded. grounded. Practically, Practically, it it is is convenient convenient to to subdivide subdivide the the the channel channel into into N fragments fragments and and consider consider the the charge charge density density in in each each fragment fragment to be dependent time, thus by to be dependent only only on on time, thus replacing replacing the the integral integral equation equation of of (4.7) (4.7) by set of of linear linear algebraic algebraic equations. equations. Each Each of of them them will relate the the potential will relate potential aa set U(xk) at at the the middle middle point xk of of the fragment to to the the intrinsic intrinsic and and all all U(Xk) point Xk the kth kth fragment other linear linear charges. charges. After After integrating integrating (4.7), (4.7), one one can can easily easily see see that that radius radius other with R enters enters logarithmically logarithmically the the factors factors of of the the set set of of equations equations (compare (compare with (4.2)), justifying the (4.2)), thereby thereby justifying the use use of of linear linear leader leader charge charge Tr instead instead of of its its cover space space charge. charge. The The set set of of algebraic algebraic equations equations for for U(Xk) U(xk) and and T(Xk) '(Xk) is is cover solved in in time time at at each each step, step, and and the the progress is made equations solved progress is made by by using using equations (4.6). presenting the (4.6). We We are are presenting the result result of of this this solution. solution. As the the tip As the leader leader tip tip approaches approaches the the cloud, cloud, the the external external field field at at the tip site site becomes stronger stronger and and the ever increasing increasing portion portion of of the the channel channel finds finds itself itself becomes the ever in non-uniform field. the velocity in aa strongly strongly non-uniform field. Since Since the velocity and and current current are are largely largely defined by by the Uo(L) at at the the tip site, formulae formulae (4.5), (4.5), in in which is defined the potential potential Uo(L) tip site, which Eo Eo is an average average field, field, remain remain valid. valid. In In aa simple simple model of aa cloud cloud with with aa spherical spherical an model of unipolar charged region, region, the the potential distribution in in the the space space free free from from unipolar charged potential distribution charges point charge. height of charges is is the the same same as as for for aa point charge. If If H is is the the height of the the spherical spherical charge centre, centre, Qe, Q,, the the potential potential at at height at the the point displaced from from the the height x at point displaced charge vertical vertical charge charge axis axis for for distance distance r (with (with the the account account of of the the mirror mirror reflection reflection by the plane) is by the earth's earth's plane) is
"{~ {
}.
1 1 _ 1 } 2]'/2 - [(H 2]'/2 . = 47rco 4T&o [[(H ( H-- xx)2 ) 2 + rr2I1l2 [ ( H+ x)2 + ry2I1l2
Uo(x) =
(4.8) (4.8)
Figure presents the the parameter parameter calculations Figure 4.1 4.1 presents calculations for for an an ascending ascending leader leader propagating were made made from propagating in in such such aa non-uniform non-uniform field. field. The The calculations calculations were from
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144 144
lightning discharge Physical processes in aa lightning 3300 00
- 1.4 1.4
33
-<
~
- 1.2 e~
EE .
Y ..>:: .r:;:."
200
s Ii
2i
5 U U
-g 3
-
EE 0 1.0
*z
6 i-8.... 3 -0.8 0.8 'g 'u .
8t:
~
;:l
o
2...:I
100
11
o~~~:;::::'-""--'r--"'----"---"-r"---l- 0 o 10 20 30 40 50
d
- 0.6 0.6
"E?.... L
3~
0.4 2 j - 0.4
- 0.2 0.2
Time, ms
Figure 4.1. The The propagation propagation of of an an ascending ascending leader leader from from a a grounded grounded object object in Figure in aa negative cloud cloud field. field. • Indicate Indicate current current iiLL calculated from (4.5); Qc == 55C, C, negative calculated from (4.5); Q, = 3km, r = = 0.5km. 0.5km. H=
the set of equations equations (4.6) and (4.7), described above. above. The The current current at the the set of (4.6) and (4.7), as as described at the channel base base is is defined defined by by the the total total charge charge Q and the the velocity velocity by by expression expression channel Q and (4.5): (4.5): . I
dQ
dQ
="dt= vL dL ,
Q(L) =
J~ T(X) dx .
(4.9)
For For comparison, comparison, the the current current was was also also calculated calculated from from (4.5). (4.5). The The results results show show the the good good accuracy accuracy of of this this simple simple formula, formula, so so the the use use of of average average linear linear capacitance can justifiable in capacitance C C1 can be be considered considered justifiable in the the calculation calculation of of j T(L) T ( L )== C CIAUo(L) and in in the the case case of of aa sharply sharply non-uniform non-uniform field. field. j 6.Uo(L) and
4.1.3 4.1.3 Penetration into the cloud and halt There There are are two two questions questions to to be be answered answered here: here: how how high high the the maximum maximum leader leader current current is is and and where where the the leader leader halts. halts. To To answer answer the the first first question, question, one one should should keep keep in in mind mind that that the the cloud cloud charge charge is is concentrated concentrated in in aa certain certain volume volume but but not not at at aa point. point. Suppose Suppose itit is is aa sphere sphere of of radius radius R R,c with with the the centre centre at at height height H H in in (4.8). (4.8). Measurements Measurements made made during during flights flights through through storm storm clouds clouds indicate indicate that R,c is is most most likely likely to to be be by by an an order order of of magnitude magnitude smaller smaller than than H. H . The The that R maximum of aa uniformly uniformly charged charged sphere sphere is is by by aa maximum potential potential at at the the centre centre of factor of of 1.5 factor 1.5 higher higher than than on on its its surface surface and and equals equals U U,,,, = 3Qc/81rcoRc. 3Qc/8mORc. Omax = Penetrating Penetrating into into the the charged charged region, region, an an ascending ascending leader leader acquires acquires aa considerable considerable velocity velocity and and aa very very high high current. current. To To illustrate, illustrate, calculated calculated
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ascending positive leader An ascending
145
from (4.8) (4.8) for for H H = = 3 3 km, km, the the field field near near the the earth earth under under the the charge charge centre, centre, from Eo = = -Qc/27qH2 -Qc/27ffOH2 = = 150V/cm, 150 V/cm, is is created created by by charge charge Q, Qc = = -7.4C -7.4 C and and EO ~Uo,,,I lUomaxl = = 340MV 340MV for for R, Rc = = 300m. 300m. From From (43, (4.5), aa leader leader that that has has reached reached the VL = = 2.8 2.8 xX 105m/s 10 5 mls and and the the the charged charged region region centre centre acquires acquires the the velocity velocity wL current i,,max == 55 kA kA (the field at the sphere boundary is is EOmax Eomax = = 7.5 7.5 kV/cm, kV/cm, current (the field at the sphere boundary decreasing to to zero zero towards towards the the centre; centre; in in the the current current estimation, estimation, the the decreasing logarithm was was taken taken to to be be 1). The lifetime lifetime of this high high current is short, logarithm 1). The of this current is short, 1 about Rc/w~,,, lO-'s, ascent of about Rc/VL max "" 10s, with with the the total total duration duration of of the the leader leader ascent of 2 about 33 x 10- ss (these estimations ignore ignore the density, which about (these estimations the effect effect of of air air density, which is is 1.5 times 1.5 times lower lower than than normal normal at at aa height height of of 33 km). km). Maximum Maximum currents currents of of the the kiloampere kiloampere scale scale were were registered registered during during observations observations of of ascending ascending lightnings. lightnings. On On its its way way up up through through the the charged charged region, region, the the leader leader enters enters an an area area of of reciprocal reciprocal external external field field at at height height x > H. H. The The potential potential difference difference AU, 6..Ut is, is, at first, first, positive positive but but decreases decreases as as the the leader leader elongates. elongates. Its Its velocity velocity and and current current at now now decrease decrease with with time, time, but but this this process process has has its its limits. limits. There There is is aa region region of of positive charge of same value charge positive charge of nearly nearly the the same value high high above above the the negative negative charge region. Representing Representing it as aa sphere with the the centre centre at at height height H H +D D and and region. it as sphere with taking the the mirror mirror reflection reflection effect effect into into account, account, as in (4.8), we can can find find the the taking as in (4.8), we potential of of the the dipole dipole thus thus formed: formed: potential N
+
()
Qc
Uo x = 47ffo
_ -
{ I 1 [(H _ x)2 + r2]1/2 - [(H + x)2 + r2]1/2 +
}.
11 + 11 } 2]1/2 [(H 2]1/2 . [(H X)2) + [(H+ D D -- xx)2) + ~ rr2I1l2 [(H+ D + x ~ rr2I1l2
+
+
+
+
(4.10) (4.10)
Without 0) the the Without allowance allowance for for the the voltage voltage drop drop across across the the channel channel (U (U,t I'::j 0) leader where the the absolute absolute potential potential U U.o drops drops to to leader tip tip will will reach reach the the point point xx,,s , where 6..U is small relative AU,t I'::j E 400kV, 400 kV, remaining remaining negative negative as as before. before. Since Since 6..U AUlmn is small relative 1min regions (I U to huge potentials potentials of to huge of charged charged regions Uolm,, 100 MV), MV), aa positive positive ascendascendOImax "" 100 ing leader halts halts at at aa slightly lower height height than than the the zero zero equipotential ing leader slightly lower equipotential surface surface of the the external Because of of the the effect effect of of charges reflected by by the the earth, the of external field. field. Because charges reflected earth, the zero potential potential line line lies lies somewhat lower than than the the dipole dipole centre. For example, example, zero somewhat lower centre. For at D D = =H H,, which which corresponds, more or or less, less, to to reality, reality, we we have have x, Xs = = 1.486H at corresponds, more 1.4868 exactly on on the the vertical vertical axis axis (r = 0) 0) instead instead of as would would be be the the case with exactly (Y = of 1.5H, 1.5H, as case with aa solitary solitary dipole. dipole. With With greater greater radial radial displacement displacement r, Y, the the zero zero equipotential equipotential line line comes comes closer closer to to the the earth, earth, slowly slowly at at first first but but then then more more rapidly rapidly at at r>H H (figure (figure 4.2). 4.2). This This is is the the reason reason why why ascending ascending leaders leaders taking taking different different vertical vertical paths paths halt halt at at different different heights. heights. It just been U o ( x ,r) Y) = =0 0 It has has just been mentioned mentioned that that the the equipotential equipotential line line Uo(x, corresponds corresponds to to the the maximum maximum height height attainable attainable by by aa single single ascending ascending leader. leader. With With allowance allowance for for the the voltage voltage drop drop across across the the channel, channel, which which may AU,t drops drops to to the the threshold threshold may appear appear appreciable appreciable in in some some situations, situations, 6..U below the maximum height. This is supported by numerical value value 6..U AUc, 1min below the maximum height. This is supported by numerical N
Copyright © 2000 IOP Publishing Ltd.
146 146
Physical processes in a lightning discharge 1.50 1.25 1.00
~
0.75 0.50 0.25 0.00 0.0 0.0
0.5 0.5
1.o 1.0
2.0 2.0
1.5 1.5
rlR r/H
Figure with the the allowance allowance for for charges charges Figure 4.2. The The zero zero potential potential line line of of aa cloud cloud dipole dipole with reflected by the the earth earth for for D = = H. H. reflected by
calculations and (4.7) (4.7) and and illustrated illustrated in in calculations made made from from the the set set of of equations equations (4.6) (4.6) and the leader leader figure retardation rate. rate. As the 4.3. They They also also indicate indicate the the leader leader retardation figure 4.3. field E, Ec velocity becomes lower, lower, causing causing the the field velocity decreases, decreases, the the channel channel current current becomes to together with with the the potential potential to rise. rise. The The tip tip potential potential decreases decreases respectively, respectively, together difference /:::;.U AU,, which limits limits the the current current still still more, more, and and so so on. on. difference t , which
~ Q
E
2.0 2.0
200 200
1.5
150
-,
~o
0
1000 1000
800 800
.,-
loo
...8
la
600
z
...u '" 400 ~
100 ~
00 0 ;>
+
>
$:2
d 1.0 £. .s
'Q
s1i ;:s
$'" el -l
'"\'i8 0.5 0.5
U "0
50
el -l
0.0 3 2 length, krn km Leader length,
4
00
200 200
0
Figure 4.3. Numerical Numerical simulation simulation of of an an ascending ascending leader Figure leader propagating propagating in in aa cloud cloud (Qc = = 12 12 C, C, H = =D = =3 3 km, km, rr = = 0.5 0.5 km), dipole field field (Qc km), with with allowance allowance for for the the voltage voltage dipole drop across across the the channel. channel. drop
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An ascending positive leader
147
When aa leader leader goes goes beyond beyond the the lower lower cloud cloud charge charge region, region, the the external external When field field changes changes its its direction direction along along the the channel: channel: below below the the negative negative charge charge centre, centre, x = =H H,, its its vertical vertical component component is is directed directed upwards upwards but but above above the the is directed directed downwards. downwards. Correspondingly, Correspondingly, the the external external potential potential U, Ua centre it it is centre is non-monotonic non-monotonic and and has has an an extremum extremum at at height height H (absolute (absolute maximum). maximum). is The The leader leader continues continues to to develop develop beyond beyond the the maximum maximum point, point, as as long long as as the relation relation the
flUt = ut(L) Ut(L) - Uo(L) Ua(L) = = IUo(L)I lUa(L)I-1 Ut(L)1 > AUt,,,,, flU/min - Iut(L)/ Aut is valid. valid. But But now, now, the the leader leader velocity velocity decreases decreases continuously, continuously, because because I Uo(L)I Ua(L)1 is drops with with the the leader leader elongation elongation and and IIUt(L) rises due due to to the the rising rising channel channel drops U,(L)II rises field. It It is is clear clear that that aa leader leader can can develop develop successfully successfully in in any any other other direction, direction, field. since it it is is capable capable of of propagating propagating in in the the direction direction strictly strictly opposite opposite to to the the since external field. field. The The calculations calculations show show the the leader leader path path along along the the equipotential equipotential external line in in aa zero zero external external field. field. Here, Here, AU,, flUt , i and and wL VL decrease decrease slowly, slowly, only only due due to to line the greater greater voltage voltage drop drop across across the the channel; channel; otherwise, otherwise, the the leader leader would would the travel for for an infinitely long long time. time. travel an infinitely We have have focused this circumstance circumstance because because it it is is here here that that the the principrinciWe focused on on this pal features leader process process manifest manifest themselves themselves clearly. The external pal features of of aa leader clearly. The external field field at site is is usually cannot affect at the the tip tip site usually low low and and cannot affect the the instantaneous instantaneous leader leader velovelocity, current current and and direction direction of of motion. motion. The may vary vary randomly, randomly, aa city, The direction direction may fact well well known known to to those those making making lightning lightning observations. observations. What What is is important important fact is the the voltage voltage U created by by this this field field along along the the leader leader path, path, rather rather than than the the is U,a created field strength. The propagation propagation of of aa positive positive leader is provided provided by by the the transtransfield strength. The leader is port of high positive positive charge charge to to its its streamer streamer zone. zone. The The current of many many port of aa fairly fairly high current of streamers streamers taking taking the the charge charge out out accumulates accumulates in in the the channel, channel, heating heating it it and and providing providing its its viability. viability. But But for for many many streamers streamers to to be be excited excited off off from from the the leader leader tip, tip, the the latter latter must must possess possess aa high high potential potential relative relative to to the the unperturbed unperturbed flUt = I· This potential potential AU, = Ut(L) U,(L) -- Ua(L) U o ( L )~ x lUa(L) IUo(L)I. This is is indicated indicated by by the the absence absence of of appreciable appreciable discrepancies discrepancies between between the the current current calculations calculations made made straightstraightforwardly forwardly from from the the linear linear density density of of induced induced charge charge r(x) T(X) in in aa strongly strongly nonnonuniform flU t . uniform external external field field and and from from formula formula (4.5) (4.5)containing containing only only AU,. Therefore, Therefore, it it is is not not surprising surprising that that the the lightning lightning paths paths exhibit exhibit the the diversity diversity illustrated illustrated in in figure figure 4.4. 4.4.No No random random change change of of the the leader leader path path can can disturb disturb its its viability. viability. A leader leader can can follow follow any any direction: direction: it it can can move move along along the the external external field field or or in in the the opposite opposite direction, direction, along along the the equipotential equipotential line, line, etc. etc. -- all all ways ways are flUt > AUtmn flU/min is are open open as as long long as as the the condition condition AU, is valid. valid. But But the the leader leader acceleration acceleration does does depend, depend, of of course, course, on on its its direction direction of of motion. motion. Moving Moving along along the the field, field, the the leader leader is is accelerated, accelerated, because because the the voltage voltage drop drop is is compencompensated sated excessively excessively by by the the increase increase in in 1I U U,aI.1, When When the the leader leader moves moves in in the the opposite opposite direction, direction, itit is is decelerated. decelerated. The The maximum maximum acceleration acceleration is is achieved achieved in in the the direction Ub and AU,, and this this seems seems to to be be the the reason reason for for direction of of the the maximum maximum gradient gradient fl the the fact fact that that the the main main leader leader branch branch darts darts in in the the direction direction of of the the rising rising field, field, i.e., 5.6). i.e., towards towards aa charged charged cloud, cloud, aa high high object, object, etc. etc. (for (for details details see see section section 5.6). 1
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Physical processes in a lightning discharge
4.4. A photograph of aa well-branched well-branched lightning lightning with with the the path path bendings. bendings. Figure 4.4. photograph of
4.1.4 4.1.4 Leader branching branching and sign reversal
Leader branches are nearly always visible visible in in photographs photographs of of ascending ascending Leader branches are nearly always lightnings. Branching Branching may may start start almost almost from from the the channel channel base base or or after after the the lightnings. leader has covered many many hundreds hundreds of of metres (figure 4.4). The The currents currents of of leader has covered metres (figure branches are are summed summed up up at at the the branching so it it is is higher higher at at the the channel channel branches branching points, points, so base than than in in any any branch. branch. It It is is very very unlikely that branches would start start base unlikely that branches would jj.Ut at simultaneously potential differences simultaneously and and that that the the potential differences AU, at their their tips tips would would be the same same at at any any moment moment of of time. time. Rather, Rather, the the values values of of AU, are distributed distributed be the jj. Ut are randomly. An An abrupt abrupt decrease decrease or or even even an an entire entire cut-off cut-off of of current current in in one one of of randomly. the branches does not not at at all all mean that aa similar similar thing thing has happened in in the branches does mean that has happened another branch or in in the the base. Therefore, at another branch or base. Therefore, at least least one one of of the the 'main' ‘main’ branches will will have have aa relatively current and, and, hence, hence, aa greater greater probability probability branches relatively high high current to go go up very high and even even to to reach the maximum maximum leader leader height height x, to up very high and reach the X s than than aa single leader leader does. does. This This event event is is stimulated stimulated by by the decreasing voltage drop single the decreasing voltage drop along the leader 'stem', ‘stem’, where where the the total total branch current has has accuaccualong the branching branching leader branch current mulated and and where where the the field field is is low low (in (in accordance accordance with with the the current-voltage current-voltage mulated characteristic), especially especially if if the the stem stem is is long long and and branching occurs at at different different branching occurs characteristic), heights. heights. Branching can can send send the the leader leader up above the the zero zero equipotential equipotential surface, surface, Branching up above where its sign sign reversal reversal occurs. occurs. Imagine Imagine the the situation, situation, in in which where its which aa branch, branch,
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Ann ascending positive leader A
149 149
running far from from the the charged charged cloud cloud region, region, has has reached reached the the maximum maximum running up up far height , ( r ) and and stopped. stopped. The The channel channel plasma cannot decay decay immediately immediately plasma cannot height xxs(r) but persists for for some some time. time. During During this this time, time, another, another, luckier luckier branch has but persists branch has approached the the negative negative cloud cloud bottom and even even partly it. In In approached bottom and partly penetrated penetrated it. this region, region, the the cloud cloud has has potential U,,,,, so that the branch portion that ' this potential U so that the branch portion that Omax has entered the positive positive charge has entered it it acquires acquires the charge 27rEORe IQel IQel eiRe IUoImax:::::: In(L/R) 47rEOR e :::::: 2ln(L/R):::::: 81Qel which may be as negative cloud which may be as large large as as 8:::::: S x 10% 10% of of the the negative cloud charge. charge. Due Due to to the the partial compensation partial compensation of of the the lower lower cloud cloud charge, charge, with with the the upper upper charge charge being the zero by the being constant, constant, the zero equipotential equipotential surface surface will will become become lower lower by the length ~x, which which is that length Ax, is about about the the same same percentage percentage of of Xx,s -- H, H , so so that ~x :::::: H).) . As As aa result, upper portion portion of Ax x 8(x S(x,s -- H result, the the upper of the the first first halted halted branch branch (from (from the the tip tip down down to to the the new new zero zero equipotential equipotential surface) surface) will will be be in in aa field field directed directed downwards. downwards. The The new new external external potential potential at at the the tip tip site, site, U; :x: : IdUo/dxlxs~x, IdUo/dxlxsAx, will and the potential difference difference uh will become become positive positive and the potential AU, = Ut(L) U,(L)-- Uh(L) U ; ( L )will will be negative. For For the the example example given given in in the the previous ~Ut = be negative. previous section X s -- H :::::: km at have ~x section with with x, x 1.5 1.5 km at 8S :::::: x 0.1, 0.1, we we have Ax :::::: x 150 150 m; m; from from formula formula (4.10) with with r« r << H, H , we we have have IdUo/dxlxs:::::: IdUo/dx~xs x 600V/cm, 600V/cm, so so that eventually that eventually (4.10) Uh(L) the branch penetrating the U;(L) :::::: x 9MV. 9MV. Even Even if if the branch penetrating the cloud cloud charge charge misses misses its its centre (as centre to to enter enter aa region region with with aa potential potential several several times times lower lower than than U U,Omax ,, (as aa result, be reduced be sufficient result, Uh(L) UA(L) will will be reduced as as much), much), this this will will still still be sufficient to to revive the first revive the first leader leader branch. branch. Therefore, has aa chance revive and up to Therefore, the the halted halted leader leader has chance to to revive and move move on on up to negative leader the positive cloud the upper upper positive cloud charge charge but but as as aa negative leader this this time. time. The The leader leader position position at at the the point point of of the the first first stop stop is is unstable. unstable. Even Even aa slight slight perturbation, such perturbation, such as as aa decrease decrease in in the the lower lower cloud cloud charge charge (in (in the the example example presented, branch) may presented, due due to to the the penetration penetration of of another another branch) may stimulate stimulate its its further further growth growth with with the the opposite opposite sign. sign. As As the the leader leader develops, develops, it it will will penetrate the upper penetrate into into an an increasingly increasingly higher higher field field of of the upper charge charge and and become accelerated. passed the the upper upper charge D, it become accelerated. Having Having passed charge centre, centre, H + D, it will be retarded retarded and this time, time, at height H,,, H max > H + D, D, will be and stop, stop, for for good good this at aa height where potential of Utmin . where the the potential of (4.10) (4.10) will will drop drop to to aa relatively relatively low low value value of of ~ AUtmin. The height H may be be 10-20 km or higher if The height H,,,max may 10-20km or higher if one one accounts accounts for for the the air air density density decrease. decrease. The The currents currents flow flow in in different different directions directions in in different different portions of Above the the equipotential the current portions of this this leader. leader. Above equipotential surface, surface, the current flows flows downwards, negative leader. In the the lower portion which downwards, as as in in aa negative leader. In lower leader leader portion which serves positive branches, remains directed serves as as a stem stem for for many many positive branches, the the current current remains directed upwards. The who registers the earth, may not upwards. The observer, observer, who registers the the current current at at the earth, may not suspect the sign reversal occurring up in suspect the sign reversal occurring up in the the clouds. clouds. The The channel channel field field is is established reverses in established in in accordance accordance with with the the current. current. It It reverses in the the upper upper channel channel portion, thereby reducing the branch that portion, thereby reducing the total total voltage voltage drop drop across across the the branch that went up and went far far up and stimulating stimulating its its further further development. development.
+
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+
150 150
4.2
Physical processes in in a lightning discharge
Lightning Lightning excited excited by an an isolated isolated object
Like isolated from from the the earth earth can can become become Like aa high high grounded grounded body, body, aa large large object object isolated of lightning in a high electric field of a storm cloud. Discharges can aa source source of lightning in a high electric field of a storm cloud. Discharges can be induced by fields extending not only between the earth and a charged be induced by fields extending not only between the earth and a charged cloud lightning discharge discharge cloud but but also also between between oppositely oppositely charged charged clouds. clouds. A lightning be excited by a large aircraft, rocket or spacecraft when it travels through can can be excited by a large aircraft, rocket or spacecraft when it travels through is a serious hazard to its flight. Therefore, this the troposphere, and this the troposphere, and this is a serious hazard to its flight. Therefore, this phenomenon is of primary practical importance. phenomenon is of primary practical importance. 4.2.1 4.2.1 A binary leader
In high grounded grounded body body In contrast contrast to to an an ascending ascending leader leader starting starting from from aa high (section two leaders, leaders, one one going going along along the the (section 4.1), 4.1), an an isolated isolated body body produces produces two direction. The The physical physical external external field field vector vector and and the the other other in in the the opposite opposite direction. the same. The external external field field induces induces reason reason for for the the excitation excitation of of two two leaders leaders is is the same. The between its its potential potential and and charge charge in in the the conductor, conductor, so so that that aa large large difference difference between If the the body body is is extended extended the end. If the external external potential potential arises arises at at the the conductor conductor end. increases abruptly. abruptly. In In contrast contrast along along the the field, field, the the electrical electrical strength strength at at its its end end increases to the the situation situation with with aa grounded grounded conductor, conductor, the to the opposite opposite charge charge does does not not flow at the the other other end, end, polarizing polarizing the the flow down down to to the the earth earth but but accumulates accumulates at isolated body. A grounded conductor in an external field possesses the earth’s isolated body. grounded conductor in an external field possesses the earth's potential, while while an an isolated isolated conductor conductor acquires acquires aa potential potential, potential corresponding corresponding to to an Large differences differences between between the the an average average external external potential potential along along its its length. length. Large of the the body’s and and external external potentials potentials (of (of opposite opposite signs) signs) now body's now arise arise at at the the ends ends of body, and and both both ends ends are are capable capable of of exciting exciting leaders leaders of of the the respective respective signs. signs. A body, long conductor conductor absolutely absolutely symmetrical symmetrical relative relative to long to its its average average cross cross section section transversal to the uniform field acquires potential U equal exactly to the the transversal to the uniform field acquires potential V equal exactly to external field at the body’s centre. The distribution of unlike charges in external field at the body's centre. The distribution of unlike charges in each of of its its halves halves is is identical identical to to the the charge charge distribution distribution in in aa grounded grounded each conductor of of the the same same size size and and shape shape as as the isolated conductor the isolated conductor half. half. conductor The process described here can be easily reproduced in laboratory laboratory The process described here can be easily reproduced in conditions. Figure Figure 4.5(a) 4.5(a) shows shows streak streak pictures conditions. pictures of of leaders leaders which which have have started started from aa rod rod of of 50 50 cm cm in in length, length, suspended suspended by by thin thin plastic plastic threads threads in in aa 3-m 3-m gap gap from in aa uniform uniform field. field. One One can can see see all all characteristic characteristic features features of of aa positive positive leader leader in propagating continuously continuously to to the the upper upper negative negative plane of aa stepwise stepwise propagating plane and and those those of negative leader leader travelling travelling down down towards towards aa plane negative plane anode. anode. Generally, Generally, leaders leaders arise at at different different moments moments of of time time because of the because of the threshold threshold field field difference difference arise for the the excitation excitation of of positive positive and and negative initial streamer negative initial streamer flashes flashes or or due due to to for the difference difference in in the the curvature curvature radii radii of of the the the rod rod ends. ends. The The leaders leaders may may have have different velocities velocities because because the the same same voltage U,t creates f::::.U creates streamer streamer different voltage drop drop A zones of of different different sizes sizes at at the the positive positive and and the the negative negative ends. ends. The The instantainstantazones neous currents currents at at the the growing growing channel channel ends ends may neous may also also differ. differ. But But on on average, average, every leader leader transports transports the the same same charge, charge, since since the be every the net net charge charge remains remains to to be
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Lightning excited excited by an isolated object
151 151
Figure 4.5. photographs of placed in 4.5. Streak Streak photographs of leaders leaders from from the the ends ends of of aa metallic metallic rod rod placed in aa (a) general general view; view; (b) (b) fast fast streak streak photograph photograph demonstrating demonstrating the the uniform electric field: field: (a) uniform electric relationship between positive and negative leaders; (2), (3) (3) tip tip and and relationship between the the positive and the the negative leaders; (1) (1) rod, rod, (2), streamer streamer zone zone of of positive positive leader, leader, (4), (4),(5) (5) tip tip and and streamer streamer zone zone of of negative negative leader, leader, (6), (7) negative and positive leader flashes. flashes. (6), negative and positive leader
zero the voltage be zero in in aa system system isolated isolated from from the voltage source. source. The The discharges discharges appear appear to to be say, aa flash flash -- of of one one leader leader appreciably appreciably actiactiinterrelated. Any Any fluctuation fluctuation -- say, interrelated. vates the other: other: the the space space charge charge (e.g. (e.g. positive) incorporated in in front front of of the the vates the positive) incorporated rod the accumulation rod stimulates stimulates the accumulation of of negative negative charge charge across across the the conductor, conductor, high-speed streak thereby thereby enhancing enhancing the the field field at at its its negative negative end. end. The The high-speed streak pictures pictures in resolving individual in figure figure 4.5(b), 4.5(b),resolving individual streamer streamer flashes, flashes, show show an an activation activation of of the positive the positive leader leader channel channel following following aa negative negative leader leader flash. flash. The conditions conditions for for the the start start of of leaders leaders from from aa long long isolated isolated conducting conducting The body are body are the the same same as as from from aa grounded grounded conductor, conductor, and and they they are are also also defined defined by expression (4.4). (4.4). But But now, now, when when estimating estimating the the threshold threshold field field Eo from the the by expression Eo from value of ti.U AVimin = Eod, one should keep in mind that is a half length of an value of = Eod, one should keep in mind that d is a half length of an imin isolated isolated leader. leader. For For aa field field capable capable of of exciting exciting aa discharge discharge from from aa conductor conductor of length length 2d, 2d, we find of we find
]
E = - [1 [bln(L/R)]2/5 bln(L/R) 2 / 5 . Eoo =_1_ d3/5 27fE:oa d3/5 27qa
(4.11) (4.11)
As ratio of As in in the the illustration illustration in in section section 4.1, 4.1, we we take take the the ratio of the the channel channel length length of of young leader leader to to the the equivalent equivalent charge charge cover cover radius radius to to be be L/R M 10. 10. Then Then aa young L/R ~ we = 440 440 V V/cm for an an aircraft aircraft of of length length 2d = = 70 70 m. m. This This estimate estimate we have have Eo Eo = jcm for describes describes the the external external field field component component along along the the aircraft aircraft axis. axis. But But an an aircraft aircraft so that the threshold threshold external external field field often flies flies at at an an angle angle to to the the field field vector, vector, so that the often may be be several times higher. the lightning may several times higher. Fortunately, Fortunately, the lightning excitation excitation threshold threshold is is Copyright © 2000 IOP Publishing Ltd.
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Physical processes in a lightning discharge
not very very low, low, otherwise otherwise airline airline companies companies would would suffer suffer tremendous tremendous losses losses not from lightning lightning damage. damage. On On the the other other hand, hand, fields fields of of this this scale scale are are not not very very from rare: much much higher higher fields fields were were registered registered during during airborne airborne cloud cloud surveys. surveys. For For rare: this reason, reason, the the problem problem of of lightning lightning protection protection in in aviation aviation is is regarded regarded as as this being very serious. being very serious. Having Having started started from from an an isolated isolated body, body, each each leader leader develops develops as as long long as as the permits. This the external external field field permits. This process process is is basically basically the the same same as as that that discussed discussed in we shall in section section 4.1 4.1 for for an an ascending ascending leader. leader. Below, Below, we shall consider consider the the specific specific behaviour of behaviour of two two differently differently charged charged leaders leaders developing developing simultaneously. simultaneously. This becomes especially of aa This specificity specificity becomes especially clear clear in in aa non-uniform non-uniform field field typical typical of storm storm cloud. cloud. 4.2.2 4.2.2 Binary leader development development
The principal principal features features and and quantitative quantitative characteristics characteristics of of aa binary binary leader leader can can The be understood understood from from aa simple simple model. model. The The x-coordinate x-coordinate will will be be taken taken along along the the be leaders. The The leader leader paths paths should should not not necessarily necessarily be be straight straight lines lines but but they they may may leaders. have various various bends, bends, as as is is the the case case in in reality. reality. Denote Denote the the external external field field potenpotenhave tial along along the the leader leader lines lines as as Uo(x) Uo(x) and and their their tip tip coordinates coordinates as as Xl x1 and and X2' x2. In In tial figure 4.6 4.6 Uo(x) Uo(x) corresponds corresponds to to the the field field of of aa negatively negatively charged charged cloud. cloud. The The figure leaders were were excited excited by by aa conducting conducting body body somewhere somewhere half half way way between between leaders the cloud cloud and and the the earth. earth. The The x-axis x-axis is is directed directed upwards, upwards, the the leader leader with with the the the subscript 11 travels travels downwards downwards and and the the one one with with the the subscript subscript 2 upwards. upwards. subscript Let us us neglect neglect the the voltage voltage drop drop across across the the leader leader channels, channels, ascribing ascribing the the Let same potential potential U to to the the channels channels and and the the initiating initiating body. body. The The whole whole system system same now represents represents aa single single conductor. conductor. In In the the satisfactory satisfactory approximation approximation above, above, now in which which the the capacitance capacitance per per unit unit length length C C1 at every every moment moment of of time time was was in l at
u x
Figure 4.6. 4.6. A A schematic schematic diagram diagram of of aa binary binary leader leader channel channel in in aa cloud cloud dipole dipole field. field. Figure x1:descending descending leader leader tip tip coordinate; coordinate; X2: x2:ascending ascending leader leader tip tip coordinate; coordinate; Xo: xo:position position Xl: of of zero zero charge charge point. point.
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Lightning excited by an isolated object
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assumed be the assumed to to be the same same along along the the conductor conductor length, length, the the general general condition condition for for an an uncharged uncharged conductor conductor is is X2
J
C l1[U [ U-- UO(X)] Uo(x)]dx=O; C dx = 0,
XI
U=-/ U =
1
1
X2 x 2 -- XXl I
J Uo(x) U ~ ( Xdx. dx. ) X x* 2
(4.12) (4.12)
XI
The condition condition of of (4.12) (4.12) defines defines the conductor potential U , which generally The the conductor potential U, which generally varies in in time time during during the the leader leader development development (it (it is is constant constant only only if if the the electric electric varies field and and the the binary leader are are symmetrical symmetrical relative relative to to the the centre centre of of the the field binary leader initiating body, is also also symmetrical). symmetrical). The The conductor conductor potential is initiating body, which which is potential is equal to to the the average average external external potential potential along along its its length. length. The The leader leader velocity equal velocity can be calculated from from (4.2). (4.2). It It was pointed out out above above that that the the available available can be calculated was pointed theory the velocity theory cannot cannot provide provide aa clear clear physical physical expression expression for for the velocity of of even even aa relatively negative stepwise relatively simple, simple, continuous continuous positive positive leader, leader, let let alone alone aa negative stepwise one. It It is is this this circumstance circumstance which which makes one resort the empirical empirical formula formula one. makes one resort to to the (4.2) derived derived from from results results of of laboratory laboratory experiments experiments with with currents currents up up to to (4.2) similar lOOA and justifiable, some extent, extent, for for positive positive leaders. leaders. No 100 A and justifiable, to to some No similar measurements for negative negative leaders leaders are are available, available, and and this this is is especially especially true measurements for true So, one one has has to to rely rely on on close close experimental experimental of natural lightning observations. observations. So, of natural lightning data on on breakdown breakdown voltages voltages in in superlong superlong gaps gaps at at the the sign sign reversal of the the data reversal of high voltage voltage electrode, electrode, as as well as on on the moderate velocity velocity differences differences high well as the moderate between positive positive and and negative lightning leaders. leaders. The The deviations deviations of of their their between negative lightning measured values usually overlap. These These facts facts provide provide good good grounds grounds for for measured values usually overlap. extending expression expression (4.2), (4.2), as as aa first first approximation, approximation, to to negative negative leaders. leaders. In In extending the latter latter case, case, we mean the the average average velocity velocity neglecting instantaneous the we mean neglecting the the instantaneous so justifiable effects of of stepwise stepwise development. development. This This approximation approximation is is the the more more so effects justifiable that the the direct direct dependence dependence of of the the leader leader velocity velocity on on the difference at at that the potential potential difference raise no no doubt doubt and and that the variation variation of of the the factor factor aa or or of of the the the tip, tip, AU,, the ~Ut, raise that the power picture qualitatively. power index index in in (4.2) (4.2) cannot cannot change change the the picture qualitatively. Thus, with with the account of of the x-axis directions directions and and velocities, velocities, as as well well as as Thus, the account the x-axis at the the leader leader tips, the equations equations for for the the leader leader development development the signs signs of of AU the ~U at tips, the can be be written written as as can dt
= -a[U - Uo(x1)]lI2
’
dx2 -=a[Uo(x2) dt
- U ]112 .
(4.13)) (4.13
the two Together the evolution Together with with (4.12), (4.12), expressions expressions (4.13) (4.13) describe describe the evolution of of the two xl0 and and x20 are leaders starting starting from from the the body ends, whose coordinates XlO body ends, whose coordinates X20 are leaders given given as as the the initial initial conditions conditions for for equations equations (4.12) (4.12) and and (4.13). (4.13). The The sign sign of the the conductor, conductor, xo(t), defined by the equation equation reversal reversal point point of xo(t), defined by the UO(x0) is is displaced, displaced, during during the leader propagation, in accordance accordance U(t) == Uo(xo) U(t) the leader propagation, in nature and non-uniformity along with with the the nature and degree degree of of field field non-uniformity along the the channels. channels. Having solved solved the the equations, equations, one one can can find find the currents at at the the leader leader tips Having the currents tips = x2 x l . Generally, Generally, they they differ differ quantitatively quantitatively from from one one from (4.3) (4.3) with with LL = from X2 -- Xl. another and and from from the the current current in in other other channel channel cross cross sections, sections, including including the the another through which the total charge flows flows during during the the sign reversal reversal at at point sign point xo, Xo, through which the total charge
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Physical processes in a lightning discharge
polarization. The The current current i(xo) i(xo)== iioo is is defined defined as as polarization. l'o = - = - -
dt
dt '
Ql = = Ql
XO
C1 U -- Uo(x)] Uo(x)]dx. C dx. 1 r [ [U
J
(4.14) (4.14)
Xl XI
This This current current is is used used for for changing changing the the charge charge of of the the old old leader leader portions, portions, increasing increasing or or decreasing decreasing them them as as 6.U(x, A U ( x , t), t), and and for for supplying supplying charge charge to to its its new portions. portions. This This leads leads to to the the current current variation variation along along the the channels, channels, which which new can be be found found by by solving solving the the problem. can problem. For some some simple simple distributions distributions of of Uo(x), Uo(x),the the division division of of equations equations (4.13) (4.13) For by one by one another another
[UO(X2) - U]I/2 [U - UO(X)]1/2
(4.15) (4.15)
allows the the functional functional relationship relationship between and x1 to be found from from allows between x2 X2 and Xl to be found squaring, after after which finding the the final final result result x1 ( t ) , x2(t) reduces to squaring, squaring, which finding Xl (t), X2(t) reduces to squaring, too. This This becomes becomes possible possible if if the the cloud cloud field field is is approximated approximated by the point point too. by the charge field field Uo(x) Uo(x)"'-' lxi-I, IxI-', and and if if aa new variable zz == x2/x1 is introduced. introduced. charge new variable X2/XI is The resultant resultant formulas formulas allow allow an an analytical analytical treatment treatment of of some some characteristic characteristic The relationships. To avoid avoid cumbersome cumbersome derivations, derivations, we we invite invite the reader to to do do relationships. To the reader present some this independently, this independently, while while we, we, instead, instead, shall shall present some numerical numerical calculacalculations for tions for several several variants. variants. The prove to to be be quite The calculations calculations prove quite simple simple in in integrating integrating the the set set of of equations (4.13) (4.13) and and (4.14) (4.14) as as well well as as in in the the case case of of aa more more rigorous rigorous approach approach equations to the charge charge distribution distribution along along the the conductor conductor length length is is to the the problem, problem, when when the found from from an an equation equation similar similar to to (4.7). (4.7). Figure Figure 4.7 4.7 demonstrates demonstrates the the propapropafound gation of of vertical vertical leaders leaders in in the the field field of of aa cloud cloud dipole dipole (with (with the the allowance allowance gation for the the earth's earth's effect). effect). The The calculation calculation was was made an equation equation similar similar for made using using an to (4.7). The The initiating initiating vertical vertical body is located located between lower negative negative to (4.7). body is between the the lower charge of of the dipole and and the earth, being displaced horizontally charge the dipole the earth, being displaced horizontally by by 500m from the the charge charge line. line. As As the ascending leader leader moves its tip rY == 500 m from the ascending moves up, up, its tip approaches the the bottom bottom charge charge centre centre and and enters enters aa region region of of an an ever ever increasincreasapproaches ing field. field. The The descending descending leader leader moves moves more more slowly slowly towards towards aa weaker field. ing weaker field. The the earth rapidly The external external field field potential potential approaches approaches zero zero at at the earth but but increases increases rapidly result, the the conductor near the the charged near charged cloud. cloud. As As aa result, the negative negative potential potential of of the conductor rises with with the made made up up of of the the leader leader channels, channels, U, U , rises with time, time, with the sign sign reversal reversal Xo going the cloud. point xo going up up closer closer to to the cloud. At At the the initial initial moment moment of of time, time, the the point potential is U == -27 -27 MV MV and and the the point point is is at at an an altitude altitude xo 1603m. potential is Xo == 1603 m. When the have When the ascending ascending leader leader reaches reaches the the charged charged centre centre 17 17 ms ms later, later, we we have 2040 m m and and U == -64 -64 MV. MV. The The absolute absolute potential rise the altitude altitude xo the Xo == 2040 potential rise stimulates the the descending descending leader, leader, increasing increasing its its velocity velocity by by aa factor factor of of three three stimulates during this this time time in in spite spite of of its its propagation propagation through through an an ever ever decreasing decreasing during external external field. field. The The calculations calculations made made with with (4.13) (4.13) and and (4.14) (4.14) have have yielded yielded similar results. results. similar
-
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Lightning excited by an isolated object
155 155
3.5 3.0 2.5
] ><
2.0
~1.5
----------_ ------------
-----charge sign reversal Xo (t 0)
=
::: <: 1.0 0.5
0.0 -....---T>---, " . " I +-........-..-.........---,-.....--......... . . , . . I , 0 10 15 5 10 15 20 I
o
I
Time, mc Time, me
300 250
100 50
5
10
15
20
Time, ms
Figure 4.7. The The propagation propagation of of leaders leaders from from aa metallic body located located between Figure metallic body between the the (Q, = = -10 -1OC, =D = =3 3 km, km, rY = = 0.5 0.5 km). cloud and and the the earth earth (Qe C, H = km). cloud
Figure propagation of Figure 4.8 4.8 illustrates illustrates the the propagation of one one of of the the leaders leaders in in aa zero zero external to the has external field field and and refers refers to the situation situation when when the the descending descending leader leader has suddenly changed changed its its direction direction for for some some reason at aa certain certain height to suddenly reason at height to follow the equipotential surface, surface, i.e. i.e. along along the zero field. field. The The calculation calculation follow the equipotential the zero (4.13) and and (4.14). (4.14). Similar Similar to to the the first first variant, variant, this situation was made made with this situation was with (4.13) exhibits aa remarkable remarkable property property of of aa binary binary leader. leader. The The leader leader developing developing exhibits along aa rising rising field field sustains sustains the the other other leader, leader, which travelled in in less less along which has has travelled favourable favourable conditions, conditions, allowing allowing it it to to move move with with aa certain certain acceleration acceleration even even in aa zero zero field. field. in
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Physical processes in a lightning discharge 1.4 1.4-
2 ~
~
o0
1.21.2
.
1.01.0
.
60.8 0.8i-
'gB . ~ 7 0.6 0.6.e
i:l8
leader moving moving along leader equipotential path equipotential path
.
0.4"i 0.4
.3S
.
0.2 -
\
O.O+--......--.------.-----r-~-..,....0 . 0 I 10 15 20 10 15 25
Time, ms ms Time,
4.0
1.6
1.4 3.5
7
1.2 1.2
upward leader upward
]l 3 3.0 .0
/rd downward downward leader
J
f.s U
:;;: 7
E].
.s5
1.0 1.0 0.0 2 c: 0
-
..!:I
0.8
3 ~ ''': * c:
2.5
0.6 .!:l
2.0
.~ ::l
$$
.I
.............. __ .. _- .. _- ......... _- ....
0.4
...........
0.2 1.5 I
1'5 15
20 20 Time, Time, ms ms '
25 25
0.0
The development development of of aa leader leader pair pair from from aa metallic metallic body body at at 1.5 1.5 km km above above the the Figure 4.8. The earth in in aa cloud cloud dipole dipole field field (H ( H = 33 km, km, Qc Q, = -10 - 10 C). C ) .At At the the moment moment of of time time earth km, D = 33 km, N 10 10ms and 11 km km altitude, altitude, the the descending descending leader leader turned turned to to follow follow an an equipotential equipotential path: path: "" ms and (top) leader leader velocities; (bottom) position of the zero charge charge point 0), the the altitude altitude of of the zero point (T == 0), (top) velocities; (bottom) position of of the the portion portion along along the the equipotential equipotential path. path. the ascending leader leader tip tip and and the the length length of the ascending
Where Where an an isolated isolated conducting conducting body body may may initiate initiate aa lightning lightning discharge discharge depends, leader may may start start under under aa depends, to to some some extent, extent, on on aa mere mere chance. chance. A leader as in in the illustrations just described, or or inside inside aa cloud cloud at at the the storm cloud, cloud, as storm the illustrations just described, upper charges height of height of the the lower lower or or upper charges or or somewhere somewhere between between them. them. These These variants differ differ considerably considerably in in the the polarization polarization charge charge distribution distribution along along variants the propagation conditions. situation the conductor conductor and, and, hence, hence, in in the the leader leader propagation conditions. A situation may may arise arise when when the the positive positive leader leader penetrates penetrates into into the the field field of of the the negative negative lower transporting aa positive which positive charge charge to to the the earth, earth, which lower cloud, cloud, thereby thereby transporting
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must be 'attracted' negative cloud. two situations must apparently apparently be ‘attracted’ by by the the negative cloud. (In (In the the two situations above, to the above, it it was was the the negative negative leader leader that that travelled travelled to the earth, earth, 'naturally' ‘naturally’ extracting extracting aa negative negative charge charge from from the the cloud). cloud). This This exotic exotic situation situation arises arises when body located the when two two leaders leaders are are excited excited from from aa body located somewhat somewhat above above the negative charge charge centre. centre. The The negative negative leader leader then then goes goes up up to to aa positive negative positive cloud. cloud. The The strong strong field field created created by by the the cloud cloud dipole dipole induces induces aa large large negative negative charge in in the the ascending ascending leader, leader, displacing displacing the the positive positive charge charge down. down. The The charge average external external potential potential between between the the two two leader leader tips, tips, U < 0, 0, appears appears to to average be ‘more positive' positive’ than external potential potential of of the lower tip, tip, Uo(xd Uo(xl)< 0, 0, be 'more than the the external the lower so that = U -- UO(XI) Uo(xl)> 0. 0. This This is is the the reason reason why why the the descending descending so that AU,, D.Utl = leader is is positive. positive. With With time, time, when the ascending ascending tip comes closer closer to to the when the tip comes the leader positive charge, become slightly positive charge, the the potential potential U of of the the binary binary system system does does become slightly positive (3-3.5 MV), MV), making up several several percent of IUol 1 Uol,,,.max . Then Then it it persists positive (3-3.5 making up percent of persists as such, such, sustaining sustaining the the descending descending leader leader travel to the the earth. earth. By By the as travel to the moment positive charge moment of of contact contact with with the the earth, earth, the the positive charge is is distributed distributed along along the channel the channel in in about about the the same same way way as as in in aa grounded grounded conductor conductor in in aa negative negative cloud being mainly the height height of cloud field, field, being mainly concentrated concentrated at at the of this this charge. charge. For For this this reason, the which only to the the reason, the return return stroke stroke current, current, which only slightly slightly contributes contributes to charge after after the contact, is is weak. weak. The The return stroke can can be said to to make charge the contact, return stroke be said make no no contribution to charge redistribution, redistribution, since since the the channel channel potential contribution to the the charge potential should be corrected only only slightly slightly (as (as compared compared with I Uolm,, x~ 100 100 MV), MV), by should be corrected with lUolmax by reducing it reducing it from from 33 MV MV to to zero. zero. This This reduction reduction enhances, enhances, though though only only slightly, slightly, the travels on high above the ascending ascending leader, leader, which which travels on until until it it stops stops high above the the positive positive charge charge of of the the storm storm cloud. cloud. When body initiates When an an isolated isolated body initiates two two oppositely oppositely directed directed leaders, leaders, it it does does not reaches the For not always always happen happen that that the the descending descending positive positive leader leader reaches the earth. earth. For the contact contact with earth to to take of the conductor the with the the earth take place, place, the the potential potential U of the conductor made of the the two two leaders leaders must become positive at aa certain certain moment. Othermade up up of must become positive at moment. Otherwise, the the descending will stop point xls, Xis, where where the the negative negative wise, descending leader leader will stop at at the the point leader tip tip potential U,, will will be be by small value value of of D. AVtm,, than the the leader potential Uti by aa small Utmin higher higher than D.Utmin == 0, when U is negative potential of negative potential of the the external external field field (assuming (assuming AVtm,, 0, when is equal to to UO(Xls))' U o ( x l s ) )The The . condition condition for for the the average average conductor conductor potential equal potential to to be be the earth by the positive at the moment positive at the moment of of contact contact with with the earth is is described described by the inequality inequality
JZ
°
J:2 Uo(x) dx u0(x) dx > o
(4.16) (4.16)
which potential given by which follows follows from from (4.12). (4.12). Here, Here, Uo(x) Uo(x) is is the the cloud cloud dipole dipole potential given by (4.10) with with the the allowance allowance for for its its reflection reflection from from the earth, and and x2 is the the (4.10) the earth, X2 is principle, altitude has reached by that that moment. altitude the the ascending ascending leader leader tip tip has reached by moment. In In principle, there are there are no no reasons reasons for for this this inequality inequality to to be be violated, violated, since since the the integral integral of of (4.16) in in the the limit limit x2 = 00 CO is is necessarily necessarily positive (and equal equal to to (4.16) X2 = positive (and -21n(l + D/H)Q,/4mo at r« r << H for the the negative lower cloud cloud charge charge -2ln(1 D/H) Qe/47r€O at H)) for negative lower (Q, < 0). 0). This This means the descending descending positive positive leader leader has has aa chance chance to (Qe means that that the to reach the the earth this only the ascending only requires requires that that the ascending leader leader should should reach reach reach earth -- this
+
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Physical processes in a lightning discharge
sufficient altitude. altitude. If If the the horizontal horizontal channel channel displacement displacement from from the the vertical vertical aa sufficient << H, H , we we obtain obtain x2 2 . 4 7H8 line crossing crossing the the centres centres of of the cloud charges, charges, rY « line the cloud X2 > 2.47 from H. The must cover from (4.16) (4.16) and and (4.10) (4.10) at at D = = H. The ascending ascending leader leader must cover aa distance distance 0.5H above above the positive charge charge centre. centre. of about about O.5H of the upper upper positive
4.3 4.3.1
descending leader of the first lightning lightning component component The descending The origin in the clouds clouds
Although the propagation Although lightning lightning observers observers are are familiar familiar with with the propagation of of aa descenddescending (negative) (negative) stepwise stepwise leader, leader, the the conditions conditions and and the the mechanisms mechanisms of of its its ing one has has ever ever observed observed the the lightning lightning start start or or origin are are literally literally foggy. foggy. No origin No one its development development in in the the clouds. clouds. Its Its origin origin cannot cannot be be totally totally reproduced reproduced in in its laboratory conditions, conditions, although although negative negative stepwise stepwise leaders leaders have have been been prolaboratory pro2.7). But But the the conditions conditions for for their their initiation initiation duced experimentally experimentally (section (section 2.7). duced by aa high-voltage high-voltage metallic metallic electrode electrode connected connected to to the the condenser condenser of of aa impulse impulse by generator have have little little in in common common with with what what actually actually occurs occurs in in the the clouds clouds -- aa generator cloud is is not not aa condenser condenser winding and, of of course, course, not conductor. The The cloud winding and, not aa conductor. negative cloud cloud charge charge is is scattered scattered throughout throughout the the dielectric dielectric gas gas on on small small negative hydrometeors. perceive how hydrometeors. It It is is very very hard hard to to perceive how the the charges, charges, fixed fixed to to particles particles with low low mobility mobility and and dispersed dispersed in in aa huge huge volume, volume, can can come come together together to to form form with channel in in aa matter matter of of aa few few milliseconds. milliseconds. aa plasma plasma channel In our our terrestrial terrestrial practice, practice, we we encounter encounter events events somewhat somewhat similar similar to to the the In spark initiation initiation in in the the clouds. clouds. Investigation Investigation of of what what has has caused caused an an explosion explosion spark or aa fire fire in in industrial industrial premises containing an an abundance abundance of of electrostatic electrostatic dust dust or premises containing spark discharge discharge arising arising in in aa particles or or droplets droplets can can provide provide evidence evidence for for aa spark particles medium with with aa dispersed dispersed charge. charge. Lately, Lately, there have been of studies studies medium there have been reports reports of with gas jet generators ejecting ejecting into into the atmosphere miniature electrically with gas jet generators the atmosphere miniature electrically [3,4]. Sometimes, Sometimes, extended extended bright structures of of about about 10 10 cm cm charged clouds clouds [3,4]. charged bright structures in size size were observed along along aa charged charged spray spray boundary; on some some occasions, occasions, in were observed boundary; on they observed to to form form spark spark channels channels of of about about I1m in length. length. UnfortuUnfortuthey were were observed m in so nately, no measurements could could be be made made of of the field at at the the discharge discharge start, start, so nately, no measurements the field the fact fact of of discharge discharge excitation excitation was only stated. stated. Therefore, Therefore, one one can can do do the was only nothing more more than than just make conjectures conjectures about about the the excitation excitation mechanisms mechanisms nothing just make of of lightning lightning in in the the clouds clouds and and of of sparks sparks in in laboratory laboratory sprays. sprays. Speculations concerning concerning these these mechanisms (the only only type of conclusion conclusion Speculations mechanisms (the type of we can can draw draw today) have to to be arrived at at via via the process of of elimination. elimination. A we today) have be arrived the process A cloud medium medium cannot cannot be be considered considered as as being being conductive conductive when we speak speak of of cloud when we current not transported current supply supply to to the the leader leader channel. channel. Common Common charges charges are are not transported directly to to the the leader, leader, nor do they they leave leave the the cloud cloud by by themselves themselves during during the the directly nor do fast leader leader process. process. Therefore, Therefore, the the cloud cloud charges charges play different role role -fast play aa different they are are the the source source of of electric electric field field which ionizes the the air air molecules, molecules, producing they which ionizes producing To fulfil fulfil the first task, task, the initial initial plasma, plasma, and and then then sustains sustains the leader process. the the leader process. To the first the field field somewhere somewhere in in the charged region is to exceed the ionization to exceed the ionization the the charged region is
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The first lightning component The descending leader of of the theJirst
159
threshold (E (Eii ::::::: M 20-25 20-25 kV/cm kV/cm at at the the height of the cloud charge) charge) or or the the cloud cloud threshold height of the cloud is the field polarization charge. is to to contain contain inclusions inclusions enhancing enhancing the field locally locally via via the the polarization charge. It seems seems that that neither neither mechanism should be discarded entirely, entirely, although although cloud cloud It mechanism should be discarded probing rarely registered registered fields fields exceeding exceeding several several kilovolts centimetre. probing rarely kilovolts per per centimetre. These results do testify to higher fields, because most These results do not not testify to the the absence absence of of higher fields, because most of of the concerned fields fields averaged averaged over over lengths lengths of of several several dozens the measurements measurements concerned dozens of metres. at the of lightning lightning initiaof metres. No No measurements measurements were were made made at the moment moment of initiation, of aa detector detector registering registering aa field field at at the the right tion, because because the the probability probability of right place at the extremely low. low. On On the other hand, conditions place at the right right moment moment is is extremely the other hand, the the conditions necessary for the the excitation excitation of of aa leader leader process cloud are are quite quite rare; necessary for process in in aa cloud rare; otherwise, the number number oflightning per square kilometre of the earth's otherwise, the of lightning strikes strikes per square kilometre of the earth’s surface would greatly exceed exceed 2-5 2-5 per per storm storm season. season. surface would greatly Let us estimate the to be be occupied occupied by cloud charge charge capable capable of of Let us estimate the volume volume to by aa cloud creating an an ionization ionization field. field. It It was was mentioned mentioned above above that that the the field field Eo at the the creating Eo at earth was was often often found found to to be lOOV/cm during thunderstorms. thunderstorms. This This value value earth be 100 V/cm during should to be be the near-earth should not not be be considered considered to the cloud cloud dipole dipole field, field, since since the the near-earth charge by microcoronas charge provided provided by microcoronas from from various various pointed pointed objects objects attenuates attenuates the cloud field field at at the the earth. earth. A A similar similar value value is is obtained obtained from from aa small small positive positive the cloud charge under the principal negative charge supposed supposed to to lie lie under the principal negative charge charge [5]. [5]. Taking, Taking, for for estimations, the the intrinsic intrinsic dipole dipole fields fields Eo to be be 200 200V/cm and the the heights of Eo to V/cm and heights of estimations, the lower the upper upper (positive) be x x == H == 33 km km the lower (negative) (negative) and and the (positive) charges charges to to be and we find, and H + D = =6 6 km, km, respectively, respectively, we find, from from (4.10), the the dipole dipole charges charges Q, = = 13.3 13.3C. C. These These values values will will serve serve as as guidelines guidelines in in further further numerical calcuQc numerical calculations. The The charge charge Qc Q, can can create create field field Ei M 25 kV/cm at its its boundary if it it is is lations. E i ::::::: kV/cm at boundary if distributed throughout sphere of of radius radius R, = 220 220 m. m. Measurements Measurements show show distributed throughout aa sphere Rc = that the but one that the charged charged region region is, is, in in reality, reality, 2-3 2-3 times times larger, larger, but one should should not not discard possibility of discard the the possibility of aa short short accidental accidental charge charge concentration concentration in in aa smaller smaller to the the clouds. volume due volume due to the action action of of some some flows flows in in the clouds. More probable is macroscopically averaged maxiMore probable is the the situation situation when when aa macroscopically averaged maximum field E i and mum field of of cloud cloud charge charge is is several several times times lower lower than than Ei and local local fields, fields, enhanced E i ::::::: near polarized polarized macroparticles. macroparticles. Note enhanced to to Ei x 25 kV/cm, kV/cm, arise arise near Note that that the near aa metallic polarized in the maximum maximum field field near metallic ball ball polarized in an an external external field field E is is E max = water E,,, = 3E. 3E. Similarly Similarly enhanced enhanced is is the the external external field field of of aa spherical spherical water droplet, since since water water possesses very high dielectric permittivity = 80 80 and and droplet, possesses aa very high dielectric permittivity cE = E, 3 E ~ / (+ 2 c). E ) . Therefore, Therefore, if if charge charge Qc Q, is is concentrated concentrated in in aa sphere sphere of of E,max == 3Ecj(2 & times larger larger radius, 380 m, m, the the field field three-fold three-fold enhanced enhanced by by polariJ3 times radius, R, R c == 380 polarization the ionization zation can can achieve achieve the the ionization ionization threshold. threshold. Following Following the ionization onset, onset, streamers be produced possible streamers may may be produced around around large large droplets, droplets, giving giving rise rise to to aa possible leader, because streamers may may be be branched and extended extended in in an an average average leader, because streamers branched and field of of ""IOkV/cm. -10 kV/cm. field Leaving aside aside the the mechanisms mechanisms of of ionizing ionizing fields fields and and leader leader origin, origin, Leaving because are still still poorly we shall shall take take for for granted granted only only the because they they are poorly understood, understood, we the mere fact that that aa leader leader does does occur. occur. At its start, start, aa descending descending leader leader is is mere fact At its devoid of of the the possibility of taking taking the charge it it needs away from from the the cloud. cloud. devoid possibility of the charge needs away
+
+
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Physical processes in a lightning discharge
Observations Observations show show that that this this charge charge is is quite quite large: large: an an average average negative negative leader leader transports QL R~Z -5 -5 C C and, and, sometimes, sometimes, it it is is as as large large as as transports to to the the earth earth aa charge charge QL -20 C C [1], [l], aa value value close close to to the the evaluations evaluations of of Qc Q, for for the storm cloud. cloud. But But if if the the -20 the storm cloud the only that can provide the cloud charge charge remains remains 'intact', ‘intact’, the only thing thing that can provide the charge charge balance is to develop balance is the the ascending ascending leader leader of of opposite opposite sign, sign, which which is is to develop simultasimultaneously which [6],which neously with with the the descending descending leader. leader. This This idea idea was was suggested suggested in in [6], presented qualitative distribution distribution of of the the charge charge induced induced along along aa vertical vertical presented aa qualitative conductor up of of two two leaders leaders prior prior to to and and following following its its contact contact with with conductor made made up the earth. What What happens happens is is principally principally the the same same as as in in the the excitation excitation of of two two the earth. leaders by an leaders by by aa conducting conducting body body isolated isolated from from the the earth earth and and is is affected affected by an external 4.2). This This process process is is independent independent of of the the descending descending external field field (section (section 4.2). leader sign; sign; therefore, therefore, one one should should not not think think that negative cloud cloud can can produce that aa negative produce leader only positive cloud produces aa positive positive one. only aa negative negative leader leader while while aa positive cloud always always produces one. In two oppositely produced simultaneously, In any any case, case, two oppositely charged charged leaders leaders are are produced simultaneously, and which of them them will travel to to the earth depends depends on on the the charge charge position and which of will travel the earth position in the the cloud cloud and and on on the the leader leader starting starting point. in point. leader is is most most likely likely to to be be initiated initiated near near the external boundary A binary binary leader the external boundary of the charged region, field there there is is highest. The field field at at the the of the charged region, because because the the field highest. The centre of of an an isolated isolated charged charged sphere sphere is is zero. zero. In In the the case case of of aa uniform centre uniform charge volume, the rises along charge distribution distribution throughout throughout its its volume, the field field rises along the the radius radius 2 as E", r but but decreases with the the maximum maximum decreases from from the the outside outside as as e'" e rrP2 with as E E,,max == Qc/47r~& at the boundary. For For aa dipole dipole configuration configuration of of real E Qc/47rfOR~ at the boundary. real charges, the boundary boundary surface charges, the the field field does does not not practically practically vary vary across across the surface of of H and R,, the field field at at the the the charged region. For the above values values of of D, the charged region. For the above D, Hand R c' the 5% higher than at at the lower upper point of of the lower sphere sphere is is about about 5% upper point the lower higher than the lower point. The probability of aa binary binary leader leader being being initiated initiated at at either either point point is is point. The probability of nearly the same. same. However, However, the final result result of of the leader development development nearly the the final the binary binary leader will was essentially will differ differ radically, radically, and and this this circumstance circumstance was essentially demonstrated demonstrated in in 4.2. If If both leaders are are initiated initiated at at the the bottom edge of of the the lower lower section 4.2. section both leaders bottom edge negative charge, negative leader positive one negative charge, the the negative leader will will go go down down and and the the positive one will will go up. up. The The negative negative leader leader has chance to to reach reach the the earth earth with with aa high high go has aa real real chance negative equal to to that that of of cloud cloud charges charges averaged averaged over over the the whole whole negative potential potential equal conductor length. length. The The conductor conductor is is mostly in the the region of high high negative conductor mostly in region of negative potential, nowhere the positive potential domain. of potential, nowhere entering entering the positive potential domain. The The closing closing of this the earth the wave processes of this highly highly charged charged channel channel to to the earth leads leads to to the wave processes of charging and and charge charge exchange exchange (the (the return stroke) involving involving high high current. current. charging return stroke) The represents aa real real hazard. what happens The latter latter represents hazard. This This is is what happens in in the the case case of of aa negative lightning. If If aa binary binary leader leader is is initiated initiated at at the the upper of negative lightning. upper boundary boundary of the lower lower charge, charge, the the positive leader goes goes down down to to the the earth earth and and the the negative the positive leader negative one positive descending reach the unless it descending leader leader can can never never reach the earth earth unless it one goes goes up. up. A positive acquires aa positive For this, this, its its ascending ascending partner partner must must necessarily necessarily acquires positive potential. potential. For go beyond the the zero potential point, point, closer to the go beyond zero potential closer to the upper upper positive positive charge charge of of the the dipole. to the dipole. Owing Owing to the compensation compensation of of positive positive and and negative negative charges charges at at various sites the path, path, the various sites along along the the average average potential potential transported transported down down to to the the N
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N
The descending leader of of the first lightning component
161 161
earth is is quite quite low. low. This This actually actually cancels cancels the the return return stroke stroke current. current. Positive Positive earth lightnings very low present no lightnings with with very low currents currents of of about about 1kA 1 kA present no danger danger and and are are quite number of by the the observer of their their registrations registrations by observer increases increases quite frequent. frequent. The The number with increasing increasing sensitivity sensitivity of of the the detectors detectors used. used. with 4.3.2 4.3.2 Negative leader development development and potential potential transport transport
The propagation pattern has been been believed believed by by many many to to be The stepwise stepwise propagation pattern has be the the principal problem principal problem for for aa theoretical theoretical description description of of aa negative negative leader leader [7]. [7]. However, However, it it is is oflittle of little importance importance to to the the leader leader evolution evolution whether whether it it develops develops continuously by relatively velocity are continuously or or by relatively short short steps. steps. The The leader leader current current and and velocity are averaged averaged over over many many steps. steps. Averaged Averaged also also is is the the channel channel energy energy balance, balance, although the although the the energy energy release release at at aa distance distance of of several several step step lengths lengths from from the tip tip has has aa well well defined defined periodic periodic pulse pulse character. character. We We shall shall discuss discuss the the stepwise stepwise effect return stroke, of the the return stroke, since since effect in in section section 4.6, 4.6, following following the the consideration consideration of this process is this process is involved involved in in every every step step as as the the main main component. component. The binary leader The evolution evolution of of the the descending descending channel channel of of aa binary leader is is intimately intimately related to that brother' -- the the ascending ascending leader. leader. (In (In this t h s sense, sense, the the related to that of of its its 'twin ‘twin brother’ term be more characteristic feature feature of of term 'Siamese ‘Siamese twins' twins’ would would be more appropriate.) appropriate.) A characteristic the break-off of the twins twins is is the the break-off of their their potential, potential, which which varies varies but but little little along along their their highly highly conductive conductive channels, channels, from from the the external external potential potential at at the the start. start. In In this this respect, respect, aa lightning lightning leader leader differs differs considerably considerably from from aa laboratory laboratory leader leader starting starting from from an an electrode electrode connected connected to to aa high-voltage high-voltage source. source. Being Being 'tied ‘tied up' well conducting up’ to to the the electrode, electrode, aa laboratory laboratory leader leader with with aa well conducting channel channel carries carries the electrode which may the electrode potential, potential, which may be be close close to to the the source source emf. emf. Generally, Generally, it it is is lower lower than than the the emf emf by by the the value value of of the the voltage voltage drop drop across across the the external external circuit circuit impedance impedance when when aa discharge discharge current current is is flowing flowing through through it. it. The The underestimaunderestimation between aa laboratory tion of of the the principal principal difference difference between laboratory spark spark initiated initiated from from aa high-voltage natural electrodeless high-voltage electrode electrode and and aa natural electrodeless lightning lightning leads leads to to erroneous erroneous attempts attempts to to derive derive from from observations observations the the voltage voltage drop drop value value across across the the leader leader channel. VI, transported by aa transported by channel. The The reasoning reasoning is is usually usually as as follows. follows. The The potential potential U lightning to the be estimated lightning leader leader to the earth earth can can be estimated from from the the return return stroke stroke current current and potential and the the characteristic characteristic channel channel impedance impedance (section (section 4.4). 4.4). The The cloud cloud potential V be estimated U,,OR can can also also be estimated (see (see formula formula (4.17) (4.17) below). below). The The leader leader channel channel base potential -- as the cloud were an as if if the cloud were an electrode. electrode. Therefore, Therefore, base has has the the same same potential the the cloud the voltage voltage drop drop across across the the leader leader length, length, from from the cloud to to the the earth, earth, is is AU, lull, and and the average field field in in the the channel channel is is expressed expressed as as b. V c == IlUoRl V OR I -- lUll, the average Ec = H,, or E, = b.V A UcJ/ LL,, where where L is is the the leader leader length length (L ( L ;:::: MH or 30-50% 30-50% greater greater with with the allowance path bendings). bendings). Such the allowance for for the the path Such estimations estimations lead lead to to incredibly incredibly large E c ;:::: A U,c ;:::: 100 100 MV MV and and E, M 1 1 kV/cm. kV/cm. A mature mature leader leader channel channel large values values of of b.V with i;:::: current with i ~ 11000 A 0A current cannot cannot have have such such high high fields. fields. Its Its state state is is very very much much like plasma in 1-2 like that that of of the the quasi-equilibrium quasi-equilibrium hot hot plasma in an an arc, arc, which which has has aa field field 1-2 orders magnitude lower. theory and orders of of magnitude lower. This This follows follows from from theory and from from evaluations evaluations of of fields fields in in supedong superlong laboratory laboratory sparks. sparks.
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Physical processes in a lightning discharge
The problem using using the the expression expression The attempts attempts to to solve solve the the 'inverse' ‘inverse’ problem
I1 VORl potential have have also also failed. failed. U,,l = = D..V AU,e + I1 VII U 11 to to calculate calculate the the storm storm cloud cloud potential
When correct values values of of arc arc When one one includes includes in in this this expression expression the the generally generally correct field potentials V inconsistent with with field E E,, one gets gets unjustifiably unjustifiably low low cloud cloud potentials U,,OR inconsistent e , one atmospheric calculations. atmospheric probing probing measurements measurements and and other other calculations. The is due due to to the the rigid rigid relation relation The methodological methodological error error of of both both approaches approaches is of to the the external external field field potential potential at at of the the base base potential potential of of aa descending descending leader leader to the potential undergoes undergoes aa considerconsiderthe leader leader start. start. In In actual actual reality, reality, the the channel channel potential able by the the polarization polarization charge charge distribution distribution able time time evolution, evolution, being being determined determined by along the descending leader approaches approaches the the along the the binary binary leader leader length. length. When When the descending leader earth, from the the potential potential created created earth, its its base base potential potential may may differ differ significantly significantly from of start. start. by the moment moment of by the the cloud cloud charge charge at at the the start start site site at at the can be be made made from from A simple simple calculation calculation of of the the leader leader development development can be obtained obtained equations rigorous solution solution can can be equations (4.13) (4.13) and and (4.12), (4.12), but but aa more more rigorous from were used used for for that that purpose purpose in in section section from equations equations (4.13) (4.13) and and (4.7), (4.7), which which were 4.2. both leaders leaders start start from from the the 4.2. One One should should also also bear bear in in mind mind that that if if both least one one of of them them will will enter enter the the boundary boundary of of aa charged charged cloud cloud region, region, at at least charged through its centre. Then, Then, we we have have to to charged volume volume and and may may even even pass pass through its centre. make the the next next approximation approximation discard discard the the point point model model of of aa cloud cloud dipole dipole and and make uniformly with with the the density density by by assuming assuming that that charge charge Qe Q, is is distributed distributed uniformly 3Qe/47l'R~ R e. Inside the sphere, the potential potential of of its its 3QC/47rRf in in aa sphere sphere of of radius radius R,. Inside the sphere, the at point point r to to intrinsic intrinsic charge charge is is radially radially symmetrical symmetrical and and is is equal equal at 3Qe 3Qc V Om =7l'R' co8 e
r d R,.
(4.17) (4.17)
The potentials potentials from from the the upper upper dipole dipole charge charge and and from The from charges charges reflected reflected by by the earth earth can can be be found found as as from from point charges. They the point charges. They do do not not contribute contribute of aa negative negative sphere sphere with with much U,$. For For example, example, for for the the centre centre of much to to Vos' = -13.3C -13.3C and and R R,e = = 500m, 500m, we we have have V U,,Om = Q, = Qe = -360MV -360MV and and at at the the boundary VU,,OR = = ~ VU,, = -240 MV. With all other charges taken into = -240 MV. With all other charges taken into boundary Om account, we we get get VOH U,, = = -196 - 196 MV MV for for the the bottom bottom edge edge of of the the lower lower sphere sphere account, at H = D = 3km. 3km. at Figure 4.9 4.9 presents presents the the results results of of this this calculation calculation including including those those for for the the Figure charge distribution distribution along along the the conductor conductor length length from from an charge an equation equation similar similar to to (4.7). We We have have evaluated evaluated the the development development of of both both leaders leaders along along the the dipole dipole (4.7). axis, following following the the start start from from the the bottom of the the lower lower negative negative axis, bottom edge edge of
5
(Opposite) The The model model of of aa descending descending leader Figure 4.9. (Opposite) leader from from the the lower lower boundary boundary (Qc = = -12.5C, -12.5C, H = of the the negative negative dipole dipole charge charge (Qe of = 33 km, D = = 33 km, km, R, Rc = = 0.5 0.5 km). Vertical channels channels have have no no branches: branches: (top) (top) tip tip positions Vertical positions of of the the negative negative descending descending x l , and and its its positive positive ascending ascending partner, partner, x2, leader, Xl, X2, with with the the points points of of zero zero potential potential leader, xo; (centre) (centre) charge charge distribution distribution along along the differences, Xo; the leader leader channel; channel; (bottom) (bottom) differences, potential and and velocity velocity of of the the descending descending leader. leader. potential
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The The descending leader of of the first lightning component 4
O+--~--,.-~-..,....-..------,~~
o
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15
2-
1-
E ..§ --. U 00 u
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E
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-
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e
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~
'u
~.
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Il.l
-g ~
~
~
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~
8
120 0.5 0
Copyright © 2000 IOP Publishing Ltd.
5
100 10 Time, ms
15
163 163
164 164
Physical processes in a lightning discharge
sphere. was used used as as Uo(x), except for for the the sphere. The The dipole dipole potential potential from from (4.10) (4.10) was Uo(x), except length (4.17) was was employed employed with with length in in the the charged charged region, region, where where expression expression (4.17) rr = binary system system is is accelerated accelerated = Ix jx -- HI. HI. The The descending descending negative negative leader leader of of aa binary quickly m, it it travels travels farther farther to to the the quickly after after the the start. start. Having Having covered covered about about 500 500 m, 5 earth VL:::::; (1.6-1.7) x 10 m/s, aa value value x (1.6-1.7) 105m,’s, earth with with aa slightly slightly decreasing decreasing velocity velocity wL close the earth earth in in 16 ms. By By that that time, time, close to to observations. observations. The The leader leader strikes strikes the 16ms. the ascending leader has reached the height X2 :::::; 3.6 km. This is far even the ascending leader has reached the height x2 = 3.6 km. This is far even from the zero potential point located at x:::::; 4.5 km, let alone from the from the zero potential point located at x x 4.5 km, let alone from the upper of 6km. 6 km. The The descending descending upper positive positive charge charge located located at at an an altitude altitude of leader, the local potential -185 MV, transtrans185 MV, leader, which which started started from from aa site site with with the local potential ports :::::; -lOS MV, in in spite spite of of the the U1j % 105 MV, ports to to the the earth earth nearly nearly half half of of this this value, value, U initial the channel channel assumed assumed to to initial assumption assumption of of the the zero zero voltage voltage drop drop across across the be be aa perfect perfect conductor. conductor. The by the the large large calculated calculated value value of of The reader reader should should not not feel feel discouraged discouraged by U which is more appropriate to record strong lightnings rather than to , U1, which is more appropriate to record strong lightnings rather than to aa 1 common that the the cloud cloud paramparamcommon lightning lightning discharge, discharge, especially especially considering considering that eters moderate. It will be be demonstrated demonstrated eters taken taken for for the the calculation calculation were were quite quite moderate. It will in section 4.3.3 that leader branching, which is a rule rather than an excepexcepin section 4.3.3 that leader branching, which is a rule rather than an tion, reduces considerably the potential transported down to the earth. It tion, reduces considerably the potential transported down to the earth. It is quite quite likely, likely, however, however, that that lightnings lightnings of of record is record intensities intensities are are produced produced in ordinary ordinary clouds clouds rather rather than than in in those those having in having aa record record high high charge, charge, but but only branch (or does so so slightly). slightly). only if if the the descending descending leader leader does does not not branch (or does The with unbranched unbranched leaders leaders is is The above above calculation calculation for for an an ideal ideal situation situation with interesting First, one one should should understand understand the the interesting and and useful useful for for two two reasons. reasons. First, physics before one one turns turns to to its its complex complex physics of of aa simple simple observable observable phenomenon phenomenon before modifications. The The other other reason reason is, is, probably, probably, more modifications. more important. important. Practical Practical lightning protection protection requires requires the the knowledge knowledge of lightning of both both typical typical average average lightning lightning parameters and their record high values. It is the latter that become more parameters and their record high values. It is the latter that become more important in in designing designing prospective prospective measures measures for for especially important especially valuable valuable conconAs was was pointed pointed out out above, above, the structions and and objects. objects. As structions the case case of of an an unbranched unbranched leader just just discussed discussed is is likely likely to to be be one one of of the leader the rare rare but but most most hazardous hazardous phenomena. phenomena. U11 transported transported by by aa lightning lightning leader The potential potential U The leader to to the the earth earth is is an an important parameter for practical lightning protection. The return stroke important parameter for practical lightning protection. The return stroke current (section (section 4.4), 4.4), the the most most destructive destructive force force of current of lightning, lightning, is is proportional proportional U1. The nature of U1 becomes clear from the above conception of to U to • The nature of U becomes clear from the above conception of j j descending leader leader development development in in aa binary leader process. descending binary leader process. Ideally, Ideally, potential potential U1j is is that that of of aa perfect perfect conductor, conductor, made made up of two U up of two leader leader channels, channels, at at the the of its its contact contact with with the the earth. earth. But moment of moment But the the ascending ascending and and descending descending leaders develop develop differently, differently, because because their leaders their paths paths cross cross regions regions possessing possessing Uo(x). The different distributions distributions of of cloud cloud potentials potentials Uo(x). different The descending descending leader leader travels nearly nearly without without retardation retardation because because the travels the potential potential difference difference at at its its AU,, = Uo(xl), remains almost constant (a decrease in 1 U1 is largely tip, 6.U tip, = U Uo(xd, remains almost constant (a decrease in I U! is largely tj 1 Uo(x)I decreasing towards towards the compensated by by lUo(x) compensated I decreasing the earth). earth). The The ascending ascending
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of the first lightning component The descending leader of
165 165
I”
u
x
Estimation of of the the potential potential transported transported by by aa negative negative leader leader to to the earth. Figure 4.10. Estimation the earth.
positive leader leader moves moves 'against' ‘against’ the the field field and and soon soon enters enters aa region region of of aa rapidly rapidly positive rising potential; as D.Ut2 == U -- U (X2) becomes relatively rising external external potential; as aa result, result, AU,, U0(x2) becomes relatively O low soon soon after after the the start. start. This This leads leads to to aa lower lower velocity of the the ascending ascending low velocity of leader, now goes being affected by its leader, which which now goes up up 'unwillingly', ‘unwillingly’, being affected by its more more active active twin which which moves moves faster, faster, pumping pumping its its charge charge into into it. it. For For this this reason, reason, just twin just before the the total before the descending descending leader leader contacts contacts the the earth, earth, the total potential potential of of the the system nearly nearly coincides coincides with with the the external external field field potential potential U U0(x2) at the site system (X2) at the site O of the the ascending ascending leader leader tip tip (D.U/ (AU,,2 = = U -- U Uo(x2) << lUI). 1 Ul). of O(X2) « This the transported This circumstance circumstance makes makes it it possible possible to to determine determine the transported potenpotential U1j = = U just from the the condition condition U = =U U0(x2) at x 1 = 0. The condition has tial U just from (X2) at Xj = O. The condition has O clear geometrical geometrical interpretation interpretation (figure (figure 4.10). 4.10). The The shaded shaded regions regions between aa clear between the external potential curve Uo(x) Uo(x) and the horizontal line line intercepting intercepting it it must must the external potential curve and the horizontal be identical both sides the left-hand point (the point of be identical on on both sides of of the left-hand interception interception point (the point of the the conductor xo). This conductor sign sign reversal, reversal, xo). This results results from from the the net net polarization polarization charges charges of of both signs being identical; proportional to both signs being identical; they they are are proportional to the the shaded shaded regions regions (see (see formula (4.12)). (4.12)). This This approach approach can can be to find find U U1j in in different different charge charge formula be used used to distribution models and for for different different horizontal horizontal deviations deviations of of the the vertical vertical distribution models and leader path from the dipole both leaders propaleader path from the dipole axis. axis. In In aa simple simple case case when when both leaders propagate with D == Hand gate along along the the dipole dipole axis, axis, formulae formulae (4.12) (4.12) and and (4.10) (4.10) with H and r == 0, 0, together with together with expression expression (4.17), (4.17), yield yield aa dimensionless dimensionless equality equality for for finding finding point and then then U j1: : point x2 X2 and Rc H'
~=
(4.18) (4.18)
i,
= expressions (4.18) (4.18) give give For the the variant variant shown shown in in figure figure 4.9 4.9 with with n For ~ =~, expressions and U / U O R== 0.63, 0.63, in in aa fairly fairly good good agreement agreement with with the the and UOR calculations calculations of of the the leader leader evolution evolution (note (note that that U U,,OR :::::: RZ U U,,OH is is the the external external
e2 = 1.27 1.27 ~2 =
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Physical processes in a lightning discharge
field field potential potential at at the the bottom bottom edge edge of of the the lower lower cloud cloud charge charge that that has has triggered triggered both both leaders). leaders). A negative negative descending descending leader leader can can start start from from any any point point on on the the lower lower A of hemisphere of of the the bottom bottom negative negative charge charge of of the the cloud. cloud. The The location location of hemisphere this point point is is quite quite likely likely to to be be aa matter matter of of chance, chance, since since the the field field across across the the this surface is is nearly nearly uniform. uniform. Depending Depending on on the the location location of of the the starting starting point, point, surface of different different the ascending ascending twin twin crosses crosses the the charged charged region region along along chords chords of the lengths, lengths, and and this, this, along along with with the the other other factors, factors, affects affects the the potential potential transtransported at the moment of contact U1 at the moment of contact ported to to the the earth. earth. The The maximum maximum potential potential U 1 with with the the earth earth is is characteristic characteristic of of aa leader leader that that has has started started from from the the lowermost lowermost point point of of the the charged charged sphere, sphere, when when the the paths paths of of both both twins twins cross cross the the regions regions of of maximum maximum potentials potentials and and the the ascending ascending leader leader path path in in it it is is longest. longest. The The value value of U1 max is is about about 60% 60% of of the the external external potential potential U U,,OR at at the the start start and and 40% 40% of of of U 1 max the U,,Om value value at at the the centre centre of of the the negative negative cloud cloud charge. charge. But But even even the maximum maximum U if if the the descending descending leader leader is is initiated initiated near near aa lateral lateral point point of of the the hemisphere hemisphere located located at at aa maximum maximum distance distance from from the the dipole dipole axis, axis, it it transports transports aa max ~ x O.4U O.4UoR. considerable potential potential found found from from calculations calculations to to be be 0.65U 0.65U1 considerable 1max OR ' Therefore, Therefore, an an unbranched unbranched negative negative leader leader transports transports to to the the earth earth aa high high potential, potential, (0.6-0.4)U (0.6-0.4)UOR, no matter matter where where it it has has started started from from the the lower lower OR , no hemisphere. hemisphere.
4.3.3 The The branching effect effect 4.3.3 Measurements of of return return stroke stroke current current show show that that aa descending descending negative negative Measurements 100MB lightning rarely rarely transports transports to to the the earth earth aa potential potential as as high high as as 100 MB lightning (Z, == U1/Z, U 1 / Z ,where where Z 2 is is the the channel channel impedance; impedance; see see section section 4.4.2). 4.4.2). The The (1M reason for for this this is is not not the the supposedly supposedly lower lower potentials potentials of of most most clouds. clouds. The The reason value "'-' 100 MV is value of of '-100MV is characteristic characteristic of of cloud cloud charges charges moderate moderate in in size size and and density. The The reason reason is is most most likely likely to to be be the the leader leader branching, branching, since since an an density. unbranched leader leader is is an an exception exception rather rather than than the the rule. rule. Numerous Numerous downward downward unbranched branches of of aa descending descending leader leader can can be be well well seen seen in in photographs. photographs. Although Although branches ascending leaders leaders are are screened screened by by the the clouds, clouds, their their branching branching can can be be ascending registered [8- lo]. However, However, the the potential potential registered by by radio-engineering radio-engineering instruments instruments [8-10]. is affected affected by by the the branches branches of of aa descending descending negative negative leader leader rather rather than than U 11 is of its its positive positive ascending ascending twin twin brother. brother. of us make make sure sure first first that that the the branches branches of of an an ascending ascending leader leader do do not not Let us Let change change the the situation situation much. much. In In the the limit limit of of aa very very intensive intensive branching, branching, the the negative cloud cloud bottom, bottom, pierced pierced by by numerous numerous conductive conductive channels, channels, is is electroelectronegative statically statically identical identical to to aa continuous continuous conductive conductive sphere sphere of of capacitance capacitance 4mORc, whose charge has been pushed out on to the surface. The net net R whose charge has been pushed out on to the surface. The C,c == 41l"E ' O c charge of of aa system system made made up up of of aa sphere sphere and and aa negative negative leader leader attached attached to to charge -Qc. With the the neglect neglect of of the the voltage voltage it remains remains equal equal to to the the initial initial charge charge -Qc. it With drop across across the the descending descending channel, channel, the the binary binary system system possesses possesses the the same same drop potential U along along its its length. length. At At the the moment moment of of contact contact with with the the earth, earth, potential
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The descending descending leader of of the first lightning component The
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when the leader capacitance capacitance C CL corresponds to to the length LL x H , this this when the leader the length ~ H, L corresponds potential is potential is
vU=-=-~
Qc cceV UOR OR ~ Qe = C V OR C C I1 + L/2R, L/2Re In(L/ R) Cc C,L ln(L/R) L e+C cc CL e+C
+
+
-
+
(4.19) (4.19)
where UoR would be be the the boundary boundary potential of aa charged charged sphere, sphere, were it where V potential of were it OR would isolated (the small capacitance gain due to charges induced in the earth are isolated (the small capacitance gain due to charges induced in the earth are ignored). For For L/R x 100 100 and and H/Rc = 6, 6, as as in in the numerical ignored). L/R ~ H / Re = the previous previous numerical illustration, the potential U1 x O.6UoR is nearly the same as that transported illustration, the potential VI ~ 0.6VOR is nearly the same as that transported to the the earth earth in in the the absence absence of of ascending ascending leader leader branching. The potential of to branching. The potential of the cloud-leader system system drops drops because because of of the the outflow outflow of of some some of of the cloud the cloud-leader the cloud charge to to the the new capacitance of of the the descending descending leader leader just charge new capacitance just produced. produced. A similar effect effect has has been observed in in long long laboratory laboratory sparks. sparks. The The capacitance capacitance similar been observed of an an extremely extremely long long spark spark is is often often only only one one order order of of magnitude magnitude smaller smaller of than the output capacitance of a impulse voltage generator, connected than the output capacitance of a impulse voltage generator, connected directly to to aa gap gap without without aa large large damping damping resistor. resistor. The The charge charge inflow inflow into into directly the leader leader is is quite quite appreciable appreciable and and reveals reveals itself itself as as aa voltage voltage drop drop across across the the the gap. gap. branched descending descending leader leader possesses possesses aa larger larger capacitance capacitance than than an an A branched unbranched one; one; it it takes takes away away aa higher higher charge charge from from the the cloud cloud and and decreases decreases unbranched the potential potential more. more. To To estimate estimate this this effect, effect, let let us us represent represent aa branched branched leader leader the as aa bunch bunch of of n identical identical conductors conductors of of radius radius R R and and length length L, L, spaced spaced at at as distance dd (L (L > dd » >> R). R). Supplied Supplied by by the the same same power power source, source, they they possess possess distance the same same potential potential V U and and linear linear charge charge 7. T . The The potential potential at at the the centre centre of of the any of of these these conductors conductors is is found found by by summing summing the the potentials potentials of of all all charges charges any of all all conductors, conductors, including including the the intrinsic intrinsic potential. potential. Integration Integration with with the the of neglect of of the the small small effect effect of of the the earth earth yields yields neglect V =
_7 [In ~ + (n _ 1) In ~] 21l'co R d
=
~ In [( ~) (~) n - I] 21l'co
R
d
of the the n conductors, conductors, C Ctn = n7 nrL/U, is larger larger than than that that The total total capacitance capacitance of The L/ V, is tn = of aa single single isolated isolated conductor, conductor, but but this this gain gain is is less less than than n-fold: n-fold: of
Cttnn C C cttl 1
nln(L/R) nln(L/R) l In[(L/R)(L/dt] ln[(L/R)(L/d)n-']
The reduction reduction in in the the potential potential transported transported to to the the earth earth roughly roughly follows follows The the distribution distribution of of the the cloud cloud charge charge Qe Q, between between the the capacitances capacitances of of the the the charged cloud cloud cell, cell, C C,, and of of the the leader, leader, C Ctn, described by by the the first first equality equality charged e , and tn , described of (4.19). (4.19). For For n = 10 10 branches branches separated separated at at distances distances dd = L/3 L/3 and and of L/R ~ x 100 100 derived derived from from photographs, photographs, the the capacitance capacitance is is CtlO Ctl0~ x 3.2C 3.2Ct1. This L/R tl . This well-branched leader leader will will transport transport to to the the earth earth potential potential VI U1 ~ x 0.3V 0.3Uo~. In well-branched OR ' In of aa leader leader (especially, (especially, aa well-branched well-branched one) one) which which is is view of of the the real real length length of view about about 1.5 1.5 times times longer longer than than the the charge charge height height H, H , i.e. i.e. LL ~ M 1.5H, 1.5H, the the
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Physical processes in a lightning discharge Phjsical
potential decreases to 0.2U with the potential decreases to O.2UoR, in good good agreement agreement with the data data on on negative negative OR ' in lightning currents. currents. The The number of branches and their their lengths lengths vary randomly branches and vary randomly lightning number of U1I determining determining the return stroke stroke current current with the the lightning. lightning. The The potential with potential U the return vary with them. to produce produce aa wide vary together together with them. This This variation variation is is likely likely to wide range range of variation of be of current current amplitudes. amplitudes. The The variation of storm storm cloud cloud charges charges seems seems to to be less significant. significant. less There is is another another source source of of reduction reduction in in the the potential transported by by aa There potential transported negative leader leader to to the the earth. earth. In In more more complex complex models models than than the the vertical vertical negative dipole to aa low to be be dipole variant, variant, the the reduction reduction is is due due to low positive positive charge charge assumed assumed to present at bottom of positive [5]. Calculations Calculations show show that that if if aa positive present at the the very very bottom of aa cloud cloud [5]. 4C km radius radius with the centre 4 C charge charge of of 0.25 0.25 km with the centre at at 22 km km above above the the earth earth is is added added Q, = = ±13.3C, 113.3C, R, = 0.5 0.5 km km and and H =D =3 3 km, km, the the to the the above above dipole dipole with Rc = H = D= to with Qc negative leader leader initiated initiated from from the the bottom edge of of the charge will will negative bottom edge the negative negative charge transport half of of the the potential potential to to the the earth. earth. transport half
4.3.4 Specificity of a descending descending positive positive leader leader 4.3.4 Positive leaders leaders do do not not occur occur very very frequently. frequently. Statistics Statistics indicate indicate that that in in Positive 10 times times smaller smaller than than that that of of negative negative ones. ones. But But it it Europe their their number number is is 10 Europe is quite quite likely likely that that their their actual actual number number is is larger larger than than the the number number of of their their is 4.3.1 that that aa descending descending positive positive registrations. It It was was pointed pointed out out in in section section 4.3.1 registrations. leader does does not not carry carry high high potential potential to to the the earth earth and and that that its its return return stroke stroke leader current is is low. low. For For this this reason, reason, the the electromagnetic electromagnetic field field of of aa positive positive current lightning discharge discharge can can be be detected detected at at aa much much shorter shorter distance distance than than that that of of lightning negative discharge discharge and, and, probably, probably, not not all all of of them them are are registered. registered. aa negative If the the bottom bottom charge charge of of aa cloud cloud dipole dipole is is negative, negative, aa positive positive descending descending If leader may may start start either either from from the the upper upper negative negative hemisphere hemisphere or or from from the the leader bottom hemisphere hemisphere of of the the upper upper positive positive charge. charge. The The leader leader will will reach reach the the bottom earth, transporting transporting to to it it aa positive positive potential, potential, provided provided the the condition condition of of earth, (4.16) is is met. met. With With aa small small deflection deflection of of the the leader leader vertical vertical axis axis from from the the (4.16) (Y « << H), H ) , the the transported transported potential potential found found from from (4.12) (4.12) and and (4.10) (4.10) dipole axis axis (r dipole with xx = =0 0 will will be be with U=
-~ln H~(X2 + H I )(X2 - H 2). 41TcoX2 HI (X2 + H 2)(X2 - Hd .
(4.20)
where HI H1and and H H2 are the the heights heights of of the the bottom bottom and and top top charge charge centres centres and and X2 x2 where 2 are is the the ascending ascending leader leader height height at at the the moment moment the the descending descending leader leader contacts contacts is 4.2.2 that that an an ascending ascending leader leader the earth. earth. We We mentioned mentioned at at the the end end of of section section 4.2.2 the x2 = = 2.47H 2.47H1 at H2 = 2H1; then we have = O. 0. must go go up up at at least least to to X2 j; must at H = 2H then we have U = 2 I At X2:::::> x 2 M 4Hj, 4H1, the the function function U(X2) U ( x 2 ) crosses crosses the the smooth smooth maximum, maximum, At U,,,max :KZ -Qc/207r~OH1:::::> M 8MV, 8MV, if if Qc Q, == -13.3C -13.3C and and H H1 3 km, as as in in the the U : :> -Qc/201TcoHj j == 3km, previous examples. examples. Even Even the the maximum maximum potential potential transported transported to to the the earth earth previous of aa descending descending positive positive is small. small. This This means means that that the the return return stroke stroke current current of is leader travelling travelling along along the the dipole dipole axis axis will will be be low. low. The The potential potential and and the the leader
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The descending leader of first lightning component of the the$rst
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current will be still still lower, lower, with with the the real real voltage drop U U,c across across aa channel channel of of current will be voltage drop 4H x 10 lOkm taken into into account. account. Even Even for for the the channel channel field field total length 4H:::::; total length km taken E x 10 10 V/cm, V/cm, the the value value of of U U,c :::::; 10 10 MV MV is is comparable comparable with with U U,max ,. ' This This lightlightE:::::; so weak weak that that it it has has little little chance chance of of being and included included in in the the ning is is so being registered registered and ning statistics. statistics. Vertical channels channels demonstrate demonstrate maximum maximum positive positive potentials transported Vertical potentials transported to the the earth. earth. They They go go through through more more or or less less identical identical regions regions of of negative negative (at (at to the bottom) and positive positive (at (at the the top) top) external external potentials, and the the respective respective the bottom) and potentials, and contributions to to the the integral integral of of (4.12) (4.12) are are mutually mutually compensated. compensated. Positive Positive contributions lightnings, however, can possess possess very very high high currents. currents. With With the foregoing lightnings, however, can the foregoing taken into into account, account, one one can can suggest suggest at at least least two reasons for for this. One is is aa taken two reasons this. One favourable random random deviation deviation of of the the channel channel path from the the vertical line. favourable path from vertical line. Suppose the the ascending ascending leader leader of of aa binary system, starting starting from from the Suppose binary system, the upper upper positive charge point point closest closest to earth, xo = H2 moves up vertically, positive charge to the the earth, Xo = H 2 -- R,, R c , moves up vertically, while the the other other leader, leader, having having descended descended to to the the zero zero potential potential point while point between between the charges, charges, turns turns aside aside and and goes goes along along the the zero zero equipotential equipotential line. line. After After it it the has deviated for for aa large large distance distance r from from the the dipole dipole axis, axis, it it turns down has deviated turns down vertically to to contact contact the the earth earth this this time. time. In In this this case, case, the the descending descending leader leader vertically misses to misses the the region region of of high high negative negative potential, potential, and and positive positive contribution contribution to the integral of (4.12) remains uncompensated. Calculations with formulae the integral of (4.12) remains uncompensated. Calculations with formulae (4.12), made at km, H (4.12), (4.10) (4.10) and and (4.17) (4.17) made at HI H I == 3 km, H22 == 6km, 6km, R R,c == 0.5 0.5 km, km, and and r == 1km transport to 1 km show show that that the the descending descending leader leader will will transport to the the earth earth aa potenpotential 4.3 times greater than than that to be be transported transported along along the the dipole dipole axis. axis. tial 4.3 times greater that to Another principal possibility possibility is Another principal is the the deviation deviation of of the the dipole dipole axis axis itself itself from with the vertical leader path preserved. from the the vertical vertical line, line, with the vertical leader path preserved. The The centres centres of top and bottom charges be shifted vertical line of the the top and bottom charges can can be shifted from from the the same same vertical line because of the difference the wind wind forces the because of the difference in in the forces at at different different heights. heights. Then Then the leader that has started started up up vertically vertically from from the the top top charged charged region region passes leader that has passes through the region of of high high positive positive potential, potential, while its twin, twin, descending descending ververthrough the region while its tically, will will appear appear to to be shifted aside aside relative to the the bottom bottom charge charge and and go go tically, be shifted relative to through the the region of low low negative negative potential. potential. The The effect effect will will be the same same as as through region of be the in the the first first case. case. Quantitatively, Quantitatively, it it may may even even appear appear to to be be stronger, stronger, since since in the length length and and capacitance capacitance of of the the descending descending leader leader are are smaller smaller due due to to the the the lack of of an an extended extended path path along along the the zero zero potential potential line. line. lack 4.3.5
A counterleader counterleader
The descending descending lightning lightning leader leader does does not not reach reach the the earth earth or or aa grounded grounded body, body, The the electric because because it it is is captured captured by by the the ascending ascending leader leader developing developing in in the electric field field of of cloud cloud and and earth-reflected earth-reflected charges. charges. This This field field is is enhanced enhanced by by the the charge charge of of the descending descending leader leader approaching approaching the the earth. earth. This This can can also also happen happen in in laboralaborathe tory conditions, conditions, especially especially if if the the descending descending leader leader is is negative. negative. Then Then the the tory counterleader is is positive and requires requires aa lower lower field field for for its its development. development. positive and counterleader Streak Streak pictures pictures of of laboratory laboratory sparks sparks clearly clearly show show the the counterleader counterleader start start
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Physical processes in a lightning discharge Physical
Figure 4.11. A streak photograph of with aa counterleader streak photograph of aa long long spark spark with counterleader coming coming from from aa grounded electrode. electrode. grounded
and motion towards the the descending descending leader leader (figure (figure 4.11). 4.11). The The altitude altitude at at which which and motion towards their encounter encounter occurs occurs depends depends on on the the descending descending leader leader sign sign and and charge. charge. their The length length of of the the counterleader counterleader at at the the moment of their their contact contact is is important important The moment of for lightning lightning protection protection practice, because it it defines defines the the number number of of strikes strikes at at for practice, because bodies bodies of of different different heights heights and, and, to to some some extent, extent, the the current current rise rise parameters parameters of body. of the the return return stroke stroke from from the the affected affected body. Let which the Let us us estimate estimate the the altitude altitude z, which the descending descending leader leader tip tip is is to to reach to to be be able produce aa reach able to to create create aa field field at at the the earth earth high high enough enough to to produce viable counterleader. The The latter latter does does not not differ differ from from any any other other ascending ascending viable counterleader. leader, body of requires that that the the leader, and and its its development development from from aa body of height height d requires near-terrestrial field should should exceed exceed the the value value of of Eo Eo from from formula formula (4.11). (4.1 1). near-terrestrial field For premises, the For the the height height d == 30m 30m characteristic characteristic of of industrial industrial premises, the field field must Eo ~ x 480 480V/cm. If the the cloud cloud field field is is '" -lOOV/cm, field must be be Eo Vfcm. If 100 Vfcm, the the field AE == 380Vfcm 380V/cm must must be created by the descending descending leader leader with with charge. charge. t::..E be created by the The main main contribution contribution to the near-terrestrial near-terrestrial field field is is made made by by the the charge charge The to the concentrated at at the the leader leader channel channel bottom. bottom. Therefore, Therefore, the the calculation calculation of of concentrated the field field t::..E AE under very long long vertical vertical conductor conductor should should utilize constant the under aa very utilize the the constant value of of 7r averaged averaged over over this this bottom bottom of of length length "'z, -z, rather than the the linear linear value rather than the charge by the density density of of the the non-uniform non-uniform charge charge 7(X). r(x).With With the charge reflected reflected by the earth, earth, we we have have
-
r "dx r_ ~ U u-uo - _7 t::..E z ~ ~Jx dx = - Uo ~ N
*E(z) ()
U U
=GIZ - z l n ( zLIn(L/ / R ) R) -zln(H/R) 27rco = T x - G 27rcoz z In(H/ R) 2
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(4.21)) (4.21
Return stroke
171 171
where where U is is the the channel channel potential potential and and LL :::::; xH H is is its its length, length, which which is is about about the the << H. Here, we we have have used conventional cloud height height at at the the tip height zz « tip height H. Here, used the the conventional cloud expression 7T == C C1 ( U -- Uoo)) with average linear linear capacitance capacitance and and accounted accounted expression with average 1 (U Uoi « << I1 UI. UI. For For an an unbranched for the the near-terrestrial of the the cloud, cloud, I1 Uol for near-terrestrial potential potential of unbranched x 50 50 MV, MV, we we obtain obtain zz :::::; x 260 260 m m descending leader leader carrying carrying high high potential potential U :::::; descending at In(H/ ln(H/R) x 5. 5. at R) :::::; The counterleader counterleader arises arises at at the the last last stage stage of of the the descending descending leader leader The development, i.e., i.e., near near the earth. Its Its velocity velocity is is not not high high and and is is equal equal to to development, the earth. 4 wL,1 :::::; x 22 Xx 10 104m/s from the the first first formula formula of of (4.5), (4.5), because the potential VL m/s from because the potential difference A U x Eod :::::; x 1.5 1.5 MV. MV. The The descenddescenddifference on on the the leader leader tip tip is is quite quite low, low, 6.U:::::; ing has an higher velocity. reason, the ing leader leader has an order order of of magnitude magnitude higher velocity. For For this this reason, the x (VL (wL, /vL)z x 25 m by the moment of counterleader acquires acquires the the length length LL11 :::::; counterleader /VL)Z :::::; 25 m by the moment of 1 is summed summed with with the the encounter. This This is is aa large large value, value, since since the the length length LL11 is encounter. d, so so that that the the total height of of the the grounded grounded conductor conductor becomes becomes body’s height body's height d, total height nearly doubled. This This affects affects the the frequency frequency of of the the body’s damage by by lightning lightning nearly doubled. body's damage strikes. strikes. It z,to to which which the the It follows follows from from formulae formulae (4.21) (4.21) and and (4.11) (4.11) that that the the height height z, leader leader descends descends before before it it can can initiate initiate aa counterleader, counterleader, is is greater greater for for higher higher 3 5 although this depenpremises, from which which the the counterleader counterleader starts, starts, Zz '" dd3I5, / , although premises, from this dependence the altitude body and dence is is not not very very stringent. stringent. It It is is important important that that as as the altitude of of aa body and zZ become greater, become greater, the the counterleader counterleader has has more more time time for for its its acceleration acceleration and and can can acquire acquire aa longer longer length. length. It It is is important important for for applications applications that that it it is is not not only only the the L 1 which which increases but also L) / d ratio. ratio. length increases but also the the L1 length L1 The be refined The simple simple estimation estimation obtained obtained from from (4.21) (4.21) and and (4.11) (4.11) can can be refined by by accounting non-uniformity in T ( X ) non-uniformity in the the integral integral of of (4.21) (4.21) arising arising from from accounting for for the the 7(X) U -- Uo(x) rejecting the the proportionality 7'" T U 0 ( x ) and and by by rejecting the approximation approximation of of the proportionality constant In the latter case, 7(X) should be found . C1. In the latter case, T ( X ) should be found constant linear linear capacitance capacitance C 1 from from equation equation (4.7). (4.7). Calculations Calculations show show that that the the two two corrections corrections are are rather rather small, be considered be satisfactory. so the the estimations estimations above above can can be considered to to be satisfactory. small, so N
N
4.4
Return Return stroke
4.4.1 The basic mechanism 4.4.1 process oflightning begins at the A return A return stroke, stroke, or or the the process of lightning channel channel discharging, discharging, begins at the moment the the cloud-earth the moment cloud-earth gap gap is is closed closed by by aa descending descending leader. leader. After After the contact with the body, the will be contact with the earth earth or or aa grounded grounded body, the leader leader channel channel (it (it will be taken to to be negative for must acquire potential, since the taken be negative for definiteness) definiteness) must acquire zero zero potential, since the earth's Zero potential potential is by the earth’s capacitance capacitance is is 'infinite'. ‘infinite’. Zero is also also acquired acquired by the ascendascendtwin brother. brother. The ing which is ing leader, leader, which is aa continuation continuation of of its its descending descending twin The grounding the leader high potential potential leads to aa dramatic grounding of of the leader channel channel carrying carrying aa high leads to dramatic charge redistribution along distribution prior prior charge redistribution along its its length. length. The The initial initial channel channel distribution T~ = = CdU C1[Ui U 0 ( x ) ]Here Here . and below, below, the the potential potential to stroke was was 70 to the the return return stroke and j -- Uo(x)]. transported potential for transported to to the the earth, earth, which which acts acts as as the the initial initial potential for the the return return
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Physical processes in a lightning discharge
Figure 4.12. Schematic Schematic recharging recharging of of aa lightning lightning channel channel after after the the contact contact of of the the
descending the earth. regions, charge; descending leader leader with with the earth. Shaded Shaded regions, charge; (a) ( a ) moment moment of of the the leader leader contact return stroke contact with with the the earth; earth; (b) (b) the the return stroke reaching reaching the the upper upper channel channel end; end; (c) (c) charge change. change. charge
stroke, be constant stroke, will will be be denoted denoted as as U Vi. before, it it will will be be taken taken to to be constant along along i . As before, both leader the channel both leader lengths, lengths, and and the the voltage voltage drop drop across across the channel will will be be ignored ignored as parameter. We as an an insignificant insignificant parameter. We shall shall assume assume that that the the channel channel is is charactercharacter, ized by linear which does not vary along its length or C1, which does not vary along its length or in in time time ized by linear capacitance capacitance C 1 during during the the return return stroke stroke process. process. After After the the whole whole channel channel has has acquired acquired zero zero = 0, 0, the the linear linear charge charge becomes equal to to T) T~ = = -C -C11 Uo(x). Uo(x). The potential, U = potential, becomes equal The channel portion belonging belonging to just channel portion to the the negative negative descending descending leader leader does does not not just (Uo 0, r0 0, lose its its negative negative charge charge but it acquires acquires aa positive charge (U < 0, TO < 0, lose but it positive charge o T~ > 0). 0). Not only does does it it become become discharged discharged but but it it is is also also recharged. recharged. The The TI Not only twin positive channel positive charge twin positive channel high high in in the the cloud cloud acquires acquires aa larger larger positive charge (figure return stroke (figure 4.12).The 4.12).The linear linear charge charge variation variation for for the the return stroke lifetime lifetime is is AT == TI T~ -- TO T~ = = -COU -CoUi.i . At At Ui(x) Ui(x) == const, const, the the charge charge variation is constant constant Lh variation is along the the channel channel length length and and has has such such aa value value as as if if aa long long conductor conductor (a (a along long pre-charged to becomes completely Uii becomes completely discharged discharged (as (as long line) line) pre-charged to the the voltage voltage U r0 == C) C1U Vii to to become become TI T~ = = 0). 0). if it it were were TO if It has been It has been emphasized emphasized that that the the leader leader charge charge is is concentrated concentrated in in its its cover. cover. The The charge charge in in aa non-conducting non-conducting cover cover changes changes due due to to the the charge charge incorporation incorporation from from the the conductive conductive channel, channel, owing owing to to the the streamer streamer corona corona excitation the channel excitation at at the channel surface. surface. This This is is an an exceptionally exceptionally complicated complicated proprocess, whose rate rate can be found cess, whose can be found only only from from an an adequate adequate theoretical theoretical treatment. treatment.
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Return stroke
For this reason, reason, the the assumption For this assumption of of capacitance capacitance C C1 being constant, constant, which which 1 being implies aa zero-inertia zero-inertia charge charge variation variation in in the the cover cover with with varying varying channel channel implies potential, is quite quite problematic. But if if we we discard discard this this seemingly seemingly essential essential problematic. But potential, is assumption, nothing nothing will will change change qualitatively qualitatively or or even even quantitatively. quantitatively. assumption, Indeed, suppose suppose the the cover cover charge charge does does not change at at all all during during the the time time Indeed, not change the whole whole channel channel acquires acquires zero zero potential. This is is equivalent equivalent to the assumpassumpthe potential. This to the tion that that the the channel channel capacitance capacitance is is determined, determined, during during the return stroke stroke protion the return process, by the conductor conductor radius radius rr,c rather rather than than by by the the cover cover radius radius R. Because of of R. Because cess, by the the logarithmic logarithmic dependence dependence of of linear linear capacitance capacitance (2.8) (2.8) on on the the radius, it the radius, it decreases to the value equal to about aa half half of of C C1; for example, we decreases to the value Ci ; C~ equal to about for example, we 1 obtain C C1 % IOpF/m 10pF/m and and Ci RZ 4.4pF/m at at 1=4000m, I = 4000m, R = 16m 16m and and obtain C~ ~4.4pF/m R= 1 ~ 1.5cm. cm. Then Then the the charge charge variation variation during during the the stroke stroke rY,c == 1.5
)+ ~TT== Tl -- ~TOo = A T~ = [[(C ( C1 l-- CC;)(U i ) ( U ij --U U o )o+ C ’C~(O 1 ( 0 --U oU)o])]- [- C[C 1 (1 (U U ij- -U oU)o])]
-c:ui
= -C~Ui =
(4.22) (4.22)
remains the the same same in in order order of of magnitude. magnitude. Consequently, Consequently, when when considering considering remains C: ~ % C C1 and assume the the fundamental stroke stroke mechanisms, mechanisms, one one can can take take C~ fundamental and assume 1 equivalent line line to to be be charged charged uniformly. uniformly. equivalent Measurements made made at at the the earth earth show show that that aa descending descending leader leader is is Measurements discharged with with aa very very high high current. current. For For negative negative lightnings, lightnings, the the current current discharged impulse 10-100 kA impulse of of aa return return stroke stroke with with an an amplitude amplitude 1ZM -10-100 kA lasts lasts for for 5050M "-' on the the 0.5 0.5 level. level. A short short bright bright tip tip of of the the return return channel channel well well seen seen in in 100 ps on IO011S streak photographs photographs runs runs up up for for approximately approximately the the same same time. time. Its Its velocity velocity streak v,r ~ M (0.1-0.5)c (0.1-0.5)~is is only only aa few few times times less less than than light light velocity velocity c. c. It It would would be be V natural natural to to interpret interpret this this fact fact as as the the propagation propagation of of aa discharge discharge wave wave along along the channel; channel; this this wave wave is is characterized characterized by by aa decreasing decreasing potential potential and and rising rising the current. Due Due to to an an intensive intensive energy energy release, release, the the channel channel portion portion close close to to current. the wave wave front, front, where where the the potential potential drops drops from from U U,j and and aa high high current current is is the produced, K, as produced, is is heated heated to to aa high high temperature temperature (from (from 30000 30000 to to 35000 35000K, as shown shown by by measurements). measurements). This This is is why why the the wave wave front front is is so so bright. bright. The The channel behind behind itit is is cooled cooled due due to to expansion expansion and and radiation radiation losses, losses, becoming becoming channel less bright. bright. A return return stroke stroke has has much much in in common common with with the the discharge discharge of of aa less common common metallic metallic conductor conductor in in the the form form of of aa long long line. line. The The line line discharge discharge also also has has aa wave wave nature, nature, and and this this process process was was taken taken to to be be aa model model discharge discharge in in shaping shaping the the ideas ideas concerning concerning the the return return lightning lightning stroke. stroke. lightning channel channel is is discharged discharged much much faster faster than than itit was was charged charged A lightning 2)v . But during vL ~ % (l0-3-1O(10-3-10-2)~,. But during its its development development with with the the leader leader velocity velocity VL r the the variations variations in in potential potential and and linear linear charge charge during during the the charging charging and and the the discharge are are expressed expressed as as values values of of the the same same order order of of magnitude: magnitude: TO ro"-' ~T. AT. discharge In In agreement agreement with with the the velocity, velocity, the the channel channel is is discharged discharged with with current current 11 ,M ~ M ~TVr ATV, by by aa factor factor of of Vr/VL v,/uL ~102_103 %102-103 higher higher than than the the leader leader current current M r0uL ~ ~ 1 0 0 The A . linear channel resistance I decreases approximately iiL ~ TOVL ~100A. The linear channel resistance R decreases approximately L 1 as much much during during the the leader-stroke leader-stroke transition. transition. This This decrease decrease is is due due to to the the as
-
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174 174
Physical processes in in a lightning discharge
channel channel heating heating by by high high current. current. As As aa result, result, the the plasma plasma conductivity conductivity increases increases and and the the channel channel expands, expands, making making the the conductor conductor cross cross section section larger. larger. In In this this respect, respect, aa lightning lightning discharge discharge certainly certainly differs differs from from aa discharge discharge of of aa common common conductive conductive line, line, whose whose resistance resistance remains remains constant constant (if (if the the skinskineffect is is ignored). ignored). Since Since resistance resistance is is aa plasma plasma characteristic, characteristic, its its decrease decrease can can effect be found found straightforwardly straightforwardly only only if if the the physical physical processes processes occurring occurring in in the the be channel channel are are taken taken into into account. account. (This (This situation situation will will be be analysed analysed in in section section 4.4.3.) 4.4.3.) But But this this conclusion conclusion can can be be arrived arrived at at indirectly indirectly from from general general energy energy considerations. Over Over the the return return stroke stroke lifetime, lifetime, tt,r rv H considerations. H/wr, /v" where where H H is is the the channel channel length, length, the the energy energy dissipated dissipated in in the the channel channel must must be be approximately approximately equal equal to to the the initial initial electrical electrical energy energy C C1 per unit unit length: length: 1 U:/2/2 per N
U?
2 C11U U:/2 C j /2
N rv
IhRlt, ilR 1t r
N rv
IhR1 H / v r, iiJR1H/v
N rv
ATHI-WR~. D..rHluR 1.
(4.23) (4.23)
About as as much much energy energy was was dissipated dissipated in in the the leader leader when when the the capacitance capacitance About C1 was charged. If the leader develops in the optimal mode (see section 2.6), C was charged. If the leader develops in the optimal mode (see section 2.6), j to to which which aa natural natural lightning lightning process process is, is, probably, probably, very very close, close, because because Nature Nature usually usually takes takes optimal optimal decisions, decisions, the the voltage voltage drop drop across across the the channel channel is is comcomparable with with the the excess excess of of the the leader leader tip tip potential potential over over the the external external potential. potential. parable Therefore, the the resistances resistances of of the the channel channel and and the the streamer streamer zone zone are are comparcomparTherefore, able, because because the the same same current current flows flows through through them. them. Therefore, Therefore, the the unit unit length length able, of the the leader leader dissipates dissipates the the same same energy energy C C1U,'/2 (in order of magnitude), of /2 (in order of magnitude), 1 expressed by by the the leader leader parameters parameters i], il, VL vL and and R 1L I Lsimilar similar to to (4.23). (4.23). This This expressed 2 3 yields , i.e., RdR rv 10_10. It is also found that the R I IuMrv R1Li RILiL, i.e., R 1 / R I L 10-2-10-3. It is also found that the yields Rjl 1L L average average electric electric field field in in the the leader leader channel channel and and behind behind the the discharge discharge wave wave in the the return return stroke, stroke, E E,c ~ M R1l R1IM x R1Li RILiL, have the the same same order order of of magnitude. magnitude. in L, have u ~ This This is is consistent consistent with with the the conclusion conclusion to to be be made made from from aa straightforward straightforward analysis of of the the established established states states in in both both channels. channels. The The situation situation there there is is analysis similar to to that that in in aa steady steady state state arc. arc. But But the the channel channel field field E similar E,c in in aa high high current current arc does does vary vary but but slightly slightly with with the the current current [II]. arc [ 111. M 100 100A, A, E E,c ~ z 10 10V/ Vi It follows follows from from the the foregoing foregoing that that if if aa leader leader has has iiLL ~ It 3 4 cm ~ 0.1 O/cm, the return stroke must have R ~ 10_10O/cm cm and and R ROL M 0.1 R/cm, the return stroke must have Ro x 10-3-10-4 R/cm OL o in the the steady steady state state behind behind the the wave wave front; front; the the total total resistance resistance of of aa channel channel of of in 2 several kilometres kilometres in in length length appears appears to to be be 10 lo2 R. This This value value is is comparable comparable several O. with with the the wave wave resistance resistance of of aa long long perfectly perfectly conducting conducting line line in in air, air, Z, 2,whereas whereas the total total ohmic ohmic resistance resistance of of aa leader leader of of the the same same length length is is two two orders orders of of the magnitude Z . The The ratio ratio of of the the ohmic ohmic resistance resistance of of the the line line portion portion magnitude larger larger than than Z. behind the the wave wave to to the the wave wave resistance resistance indicates indicates the the degree degree of of the the wave wave behind attenuation attenuation during during its its travel travel along along the the line line (section (section 4.4.2). 4.4.2). If If the the channel channel resistance resistance were were constant constant and and remained remained on on the the leader leader level, level, the the lightning lightning chanchannel nel discharge discharge wave wave would would attenuate, attenuate, being being unable unable to to cover cover aa considerable considerable channel channel length. length. The The current current through through the the point point of of the the channel channel closing closing on on the earth earth would would also also attenuate attenuate too too quickly. quickly. Experiments, Experiments, however, however, point point to to the the contrary: contrary: the the visible visible bright bright tip tip has has aa well-defined well-defined front, front, and and aa high high current current the is registered registered at at the the earth earth during during the the whole whole period period of of the the tip tip elevation. elevation. The The is
U?
N
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N
Return stroke
175 175
transformation of of the the leader leader channel channel during during the the wave travel decreases decreases its its transformation wave travel linear resistance resistance considerably, considerably, determining determining the the whole return stroke stroke process. linear whole return process. 4.4.2
Conclusions from from explicit solutions solutions to long line equations equations Conclusions
Long line line equations equations with with the the allowance allowance for for the the main main factor factor -- variation variation in in Long linear resistance resistance -- can can be be solved solved only only numerically numerically (section (section 4.4.4). 4.4.4). However, However, linear the nature of the the process process and and its its essential essential physical characteristics can can be be the nature of physical characteristics understood from the the analysis analysis of of well-known well-known analytical analytical solutions solutions for for simple simple understood from situations. Their Their comparison comparison with lightning observations observations indicate indicate the the imporimporsituations. with lightning tant points points for for the the formulation formulation and and solution solution of of the the real tant real problem. problem. In the absence In the absence of of transverse transverse charge charge leakage leakage due due to to imperfect imperfect insulation, insulation, aa long long line line is is described described by by the the equations equations
au au
di ai
.
- - =-L1-+Rli, L I -+R l l
ax dX
at dt
'
di
dU
dX
at
(4.24) (4.24)
-- =cl--.
They by accounting They generalize generalize equations equations (2.12) (2.12) by accounting for for inductance. inductance. The The inducinductance per per unit unit conductor conductor length, length, L I1, , as as well well as as its its capacitance capacitance C C1, can be be tance I , can assumed to approximately constant. constant. For For an an isolated isolated conductor conductor of of assumed to be be approximately radius rY,e and and length length H » >> rrc, it is is radius e , it
flo
Po H H L1 x -2 -In In-= = 0.2ln-l-lH/m. 0.2 In -pH/m L I :::::: 7T rrce rYCe 27r
(4.25) (4.25)
of an an ascending ascending to the Here, Here, the the channel channel length length H is is about about equal equal to the height height of equations (4.24) (4.24) RI = leader = 0, 0, equations leader tip. tip. For For aa perfectly perfectly conducting conducting line line with with R1 are re-differentiated to produce aa simple are re-differentiated to produce simple wave wave equation: equation:
a2u
1a2u
- 0, 21 = (L1C1)y2. (4.26) (4.26) 8x2 212 at2 If base If the the line line is is charged charged to to voltage voltage U Uii and and short-circuited short-circuited on on the the earth earth by by its its base time t == 0, voltage x= =0 0 at at the the moment moment of of time 0, aa rectangular rectangular wave wave of of complete complete voltage elimination (from (from Ui to 0) 0) and and an an unattenuated current wave wave of of the same elimination Uj to unattenuated current the same shape will propagate propagate with point (figure from the the grounding grounding point (figure 4.13): 4.13): shape will with velocity velocity vU from •
u
1
1=--
Z'
Z
=
(LC
I ) 1/2
x
'
I
~
vt.
(4.27)
While the the wave wave propagates propagates along along the the line, line, aa detector detector mounted mounted at at its its While beginning beginning will will register register direct direct current. current. If If voltage voltage U Uii is is low low and and there there is is no no charge the conductor, charge cover cover around around the conductor, the the capacitance capacitance and and inductance inductance are are by the characterized characterized by the same same radius radius rY,e in in the the logarithms logarithms of of (2.8) (2.8) and and (4.26). (4.26). In In have this this case, case, we we have
z
=
In(H/r e ) 27T
Copyright © 2000 IOP Publishing Ltd.
(flO) 1/2 = cO
60 In H n re
(4.28)
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processes in in a lightning discharge Physical processes
u ti x Figure 4.13. 4.13. Distributions Distributions of of potential potential and and current current during during the the discharge discharge of of aa perfectly perfectly Figure line. conducting line. conducting
where where cc is is light light velocity. velocity. If If one one describes describes the the capacitance, capacitance, in in contrast contrast to to the the R == 16 16m, m, then then rY,c == 1.5 1.5cm, em, inductance, with with the the leader leader cover cover radius radius R inductance, H = 4km (as (as above), above), C C1 = IOpFjm, 10pF/m, LL1 = 2.5pH/m, v=0.67C, v = 0.67C, and and H=4km I = I =2.5J..lHjm, = 500 500R. The wave wave velocity velocity is is now now lower lower than than light light velocity, velocity, but but not not 2Z = n. The much. Current Current of of amplitude amplitude ,1ZM = = 30 30 kA, kA, typical typical of of the the return return stroke stroke of of the the much. first negative negative lightning lightning component, component, arises arises at at IVii Uil = = 21 ZZM = 15 MV. The first = 15 MV. The M values values of of Vi Vi and and vw are are correct correct in in order order of of magnitude, magnitude, but but the the wave wave velocity velocity exceeds several several times times the the observable observable velocity, velocity, and and it it is is impossible impossible to to vw exceeds reduce this this discrepancy discrepancy by by varying varying the the reactive reactive line line parameters. parameters. In In aa line line of of reduce preset length, length, C C1 and L1 vary only slightly (logarithmically) with the conducpreset and L vary only slightly (logarithmically) with the conduc1 I tor radius. radius. What What remains remains to to be be done done is is to to focus focus on on the the only only parameter parameter that that tor has not not been been accounted accounted for for -- resistance resistance R R11 which which is is very very high high in in aa leader leader has but reduces reduces by by 2-3 2-3 orders orders during during aa return return stroke. stroke. but Let us us discuss discuss the the exact exact solution solution of of equations equations (4.24) (4.24) describing describing the the line line Let discharge discharge at at R R1I = const: const:
i ( x ,t ) = Z 2
=
(!2) CI
(4.29) (4.29) 1/2 '
x::::; vI
where lo(z) Zo(z)is is the the Bessel Bessel function function of of aa purely purely imaginary imaginary argumentjz: argumentjz: where
+
lo(z) ZO(Z) :::::: 1 1 + (z/2)2 (z/2)2
at at z« z << 11
+
Z lo(z) (21rz)-1/2[1 + O(z-I)] M e eZ(2~z)-1/2[1 ~ ( z - ' ) ] at at z» z >> 1. 1. z0(z) ::::::
(4.30) (4.30)
The The wave wave has has the the same same velocity velocity as as an an ideal ideal line, line, without without losses, losses, but but the the current current at at the the wave wave front front falls falls exponentially, exponentially, as as itit propagates: propagates: .
lj
(R1Xf) 22 .
Vi -at Vi = - 2 e = - 2 exp -
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(4.31 (4.31))
Return stroke
177 111
The attenuation attenuation if is is described described by the ratio ratio of of the ohmic resistance resistance R1xf Rlxf of of The by the the ohmic the line behind the the wave wave to resistance. The base current, the line behind to the the wave wave resistance. The line line base current, just the registered in which is which is just the current current registered in lightning lightning observations, observations, arises arises instantainstantaneously (with (with instantaneous instantaneous short-circuiting short-circuiting of of the the line line on on the earth) and, and, at at neously the earth) the first first moment, moment, is is determined determined exclusively exclusively by the wave wave resistance, indepenthe by the resistance, independent of of the of R i(0,O) = -Ui!Z. -Ui/Z. As As the the wave wave moves on towards the dent the value value of R II:: i(O, 0) = moves on towards the cloud, the ohmic resistance resistance the the current current has to overcome overcome becomes increasingly cloud, the ohmic has to becomes increasingly higher, so so the base current current decreases. decreases. At At at» at >> 1, 1, or or at at Rlxf/2Z» Rlxf/2Z >> 1, 1, the the higher, the base current through through the the base is current base is VV-I (4.32) (4.32) i(O, t) :::::: I 1/2 Z(27rat)
This than at wave front, This current current decreases decreases much much more more slowly slowly than at the the wave front, because because in in the negligible behind the spite spite of of the negligible front front current, current, the the line line far far behind the wave wave front front is is discharged discharged all all the the same, same, and and all all the the charge charge that that flows flows down down from from it it goes goes through the the base. base. through The The wave wave front front propagates propagates at at aa rate rate of of electromagnetic electromagnetic perturbation. perturbation. It It is independent independent of of the line ohmic ohmic resistance resistance but but is is determined determined exclusively exclusively by by is the line its parameters and its reactive reactive parameters and is is close close to to light light velocity. velocity. This This is is aa 'precursor' 'precursor' which exists under any no matter the line which exists under any conditions, conditions, no matter whether whether the line has has aa resistance or behind the precursor resistance or whether whether it it changes changes behind the wave wave front. front. The The precursor carries the line, carries information information about about the the changes changes in in the line, in in our our case case about about the the line line grounding. grounding. If If the the resistance resistance is is zero zero or, or, more more exactly, exactly, has has no no effect effect yet yet much less wave resistance resistance (Rlxf because it << Z), the the line line is is because it is is much less than than the the wave (Rlxf «Z), discharged way, and discharged in in aa resistance-free resistance-free way, and its its initial initial potential potential and and charge charge the primary primary electromagnetic practically vanish vanish right behind the practically right behind the front front of of the electromagnetic signal, precursor. When signal, the the precursor. When the the resistance resistance becomes becomes much much higher higher (practically (practically several higher) than potential disseveral times times higher) than the the wave wave resistance, resistance, the the charge charge and and potential disappear the rate reduction decreases appear gradually, gradually, and and the rate of of their their reduction decreases as as the the linear linear resistance R I increases. R I == 10 resistance RI increases. At At R1 10 n/m n / m corresponding corresponding to to the the leader leader chanchannel IlS-1 and the precursor precursor current nel resistance, resistance, the the time time constant constant is is aa = =2 2ps-l and the current decreases, decreases, in in accordance accordance with with (4.31), (4.3 l), by by an an order order of of magnitude magnitude as as compared compared with the initial value value of of i(O, i(0, 0) 0) over over the the period of time time t ""' 11 Ils, ps, for for which which the the period of with the initial precursor m (v C). Half precursor covers covers only only 200 200m (U = = 0.67 0.67C). Half way way up up to to the the cloud cloud 6 (x m), the by aa factor . According (x == 1500 1500m), the front front current current decreases decreases by factor of of 33 x 10 lo6. According 5 be 33 x 10 higher to base current lo5 times times higher to (4.32), (4.32), the the line line base current at at that that moment moment will will be than the front. the current behind the precursor than at at the front. Therefore, Therefore, the current somewhere somewhere behind the precursor will will inevitably inevitably rise rise to to aa much much larger larger value. value. Let Let us us see see where where this this happens happens and the velocity the high the and what what will will be be the velocity of of the high current current region region carrying carrying the charge charge away away from from the the line. line. relatively late the discharge process with F or the For the analysis analysis of of the the relatively late stage stage in in the discharge process with at» we shall the at >> 1, 1, we shall employ employ the the second, second, asymptotic asymptotic formula formula of of (4.30) (4.30) for for the Bessel function. function. For For the the region region xX« << xf xf = = vt ut located located fairly fairly far far from from the Bessel the be expanded. Using weak precursor, root in of 1Io0 can can be expanded. Using weak precursor, the the root in the the argument argument of N
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178 178
Physical processes in a lightning discharge
formulae formulae (4.29) (4.29) for for aa and and v, w, we we get get
i(x, t) ~ -C I Ui ( ~ )
1/2
1Tt
exp
(
X2)
- -,
(4.33)
X = --. RI Cj
4xt
This (4.22) without without the the This expression expression is is the the explicit explicit solution solution to to equations equations (4.22) inductance inductance term term but but with with the the same same boundary boundary and and initial initial conditions. conditions. The The potential potential is is
U ( x ,t)t ) ~ U vi-i r;; U(x, 2
JX/s/(4xr)’:2 (4 X I)1/2 [ exp(-e) -t2)d~ d[ == = UU,i erf erf exp(
y1T f ?i [ 0
X
(4xt)
] 1/2'
(4.34)
Expressions physical sense, Expressions (4.33) (4.33) and and (4.34) (4.34) have have aa clear clear physical sense, demonstrating demonstrating the the nature of nature of aa non-ideal non-ideal line line discharge. discharge. When the perturbation perturbation front precursor due to the the action When the front (the (the precursor due to action of of inducinductance) goes the current tance) goes far far away, away, the current decreases decreases slowly slowly from from the the line line base base to to the the time at front. front. It It also also varies varies slowly slowly in in time at every every point, point, except except for for the the region region close close to to the the front. front. This This is is the the reason reason why why the the inductance inductance effects effects in in the the main main discharge discharge region region are are very very weak. weak. With With the the neglect neglect of of the the inductance inductance term, term, equations to those heat conduction equations (4.24) (4.24) transform transform to to equations equations similar similar to those for for heat conduction or or diffusion: diffusion:
au 8(=
a au
1
ax ax' X
(4.35) (4.35)
X= RIC I '
To potential acts To use use an an analogy, analogy, the the potential acts as as temperature, temperature, current current as as heat heat flow, flow, and X as X= and x as thermal thermal conductivity conductivity (heat (heat diffusion). diffusion). We We did did not not take take x = const const out the derivative be able to come back to to this this equation, out of of the derivative deliberately deliberately to to be able to come back equation, also valid valid at at RI # const. const. also R I =IThe The process process of of line line discharge discharge is is similar similar to to the the cooling cooling of of aa uniformly uniformly heated medium, when when aa low is maintained maintained at at its its boundboundheated medium, low (zero) (zero) temperature temperature is ary, beginning with ary, beginning with the the moment moment of of time time t = = O. 0. Formulas Formulas (4.34) (4.34) and and (4.33) (4.33) describe the the diffusion diffusion of of the the earth earth potential potential along along the the channel channel (figure (figure describe 4.l4(a)). The The current-potential current-potential wave, wave, smeared smeared in in contrast contrast to to the the precursor, precursor, 4.14(a)). U
U
a
x
b
x
4.14. The The potential potential wave wave (a) in linear linear diffusion diffusion with with x X= = const const and and (b) (b) in in nonnonFigure 4.14. ( a )in X. linear diffusion diffusion with with rising rising x. linear
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Return stroke
179 179
propagates in such such aa way way that that its its characteristic characteristic point say, where the propagates in point xXl,l , say, where the potential is by half relative to Vi,j ,xi/(4Xt)I/2 x 1 / ( 4 ~ t ) ~= =/ * potential is reduced reduced by half relative to the the initial initial value value of of U 0.477, XI ::::::: with aa decreasing 0.477, obeys obeys the the diffusion diffusion law law x1 x (Xt)1/2 ( x t ) l l 2 with decreasing velocity velocity VI XI is v1 ::::::: x ! (X/ ( ~ t)/ 1/2 t ) ::::::: M ” ~xX/2xI' / 2 x 1 .From From expressions expressions (4.33), (4.33), the the current current at at point point x1 is = O. 0. Substituting Substituting the the leader leader 20% lower lower than than that that at at the the channel channel base base x 20% X = resistance R1 x 100/cm 10R/cm and and C C1 = 10 10pF/m into the the formulae, formulae, we we get get resistance R pF/m into I ::::::: I = 10 2 xX == 10 1010 m /Is.S . Over Over the time t == 10/ls 10 ps (at (at at == 20), 20), during during which weak m the time which aa weak m, the precursor will precursor will cover cover aa distance distance of of 2000 2000 m, the half-potential half-potential point point charactercharacterizing propagation of m only izing the the propagation of the the line line discharge discharge wave wave will will diffuse diffuse for for 315 3 15 m only v1 ::::::: x 0.05c. 0 . 0 5 ~ By By . that time, the base current current and will will be moving at at velocity velocity VI and be moving that time, the base i(O, will have by aa factor the initial i(0, t) t ) will have dropped dropped by factor of of 11 11 relative relative to to the initial current current i(0,O) (formula (formula (4.32». (4.32)). i(O,O) The calculated calculated values values of of xXj,l , Vj, wl, and and i(O, i(0,t) t ) can can be closer to to The be brought brought closer measurements measurements at at aa certain certain stage stage of of the the lightning lightning discharge. discharge. Instead Instead of of the the leader resistivity, resistivity, one one should should then deal with lower resistivity resistivity averaged averaged leader then deal with aa lower over region. This over the the perturbed perturbed region. This makes makes sense sense in in some some evaluations. evaluations. But But the the illusion numerical agreement with measurements illusion of of aa satisfactory satisfactory numerical agreement with measurements in in aa short stage stage of of the the process is destroyed, destroyed, as as soon soon as as we we recall one of of the the short process is recall one important return stroke important qualitative qualitative observations. observations. At At the the return stroke stage, stage, aa bright bright and wave front and well-defined well-defined wave front -- the the channel channel tip, tip, which which becomes becomes smeared smeared only time -- is moving up up to only slightly slightly with with time is moving to the the cloud. cloud. This This indicates indicates that that the the energy hence, the the current energy release release and, and, hence, current rise rise occur occur faster faster than than in in the the solution solution to (4.33). wave possessing possessing aa steep to (4.33). Clearly Clearly we we deal deal with with aa wave steep front, front, at at least least for for powerful lightnings, rather than with diffuse profiles. This powerful lightnings, rather than with diffuse current current profiles. This contracontradiction resolved by R I == const by diction can can be be resolved by rejecting rejecting the the approximation approximation R1 const and and by including, including, in in the the theoretical theoretical treatment, treatment, the the time time evolution evolution of of the the leader leader channel and and its its transformation transformation to to aa return return stroke stroke channel. channel. channel Note that that the Note the simple simple and and attractive attractive model model of of an an immediate immediate transformatransformation the leader tion of of the leader channel channel at at the the wave wave front front to to an an ideal ideal conductor conductor cannot cannot rectify the rectify the situation. situation. This This model model would would take take us us back back to to equalities equalities (4.26) (4.26) and and (4.27) (4.27) describing describing the the wave wave of of immediate immediate voltage voltage removal removal and and sustained sustained current, current, which which propagates propagates with with the the velocity velocity of of an an electromagnetic electromagnetic signal signal close possibility was was already close to to light light velocity. velocity. But But this this possibility already refused refused above. above. It It was mentioned the key was mentioned in in section section 4.4.1 4.4.1 that that the key to to the the phenomenon phenomenon of of return return stroke be the stroke should should be the analysis analysis of of the the channel channel transformation transformation dynamics. dynamics. resistance reduction reduction during The The effect effect of of the the gradual gradual resistance during the the Joule Joule heat heat release be understood release can can be understood from from equations equations (4.35) (4.35) and and (4.24) (4.24) without without the the be justifiable justifiable to by the the potential inductance inductance term. term. It It would would be to replace replace U by potential variavariation AU == U -- Uj, Ui, since since U Uii == const const in in our our approximation, approximation, so so we we get get tion dAU 1 dAU -- - aX - d A U (4.36) (4.36) i = C 1 X X ’ x = at ax ax RlCl . the amount The resistance decreases The resistance decreases while while xX increases, increases, as as the amount of of charge charge flowing flowing through becomes larger, through the the particular particular channel channel site site becomes larger, or or with with the the increase increase in in
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Physical processes in a lightning discharge
!::J.U. the rear rear sites AU. Consequently, Consequently, the sites of of the the diffusion diffusion wave, wave, where where !::J.U A U and and diffusion X are higher, propagate propagate faster are already already higher, faster than than the the front front diffusion coefficient coefficient x sites, X are A U and and x are still still low. low. To To supply supply current current to to the the region region close close sites, where where !::J.U to to the the discharge discharge wave wave front front (a (a weak weak precursor precursor is is out out of of the the question question now), now), the because of of the the small small diffusion diffusion the potential potential gradient gradient there there must must be be large large because coefficient. wave acquires coefficient. Both Both circumstances circumstances indicate indicate that that the the wave acquires aa sharp sharp front, front, its its profile profile becomes becomes steeper steeper and and convex. convex. In In contrast contrast to to the the gradual gradual asymptotic X == const, const, the the curves curves U(x) U ( x ) and and i(x) i ( x ) for for aa asymptotic approximation approximation at at x given the abscissa given moment moment of of time time look look as as if if they they stick stick into into the abscissa (figure (figure 4.14; 4.14; the be seen numerical simulation the same same will will be seen from from the the numerical simulation illustrated illustrated in in figure figure 4.17). non-linear heat heat 4.17). The The effect effect described described here here is is well well known known [12]; [12]; this this is is aa non-linear wave wave driven, driven, for for example, example, by by radiative radiative heat heat conduction, conduction, whose whose coefficient coefficient 3 approximately as as x = TT3.t drops with with decreasing decreasing temperature temperature T approximately drops X ::::: .t The which the The variant variant with with R] R I == const, const, for for which the solution solution to to (4.33) (4.33) and and (4.34) (4.34) is probably corresponds when the the energy is valid, valid, probably corresponds to to low low current current lightnings, lightnings, when energy release is too too small small to to provide an essential essential reduction reduction in in the the former former channel channel release is provide an resistance. In any any case, case, there there are are streak streak pictures pictures of of return strokes with with unclear resistance. In return strokes unclear wave fronts becoming smeared propagation for wave fronts or or those those becoming smeared after after the the propagation for aa few few hundreds of To obtain obtain conclusive conclusive evidence, evidence, stroke stroke streak streak hundreds of metres metres [13,14]. [13,14]. To pictures should be be analysed analysed at at different different currents. currents. Regretfully, Regretfully, no no simultasimultapictures should neous of currents currents and and stroke stroke waves are available. available. neous recordings recordings of waves are One can can draw draw another another conclusion conclusion from from the the solution solution to the set set of of One to the R I == const const =I# 0, 0, which which is is important important for for the the analysis analysis of of equations (4.24) (4.24) at at R] equations observations boundary conditions observations and and for for the the formulation formulation of of boundary conditions necessary necessary for for finding finding aa numerical numerical solution. solution. According According to to (4.29), (4.29), when when the the line line closes closes on on the earth earth instantaneously instantaneously the the discharge discharge current current through through the the closed closed end end also also the reaches reaches its its maximum maximum instantaneously. instantaneously. As As mentioned mentioned above, above, the the maximum maximum determined exclusively exclusively by the wave wave resistance. resistance. is independent independent of of R 1,I , being is being determined by the Clearly, be true Clearly, the the same same will will also also be true for for any any time-variable time-variable resistivity, resistivity, and and the the only question question is is how how fast fast the the current current will decrease after after the maximum. only will decrease the maximum. However, the the current current in in aa real real return return stroke stroke rises for several several microseconds, microseconds, However, rises for sometimes become sometimes for for several several dozens dozens of of microseconds, microseconds, and and this this time time may may become even with the total impulse time. Such even comparable comparable with the total impulse time. Such aa slow slow current current rise rise may may Equations (4.36) (4.36) with A U ( x . 0 ) = 0, 0, llV(O, AU(0. I)t ) = = -Vi -L'i and and no no inductance inductance terms terms allow allow selfselft Equations with llV(x,O) similar X == const. (4.33) and and (4.34) (4.34) for for x const. The The process process is is selfselfsimilar solutions. solutions. The The simplest simplest of of them them are are (4.33) b(I1 Vi U,I)" l)ntv, which corresponds corresponds qualitatively qualitatively similar in in aa more more complex complex approximation approximation for for x similar X == b( IV, which to the R] X-I evolution transformation. Constants be R I ~ x-' evolution during during the the channel channel transformation. Constants b, 6 , n, n, and and vU can can be to the R I behaviour behaviour (section (section 4.4.3): 4.4.3):n"" n x 1-2; 1-2; V"" U 0.5-1. The The wave wave front front 0.5-1. chosen chosen from from the the analysis analysis of of R] follows the relations follows the relations
-
x/ = E[b(lC:l)"]'!?t'"-'~:? Xf = ~[b(IViI)"Jl!21(v-])!2,
v[ + 1)~[b(IVil)"]]!21-(]-v)!2, Vf = = !(v i(V+ l)E[b(lC:l)"]':2t-('-V'". where to be be found by solving where ~E of of about about I1 is is to found by solving an an ordinary ordinary differential differential equation equation [12]. [12].
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Return stroke
181 181
be the commutator, whose role played by by the be only only due due to to the the properties properties of of the commutator, whose role is is played the streamer streamer zones zones of of the the descending descending leader leader and and the the counterleader. counterleader. Their Their contact contact actually actually gives gives rise rise to to the the return return stroke. stroke. The The streamer streamer zone zone field field rises, rises, as as the the streamer streamer zones zones are are reduced reduced and and the the leader leader tips tips come come close close to to each each other other or or as as the the descending descending tip tip approaches approaches the the earth earth with with no no counterleader counterleader formed. formed. 7 The mis, The streamers streamers are are accelerated accelerated to to aa velocity velocity 10 lo7 m/s, transporting transporting kiloampere kiloampere currents even even in in laboratory laboratory conditions conditions [15, [ 15,161. Thus, the rate of of current current rise rise 16]. Thus, the rate currents and the impulse front front duration duration at at the the earth, earth, tf, 9, are are determined determined by and the impulse by processes processes occurring vanishing streamer than in the former occurring in in the the vanishing streamer zone zone rather rather than in the former leader leader channel. provide indirect this, showing channel. Measurements Measurements provide indirect evidence evidence for for this, showing that that the the impulse impulse rise rise time time tf in in positive positive lightnings lightnings possessing possessing aa longer longer streamer streamer zone zone than negative negative ones, ones, at at the same voltage, voltage, is is several several times longer. than the same times longer. 4.4.3 4.4.3 Channel transformation transformation in the return return stroke
It has been been shown that electromagnetic perturbation propagates It has shown above above that electromagnetic perturbation propagates along along aa line to or line with with aa velocity velocity equal equal to or somewhat somewhat lower lower than than light light velocity, velocity, indepenindependent of of the the initial initial resistivity. resistivity. When When the the resistance resistance is is high, high, as as in in the the leader leader dent channel, the the current current and and the the potential variation induced induced by the perturbation channel, potential variation by the perturbation attenuate rapidly. rapidly. But But the the precursor precursor is is followed followed by stronger perturbation attenuate by aa stronger perturbation propagating at at aa lower lower velocity, velocity, which reduces the considerably, propagating which reduces the potential potential considerably, to zero zero with with time. The potential of aa negative negative lightning lightning drops drops to to zero zero at at to time. The potential of this channel channel site site due due to to positive charge pumping; this compensates compensates the the initial initial this positive charge pumping; this negative potential there. process is negative potential there. This This process is accompanied accompanied by by Joule Joule heat heat release release 2 with linear power which is is high at first first since since the the impulse impulse front front of of R 1, which high at with aa linear power ii2R1, the 'genuine' ‘genuine’ (not (not the the precursor) precursor) current current is is quite quite short short and and the the initial initial the (leader) resistance is relatively relatively high. high. The The processes that follow follow -- the the (leader) resistance R 1I is processes that channel heating, its the shock wave propagachannel heating, its radial radial gas-dynamic gas-dynamic expansion, expansion, the shock wave propagation, and and the the resistivity much in in common common with with those those in in tion, resistivity reduction reduction -- have have much powerful spark spark discharges discharges in in short short laboratory laboratory gaps. gaps. The The latter latter have have been been powerful extensively extensively studied studied experimentally, experimentally, theoretically, theoretically, and and numerically numerically [17-24]. [ 17-24]. Also, calculations calculations have have been been made made with with the the initial initial parameters parameters characteristic characteristic Also, of aa lightning lightning return return stroke, stroke, accounting accounting for for radiative heat exchange exchange which which is is of radiative heat especially important important in in this this large-scale large-scale phenomenon phenomenon [22-24]. [22-241. The The stroke stroke especially channel gas gas is is heated heated up to 35000 35000K. Most of of the the Joule Joule heat heat is is radiated radiated by channel up to K. Most by the highly highly heated heated gas gas in in the the ultraviolet spectrum. The The emission emission from from this this the ultraviolet spectrum. spectral region is absorbed absorbed by by the the adjacent adjacent colder colder air, air, adding adding the the newly spectral region is newly heated gas heated gas to to the the conductive conductive channel. channel. us far far from from the point of of Such aa treatment of the the process process would would take take us the point Such treatment of interest, interest, so so we we shall shall restrict restrict ourselves ourselves to to aa description description of of two two numerical numerical results results for atmospheric atmospheric air, air, obtained obtained with allowance for for radiative radiative heat heat for with aa rigorous rigorous allowance exchange both calculations, the shape parameters of exchange [23,24]. [23,24]. In In both calculations, the shape and and parameters of the the current current impulse impulse were were preset, preset, as as is is usually usually done done in in lightning lightning calculations. calculations. Of Of course, behaviour here course, the the current current behaviour here depends depends on on what what happens happens in in the the whole whole
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Physical processes in a lightning discharge
--
"7
100
S S oJ:::: Q
0
'-" b~
10
10
15
20
T, T, kK
25
30 30
35 35
The conductivity conductivity of of thermodynamically thermodynamically equilibrium equilibrium air air at at atmospheric atmospheric Figure 4.15. The pressure. pressure. of the the perturbed region. But But the formulation of of aa self-consistent self-consistent problem problem of perturbed region. the formulation requires aa combined requires combined solution solution of of aa set set of of equations equations for for aa long long line line discharge discharge and extremely extremely cumbersome cumbersome equations equations describing describing the the physical evolution of of and physical evolution each has not been each section section along along the the line. line. A problem problem of of this this complexity complexity has not been approached yet.? approached yet. t Both calculations calculations for for one-dimensional one-dimensional cylindrical cylindrical geometry geometry were were made Both made with the current impulse impulse with the current i(t) i ( t ) == IZMM t/tf t/tf
at at t
< tf, tf,
~
i(t) i ( t ) == IZMexp[-(t tf)/tp] M exp[-(t -- tf)/t p]
at t > tf 9 at!
possessing aa linearly possessing linearly rising rising front front and and exponentially exponentially decreasing decreasing tail. tail. The The calculation was made j..ls, calculation in in [23] [23] was made with with moderate moderate parameters parameters tf == 55ps, ZMM == 20 20 kA, kA, and and ttpp == 50 50 ps corresponding to to aa moderate moderate power power lightning. lightning. 1 j..lS corresponding The other other calculation calculation [24] [24] was was for for tf tf == 55 ps, = 100 100 kA, kA, and and ttpp = = 100 100 ps The j..lS, 1ZM j..lS M = of aa very very powerful lightning. It It is is generally generally believed that the the air air conductivity conductivity of powerful lightning. believed that a0 corresponds by corresponds to to its its thermodynamic thermodynamic equilibrium equilibrium and and is is determined determined by temperature temperature (figure (figure 4.15). 4.15). Figure Figure 4.16 4.16 shows shows the the evolution evolution of of pressure, pressure, gas gas density, density, temperature, temperature, and behind the powerful and radial radial velocity velocity distributions distributions behind the shock shock front front for for aa powerful current impulse impulse [24]. [24]. The The curves curves for for moderate moderate current current impulses impulses are are qualitaqualitacurrent tively tively similar. similar. problem of of aa short short laboratory laboratory spark spark is is much much simpler. simpler. The The set set should should include include aa simple simple ttThe The problem discharge bank as high-voltage source discharge equation equation for for aa capacitor capacitor bank as aa high-voltage source for for aa spark spark gap gap with with the the desired desired circuit usually usually resistance and and allowance allowance for for the the circuit circuit inductance. inductance. Note that this this kind of LRC circuit resistance Note that kind of registers damped unobservable in registers damped oscillations oscillations unobservable in lightning. lightning.
Copyright © 2000 IOP Publishing Ltd.
Return stroke stroke 2525 20
35
t = 511S 5p t=
25
~20
i$15a 15 r--.
f-<" 15
20
51:5- ~
~O
10
50
"
0
t = 511S
30
-
10 10
183 183
.),00 '
~O
3Q.O
"
"
IO 10
5
OO0
15
r,cm r, cm
20
4
t = 511S
5
0
10
r. cm
15
20 20
t = 5 llS
6
sa
4
2
! ,t......
o0 L..L-i......:.....,;.....
_
""------'-~
o
__":__ ___::'
10
15
r, cm r,cm
20
r, cm
0
Figure 4.16. The The radial distributions of of pressure density p, temperature T , and and Figure radial distributions pressure pp,, density temperature T, the cylindrical cylindrical shock shock wave wave of of aa return stroke: ,1ZM == 100 100 kA, kA, the velocity the velocity v behind behind the return stroke: tf == 5 ps, = 100 100 ps; po = =1 1 atm, atm, Po pa and and Co co are are the initial presure, air density density and and tJ J..lS, t pp = J..ls; Po the initial presure, air sound velocity; To == 300 300 K. K. sound velocity; To
The point of primary interest in in aa return return stroke stroke treatment treatment is is the the behaThe point of primary interest behaviour resistance: its linear linear resistance: viour of of the the integral integral channel channel parameter parameter -- its R Ij ==
[[f~1;
drr-'.
27rra(r) dr] 27fro-(r)
1
(4.37) (4.37)
Table 4.1 4.1 presents, among other other parameters, the linear linear resistance resistance values values presents, among parameters, the Table obtained from from T(r) T(r) data data borrowed from [23,24]. [23,24]. One One can can see see that that the the obtained borrowed from resistivity drops drops at at first first for for ll!s 1 ps but then falls falls rather rather slowly. slowly. This This decrease decrease resistivity but then ceases closer closer to to the the pulse pulse tail, tail, and and the the resistance resistance begins begins to to rise rise gradually. gradually. ceases The dramatic dramatic initial initial drop drop in in R Ij is is due due to the primary primary heating heating of of aa very very thin thin The to the initial channel channel by high density density current. current.?t As T increases increases to about 20000 20000K, initial by high to about K, the conductivity conductivity (J'a rises rises but remains nearly constant with with further further temperature temperature the but remains nearly constant The gas gas is is assumed assumed to to be in thermodynamic equilibrium at at every every moment moment of of time. time. This This assumpassumpt The be in thermodynamic equilibrium 8 tion justified by _10- 7 s) between lO-*-lO-'s) between electrons electrons and and ions, ions, resulting resulting tion is is justified by aa fast fast energy energy exchange exchange (for (for 10in aa small small difference difference between the gas gas and and electron electron temperatures. temperatures. The The ionization ionization is is of of thermal thermal in between the nature: aa Maxwellian Maxwellian distribution distribution is is established established in in the the electron electron gas, gas, and and the the amount amount of of ionizing ionizing nature: electrons is is defined defined directly directly by electron temperature, rather than than by by the field. The The electron electron by the the electron temperature, rather the field. electrons of the the gas. gas. temperature, in in turn, turn, is is determined determined by the Joule Joule heat heat release and energy energy balance balance of by the release and temperature, Equilibrium ionization ionization is is also also established established rapidly rapidly (for (for details, details, see see [Ill). [ll]). Equilibrium
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Physical processes in a lightning discharge
Table 4.1. The evolution of a return stroke channel
t (Ils)
Tmax (kK) TetT(kK) O"eff(n/Cm)-l p(atm) retT(cm) Current impulse 20 kA with tf
0.074 0.074 3.7 3.7 11 11 39 39 91 91
88 28 28 24 24 17 17 14 14
55 18 18 15 15 12 12 10 10
22 120 120 170 170 110 110 60 60
WI
(J/cm) R I (n/m)
= 5 Ils, tp = 50 IlS, Q = 1.05C >24 >24 5.5 5.5 44 22 11
0.1 0.1 0.3 0.3 0.7 0.7 1.2 1.2 1.4 1.4
1600 1600 3.1 3.1 0.38 0.38 0.20 0.20 0.27 0.27
88 26 26 38 38 46 46
Current impulse impulse 100 100 kA kA with p,ttpp = 100 100 Ils, ps, Q Q = lOC 1OC Current with tf = 55 Ils, 55 20 20 50 50 100 100 200 200 300 300
35 35 22 22 18 18 14 14 12 12 10 10
25 25 20 20 15 15 12 12 11 11 10 10
180 180 140 140 90 90 70 70 40 40 20 20
16 16 5.7 5.7 2.6 2.6 1.8 1.8 1.0 1.0 1.0 1.0
0.8 0.8 22 33 44 55 55
0.28 0.28 0.057 0.057 0.039 0.039 0.028 0.028 0.032 0.032 0.064 0.064
-
-
",ISO -150 -
Note. T T,,,max is is the the temperature temperature along along the the channel channel axis, axis, T Teff is the the average average temperature temperature in in the the Note. eff is conductive channel, channel, O'eff oeffis is an an average average channel channel conductivity, conductivity, p is is channel channel pressure, reffis is the effecconductive pressure, reff the effective radius of total energy tive radius of the the conductive conductive channel, channel, W is is the the total energy released released (no (no data data for for the the second second variant; the the given given values values was was estimated estimated as as W W >::; x ii~axRl ~ a x Rt1p t),p )and and r Q Q is is the the charge charge transported transported variant; during the the current current impulse. impulse. during
-
-
3 2 rise. with ions / , but rise. In In aa strongly strongly ionized ionized plasma, plasma, with ions of of constant constant charge charge IJ' o '" T T3I2, but 2 doubly with increasing doubly charged charged ions ions appear appear with increasing T. T . Since Since IJ' U '" ZiZC2,, where where Zj Zi is is the the ion charge charge multiplicity, multiplicity, the the two two effects effects compensate compensate each each other. other. The The resistance resistance ion of aa highly highly heated heated channel channel decreases decreases with time due due to to its its expansion expansion only. only. of with time Some pressure at Some time time later, later, however, however, the the pressure at the the channel channel centre centre drops drops to to atmospheric pressure, and atmospheric pressure, and the the expansion expansion ceases. ceases. The The conductive conductive channel channel cross reduced gradually because of the gas by cross section section is is reduced gradually because of the gas cooling cooling caused caused by thermal radiation. radiation. The resistance begins begins to because of thermal The channel channel resistance to rise rise slowly slowly because of decreasing reff and T Teff. decreasing reff and . eff The expansion expansion time time of of the the channel channel becomes longer and and its its minimal minimal The becomes longer resistance decreases for for the the stronger stronger current current impulses. impulses. Physically, Physically, the the linear linear resistance decreases W, resistance is affected affected by the energy energy released released per unit channel channel length, length, W, by the per unit resistance is rather than by the unambiguously by rather than by the current. current. This This value value is is not not described described unambiguously by the what is the current current amplitude; amplitude; what is more more important important is is the the amount amount of of the the transported charge charge Q: Q: WI W1 '" P i 2RRIt'" l t QiR QiRlI '" QE and and the the field field does does not not transported vary much. The but vary much. The calculations, calculations, however, however, deal deal with with the the current current impulse impulse but W1. Semi-quantitatively, Semi-quantitatively, the the time time dependence dependence of of resistance resistance can can be not with WI' not with be powerful cylindrical cylindrical understood using the for the the shock shock wave wave of of aa powerful understood using the relations relations for explosion. be strong explosion. The The explosion explosion can can be be considered considered to to be strong as as long long as as energy energy is is released released in in aa thin thin channel channel and and the the pressure pressure of of the the explosion explosion wave wave does does not to the pressure. In not fall fall close close to the atmospheric atmospheric pressure. In this this case, case, the the flow flow is is self-similar. self-similar.
- - -
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Return stroke
185
The shock shock front front radius radius rs r s and and pressure pressure pp in in the the affected affected region region depend, depend, within within The the accuracy accuracy of of numerical numerical factors, factors, on on W WI1 and and t as as r, r s rv ((WJ!PO)I/4 and W 1 / p o ) 1t/l/2 4 t and 1 /2 the pp rv Wl/rz Wlpo)'/2t-', where po is cold air density. The channel Wl/r; rv ((W PO)I/2 -l, where Po is cold air density. The channel t I expansion expansion is is completed completed when when the the pressure pressure drops drops to to aa certain certain value value close close to to atmospheric pressure. pressure. This This sets sets the the limit limit to to the the validity validity of of formulae formulae for for selfselfatmospheric similar motion. motion. This This means means that that they they are are still still applicable, applicable, and and the the duration duration of of similar the resistance resistance reduction reduction then then is is t rv ((W It can be be shown shown [12] [12] that that for for the W IlPO)I/2. p o ) 1 / It 2 . can aa self-similar self-similar cylindrical cylindrical explosion explosion in in the the central central region region with with the the pressure pressure equalized equalized along along the the radius radius (figure (figure 4.16), 4.16), the the internal internal specific specific energy energy depends depends and W1 WI as as E€ rv ((W?/2 where y 'Y is is the the adiabatic adiabatic exponent. exponent. on r, t and t2- rr -2)1/(r-I), on Wy/2t2-Tr-2)1/(T'I, where A point point with with fixed fixed temperature, temperature, e.g., e.g., T T:::::: K, can can be be regarded regarded as as the the M 10000 10000 K, A conductive conductive channel channel boundary, boundary, since since the the plasma plasma conductivity conductivity below below this this point ( T ) varies point is is relatively relatively low. low. The The radius radius of of aa point point with with fixed fixed T and and E€(T) varies as r rv W:'4t1-y/2, W?/4 t l- r /2, reaching reaching aa value value proportional proportional to to r,,, rmax rv W;I2 W11/ 2 with time time as with I 2 by the the moment moment the the channel channel stops stops expanding, expanding, t rv W:12. WI / . Therefore, Therefore, the the linear linear by -2 r;;;;x rv Wand channel resistance drops to aa value value proportional proportional to to Rl,, R lmm rv rmax W-'I ,, and channel resistance drops to I 2 this occurs occurs for the time time t rv W / . These relationships are qualitatively this for the W'/'. These relationships are qualitatively consistent with the the calculations calculations for for the the two two variants variants described described in in table table 4.1. 4.1. consistent with N
N
N
N
N
N
N
N
N
N
N
4.4.4 Return stroke as a channel transformation wave 4.4.4 channel transformation The first first substantiated attempt to to make make aa numerical numerical simulation of the the lightlightThe substantiated attempt simulation of ning return return stroke with allowance allowance for for the the resistance resistance variation variation was was undertaken undertaken ning stroke with as far back back as as the the 1970s The most most important important features features of of the the process, process, as far 1970s [25,26]. [25,26]. The which are due to to an abrupt conductivity conductivity rise rise at the site of intensive intensive Joule Joule heat heat which are due an abrupt at the site of release, release, became became evident evident at at once. once. The The simulation simulation showed showed that that aa weak weak initial initial perturbation perturbation (precursor) (precursor) propagating propagating up up along along the the channel channel at at an an electroelectromagnetic magnetic signal signal velocity velocity close close to to light light velocity velocity does does not not change change the the plasma plasma state state and and cannot cannot be be treated treated as as the the return return stroke stroke wave wave front front visible visible in in streak streak photographs. photographs. The The main main wave wave of of current current and and decreasing decreasing potential potential travels travels several several times times slower; slower; its its velocity velocity is is defined defined by by the the transformation transformation of of the the low low conductivity conductivity leader leader to to the the low low resistivity resistivity stroke stroke channel. channel. This This conclusion conclusion was was formulated formulated explicitly explicitly in in [25-28]; [25-281; it it reflects reflects the the nature nature of of the the lightning lightning return return stroke. stroke. Turning Turning to to numerical numerical simulation simulation today, today, we we should should like like to to formulate formulate this this problem problem in in aa simple simple and and clear clear physical physical language language and and to to try try to to outline outline problems problems to to be be solved solved within within this this model. model. An An obviously obviously essential essential aspect aspect of of the the theory theory still still is is the the resistivity resistivity dynamics dynamics of of the the lightning lightning channel. channel. An An exhaustive exhaustive formulation formulation of of this this problem problem would would involve involve aa simultaneous simultaneous solution solution of of equations equations describing describing the the propagation propagation of of aa current-voltage current-voltage wave wave and and the the channel channel dynamics dynamics at at every every point point along along its its length, length, affected affected by by the the ever ever varying varying energy energy release. release. So So we we shall shall restrict restrict the the discussion discussion to to aa simple simple model, model, having having accepted accepted aa probable probable law law for for the the linear linear conductivity conductivity rise, G == Ri R:',l , and and focusing focusing on on the the qualitative qualitative results results of of the the solution. solution. rise, G
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Physical processes in a lightning discharge
Let Let us us describe describe G G in in the the simplest simplest way way reflecting reflecting the the main main qualitative qualitative be assumed features features of of the the channel channel evolution. evolution. It It will will be assumed that that the the linear linear conducconductivity resistance tivity increases increases with with current. current. This This partly partly reflects reflects the the fact fact that that resistance decreases with with increasing increasing charge charge through through aa particular particular channel channel site. site. But But the the decreases resistance is is stabilized stabilized with time, even even though though the current continues continues to resistance with time, the current to flow. principle, the flow. In In principle, the stable stable state state of of aa lightning lightning channel channel hardly hardly differs differs in aa hhigh g h current current arc arc only only slightly slightly varies varies from that that of of an an arc. arc. The The field field E in from with current; in in other other words, words, the linear conductivity conductivity of of the the channel channel is is with current; the linear i /EE rv i. Assume equal to to the the field field E EL in the lightning leader, G,,st == i/ G Assume E to to be be equal in the lightning leader, L then the the conductivity conductivity is is whose current not low whose current is is not low on on the the arc arc scale; scale; then i / E LL. . In In aa mature mature gas-dynamic gas-dynamic process process when when the the shock shock wave is still still G,,st == i/ G wave is strong, the will drop drop with with time. As the the shock shock wave wave becomes strong, the resistance resistance will time. As becomes R11 and and the the increase increase in in G G become slower. These These weaker, the the decrease decrease in in R weaker, become slower. tendencies are described described by the relaxation-type relaxation-type formula formula tendencies are by the N
dG i/EL dG EL -- G(t) Gst(i) - G(t) G(t) - - i/ - Gst(i) - G(t) dt T T dt Tgg Tgg
(4.38) (4.38)
is the the characteristic characteristic time time oflinear of linear conductivity conductivity variation variation (relaxation (relaxation where TTg where g is const, T Tg = const, and G(0) = 0, we have have time). In In aa simple simple case case with with i == const, time). = const, and G(O) = 0, we g -t/Tg )]. G GSt[1-- exp( exp(-t/T,)]. G == Gsdl U ( x ,0) == Vi U, and and Equations (4.24) (4.24) are are solved solved with with the the initial initial conditions conditions V(x,O) Equations i ( x ,0) == 0, 0, R RI1(x, (x30) 0) == R RI,-; the reactive parameters are taken to be constant: i(x,O) ; the reactive parameters are taken to be constant: 1L C1 = 10 10 pF pF/m and LL1 = 211Hjm. 2 pH,”. The The channel channel does does not not close close on on the the earth earth C 1= jm and 1 = so through through the the time-decreasing time-decreasing resistance resistance of of the the in an an instant instant but but does does so in commutator (similarly (similarly to to the the real real lightning lightning length length decreasing decreasing through through the the commutator R,,, = R(0) exp(-cut), R(0) 10 R streamer zone). zone). The The accepted accepted values values of of R streamer = R(O) exp( -at), R(O) = 10 fl eom and acu == IIlS-1 1 ps-’ provide provide aa typical typical duration duration of of the the negative negative current current impulse impulse and FZ 511S. 5 p . The The boundary boundary condition condition at at the the grounded grounded end end of of the the line line front tf ~ front no doubt: doubt: V(O, U ( 0 .t)t ) == i(O, i(0,t)R t)R,,,. The problem of the far end up in the raises no raises The problem of the far end up in the . eom H , is is much much more more complex. complex. Conventionally, Conventionally, it it is is considered considered as as clouds, xx == H, clouds, i ( H ,t)t) == O. 0. In In reality, reality, the the situation situation is is far far from from being being being open, open, assuming assuming i(H, being self-evident. When When the the line line gets gets discharged discharged and and its its end end in in the the clouds clouds takes takes self-evident. zero potential, potential, aa high high electric electric field field must must arise arise near near itit due due to to the the voltage voltage zero AU = = -Vo(H). - U o ( H ) . This This gives gives impetus impetus to to very very intensive intensive ionization ionization difference 6.V difference processes, processes, probably probably involving involving high high current. current. This This situation situation will will be be partly partly to have have no no current. current. discussed below. below. Now, Now, we we shall shall assume assume the the upper upper end end to discussed The results results to to be be presented presented were were obtained obtained for for aa vertical vertical unbranched unbranched channel channel The H == 44 km. km. This This is is the the height height the the ascending ascending leader leader tip tip with the the total total length length H with reaches when when the the descending descending leader, leader, which which has has started started from from the the point point closest closest reaches to the the earth earth in in the the bottom bottom negative negative sphere sphere with with the the centre centre 33 km km high, high, contacts contacts to the earth. earth. It It is is normal normal practice practice to to use use the the following following averaged averaged parameters parameters the ELL == 10 10V/cm, = 100 lOOA, and of the the leader leader prior prior to to the the contact: contact: E of Vjcm, iiL A, and L = RIL = Ed EL/iL 10fljm. O/m. For For aa realistic realistic description description of of the the resistivity resistivity dynamics dynamics R lL = i L == 10 (section 4.4.3), 4.4.3), the the relaxation relaxation time time should should be be taken taken to to be be TTg = 40 40 llS, ps, when when the the (section g =
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187 187
current at at this this channel channel site site rises, rises, and and T Tg = 200IlS, 2 0 0 p , when when the current current the current g = decreases. The The model calculation reproduces the distributions distributions of of current current decreases. model calculation reproduces the i(x, potential U(x, i ( x , t) t ) and and potential U ( x ,t) t ) along along the the channel; channel; the the linear linear charge charge is is r(x, ~ ( xt) t ), == C1[ U ( x ,t) t ) -- Uo(x)]. U o ( x ) ]Generally, Generally, . the external external field field potential potential Uo(x) Uo(x) can also also CdU(x, the can vary vary in in time, time, because because we we should should not not discard discard aa possible possible partial partial neutralization of of the the charge charge in in one one of of the the regions regions of of the the cloud cloud dipole. dipole. The The neutralization latter point will not not be discussed for for the the time time being. latter point will be discussed being. The precursor traThe calculations calculations are are presented presented in in figures figures 4.17-4.21. 4.17-4.2 1. The The precursor travelling velocity 0.64e this is the velling with with velocity 0 . 6 4 ~is is damped damped so so fast fast that that this is not not shown shown in in the plots after noticeable break break from the chanplots after aa noticeable from the the principal principal wave wave re-charging re-charging the channel nel (we (we shall shall term term it it aa discharge discharge wave wave for for simplicity). simplicity). The The wave wave in in figure figure 4.17 4.17 travels along the the channel channel with velocity VU,r ~ x OAe, 0.4c, i.e., i.e., 1.6 1.6 times times slower slower than than travels along with velocity the precursor. precursor. This the This velocity velocity somewhat somewhat decreases decreases as as the the wave wave moves moves up. up. Its Its variation be conveniently well-defined variation can can be conveniently followed followed from from the the change change in in the the well-defined maximum power of the Joule R 1 (figure i2Rl (figure 4.17 4.17 (centre)). (centre)). The The maximum linear linear power of the Joule losses losses P wave wave front front power power rises rises abruptly abruptly along along aa 100-200 100-200 m m length, length, then then it it decreases decreases towards the with intensive towards the earth, earth, making making the the channel channel tip tip with intensive energy energy release release stand stand out region which out clearly. clearly. It It seems seems that that it it is is this this region which is is clearly clearly discernible discernible in in streak streak photographs. The power proportional photographs. The linear linear power proportional to to the the squared squared current current drops drops remarkably on way up up the maximum becomes remarkably on the the way the cloud, cloud, and and the the maximum becomes smeared. smeared. This with observations This is is also also consistent consistent with observations of of radiation radiation intensity intensity [14,29]. [14,29]. A A photometric study photometric study has has shown shown that that the the radiation radiation from from the the wave wave front front is is attenuated the front boundary. The attenuated and and the front loses loses its its clear clear boundary. The current current wave wave is is not not attenuated attenuated so so rapidly rapidly (figure (figure 4.17 4.17 (top)). (top)). For For the the time time of of its its earth-cloud earth-cloud the channel travel travel lasting lasting for for 341ls, 3 4 p , the the current current at at the channel base base drops drops from from the the maximum of to 24 that maximum of 35 kA kA to 24 kA. kA. This This agrees agrees with with observations observations indicating indicating that an to an average average current current impulse impulse duration duration in in aa negative negative lightning lightning is is close close to 751ls 75ps on on the the 0.5 0.5 level. level. The The wave wave front front deformation deformation depends depends on on the the initial initial potential Uii delivered delivered by by the the leader leader to to the the earth. earth. The The higher the value Uil, potential U higher the value II U i I, the the discharge the higher higher the discharge current. current. The The rate rate of of resistivity resistivity decrease decrease at at the the wave front wave front grows grows respectively, respectively, so so the the front front steepness steepness increases. increases. This This is is evident evident from from aa comparison comparison of of figures figures 4.18 4.18 (top) (top) and and 4.18 4.18 (bottom). (bottom). At At I1 Uil Uii = = 50 50 MV, MV, the the channel practically without the current current wave wave goes goes along along the channel practically without elongating elongating the the lUil = = 10 10MV it has has aa lower lower velocity and aa smooth smooth front. front. front? while at lUiI frontt while at MV it velocity and Unfortunately, there there have have been been no Unfortunately, no registrations registrations of of current current and and streak streak photographs return stroke photographs of of the the return stroke taken taken simultaneously. simultaneously. A A comparison comparison of of the provide aa good the relationships relationships between between current current and and wave wave velocity velocity could could provide good test test for for the the return return stroke stroke theory. theory. the current current impulse impulse amplitude amplitude rises, rises, the the linear linear resistance resistance falls falls more more As the quickly wave is quickly and and to to aa lower lower level, level, so so the the wave is damped damped more more slowly slowly during during its its The motion motion of of aa high high current current wave with attenuation attenuation but but without without noticeable noticeable distortions distortions t The wave with facilitates the electromagnetic field field calculation calculation necessary in many applied problems of lightning lightning facilitates the electromagnetic necessary in many applied problems of protection and methods. protection and in in substantiation substantiation of of remote remote current current registration registration methods.
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188 188
Physical processes in a lightning discharge 35 30 25
:s
20
cf
~~ 15
U
10 5 2 X, Ian
3
4
2
3
4
30 25
:> 20 ~ oJ
E 15
'0
:>
10 5
1
0
X, X,2Ian km
8p t == 81ls
600
~
18
:E 400 ..... ~ 0
~
200
o+-~-+-~-..........~-..,......->-,---..l,
o0
Copyright © 2000 IOP Publishing Ltd.
1
2
x,
X, Ian km
3
4
Return stroke
189 189
1.0
67 kA kA II,,,= max = 67 0.8
3 >-< i:l
0.6
E
U
\ ........
.3
0.4 0.4
-
0.2 0.2
-
0.0 0.0
.
I
00
,
.
.
, 2
1
.
, 3
, 4
x, kIn lan
X,
1.0 t = 8).15
I1-max = 8,15 kA
0.8 0.8 i:l
~-..e 0.6 0.6 1 ........ .I
0.4 0.4
0.2 0.2 0.0 0.0 00
2
1 X,
km
3
4
Deformation of of the the current current wave wave front front at at leader leader potential potential (top) (top) Figure 4.18. Deformation -50MV and (bottom) (bottom) -10 -10 MY; MV; for for the the other other parameters, parameters, see see figure figure 4.17. 4.17. Uii == -50 MY and
propagation along along the the channel. channel. There There is is no no damping damping at at aa very very high high current current and and propagation the impulse front front becomes steeper, as as was discussed in in section section 4.4.2 4.4.2 (figure (figure the impulse becomes steeper, was discussed 4.18). is also also observed observed in in the the current current amplitude amplitude dependence dependence 4.18). Non-linearity Non-linearity is U,i at at the the earth. earth. If If the the commutator commutator were were perfect perfect on the the initial initial potential on potential U (R,,, = 0), 0), the the current current at at the the earth earth at at the the moment moment of of contact contact would instantly (R would instantly eom = Figure 4.17. (Opposite) (Opposite) Numerical Numerical simulation simulation of of the the return return stroke stroke excited excited by by aa Figure -30 MY: MV: (top) (top) current current and and (centre) (centre) voltage disdescending leader leader with with potential potential -30 descending voltage dis10 n/m. tributions; tributions; (bottom) (bottom) the the power power of of Joule Joule losses. losses. The The initial initial leader leader resistance, resistance, 10 rl/m. V/cm. Steady Steady state state field field in in the the channel channel behind behind the the wave, wave, 10 lOY/em.
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190 190
Physical processes in a lightning discharge
r 0.6
200 1
V "...... '"
< ..I<
,, , ,, , ,
i
~ u
0.5
r ",
150
100
...
... ' ... ...
0.4
1
U
;>'"'
,,
0.3
,
50
-
0.2 0
0
20
40
60 80 Voltage, Voltage, MY MV
100
120
0.1
Figure 4.19. Calculated the current Calculated dependencies dependencies of of the current amplitude amplitude and and average average wave wave velocity in in the stroke on on the the leader leader potential Vi. velocity the return return stroke potential Uj.
take the maximum maximum value value,1ZM = = II Vil/Z, Ui l/Z,independent independent of of the the actual actual channel channel resistake the resisReam tance, and Ui. With With the the finite finite time time of of R ,,, decrease decrease to to zero, zero, the the tance, and would would be be 1ZM M ", Vj. current the channel current wave wave is is able able to to cover cover some some distance distance along along the channel and and to to include include in in the circuit the ohmic portion. For the circuit the ohmic resistance resistance of of this this channel channel portion. For this this reason, reason, the the current Z and Ui/Z and rises rises somewhat somewhat faster faster current amplitude amplitude appears appears to to be be lower lower than than Vi! than potential V (figure 4.19). It is important that the lightning current than potential Uij (figure 4.19). It is important that the lightning current amplitude be appreciably than its is found found to to be appreciably smaller smaller than its theoretical theoretical limit limit amplitude 1ZM M is Ui/Z: e.g., e.g., ,1ZM ;::::: M 0,6Vi!Z 0: 6Ui/Z at at Vi Vi == 30MV. 30 MV. This This is is another another source source of of errors errors Vi!Z: Vi == Z Z ,, M , in in particular, in the the calculation calculation in evaluations evaluations using the equality equality Vi in using the ZI particular, in of of cloud cloud potential potential from from lightning lightning current current data. data. N
4.4.5 Arising problems problems and approaches approaches to their solution
The the earth boundary condition The current current at at the earth is is independent independent of of the the boundary condition at at the the upper channel back to the earth upper channel end, end, until until the the reflected reflected wave wave comes comes back to the earth with with the processes occurring the information information about about the the processes occurring there. there. Before Before that that moment, moment, virtue of the the positive positive charge charge is is pumped pumped into into the the line line from from the the earth. earth. In In virtue of the the zero current current at at the the upper upper end end -- the current wave is boundary condition condition -- zero boundary the current wave is reflected reversal. As aa result, result, the behind the the current current behind the reflected there there with with the the sign sign reversal. reflected wave, between its reflected wave, i.e., i.e., between its front front and and the the channel channel end end (figure (figure 4.20), 4.20), decreases (it (it would would drop drop to zero in in the the absence absence of of damping). damping). The The incoming incoming decreases to zero positive charge the line making it positive charge now now re-charges re-charges the line making it positive positive (an (an ideal ideal line line The reflected wave moves moves faster faster and and is is would be re-charged to to --Ui). be re-charged VJ The reflected wave would damped because the damped more more slowly, slowly, because the linear linear resistance resistance in in most most of of the the channel channel of the the has dropped by an order order of of magnitude or more more due due to to the the action action of has dropped by an magnitude or forward current current wave. wave. forward
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Return stroke
191 191
251=::;;;;;:::::::::::: 20
10
5
0
2
1
X, x, kIn km
30
3
44
t = 34 llS
25
u. = - 30 MV 1
>
20
~ Ili 15 15-
&
-~8a 1010-1 50-
-5
-
Current and and potential distributions during during the the propagation of waves waves Figure 4.20. Current potential distributions propagation of reflected by the the cloud cloud end end of of the channel. reflected by the channel.
When the the reflected wave reaches reaches the the earth, earth, delivering delivering aa positive When reflected wave positive potential potential to it, it, aa new new discharge discharge cycle cycle begins. begins. The The newly newly acquired acquired positive positive charge charge flowflowto ing into into the earth is is equivalent equivalent to to the the extracted extracted negative charge. The The current current ing the earth negative charge. sign 4.21). In In the the absence absence of of sign at at the the grounded grounded end end is is reversed reversed (figure (figure 4.21). dissipation in in aa distributed distributed system system such such as as aa long long line, line, undamped undamped oscillations oscillations dissipation 4H/w, arise similar similar to to those those in in an an LC circuit. circuit. with aa period period T == 4H with /v r would would arise of the the kind kind is is observed observed in in lightning lightning registrations, is there there aa Nothing registrations, nor nor is Nothing of single change change in in the the current current direction. direction. This This means means that that the the discharge discharge wave wave single of the the line line or or the the reflected wave is either either not not reflected reflected by the upper upper end end of is by the reflected wave so damped damped on on the the way back to to the the earth earth that it is is unable to manifest manifest becomes becomes so way back that it unable to
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Physical processes in a lightning discharge
Imax = 30 kA
1.0 0.8 ~
~E
.-
0.6 0.4 0.2
Time, llS
0.0
20
40
0
-0.2
-0.4 -0.4
j
U
Figure 4.21. Calculated through the Calculated current current impulse impulse through the grounded grounded channel channel end. end.
itself against against the the background of other other variations variations in in the the current. current. By By changing changing itself background of the Tg or the quasi-stationary channel field EL, one can reduce , the parameter parameter T or the quasi-stationary channel field E one can reduce g L or but it or even even cancel cancel part part of of the the current current impulse impulse of of opposite opposite sign, sign, but it is is impossible portrait likelihood between the impossible to to attain attain aa portrait likelihood between the calculated calculated and and observable currents. currents. The The suppression suppression of of the the reflected wave by by raising raising the the observable reflected wave (x, t) t ) inevitably inevitably results in instantaneous values values of of the the channel channel resistance instantaneous resistance R Ij (x, results in an reduction of an excessive excessive reduction of the the impulse impulse duration duration at at the the grounded grounded end end of of the line. no way way of the line. There There seems seems to to be be no of avoiding avoiding this this even even by by changing changing the the resistivity reduction resistivity reduction law. law. The be suitable rectifying this The first first thing thing that that seems seems to to be suitable for for rectifying this situation situation is is to to question the the boundary condition at at the the upper upper end. end. This This idea idea appears appears reasonreasonquestion boundary condition able because it with lightning able because it generally generally agrees agrees with lightning current current registrations registrations at at the the earth path time jV r while model solution FZ 2H 2H/w, while the the model solution for for earth for for the the double double path time t i:::j i(0,t) t ) remains independent of of the the boundary boundary condition. condition. It It is is obvious obvious that that i(O, remains independent the open the open circuit circuit condition condition is is an an excessively excessively rough rough idealization. idealization. It It was was mentioned in the negative 4.3.3 that that if if the negative cloud cloud bottom bottom is is filled filled with with aa mentioned in section section 4.3.3 large number of branches which large number of branches which stem stem from from the the ascending ascending leader, leader, this this metallic sphere. sphere. Assuming Assuming such such aa 'metallization' 'metallization' region becomes becomes similar similar to to aa metallic region of would be be more reasonable to to consider upper end to be be of the the cloud, cloud, it it would more reasonable consider the the upper end to connected by C,e == 47rc 4 7 rOR~e~, Rdefined defined ~, by the the cloud cloud connected to to aa lumped lumped capacitance capacitance C charge radius R,, R e , instead being open. charge radius instead of of being open. The The boundary boundary condition condition at at x == H would the form form i(H, i ( H ,t) t) = =C C,e dU dU/dt. When the the current current wave reaches would have have the jdt. When wave reaches the line line end, end, the delivered current current also also discharges discharges the negative 'metallized' 'metallized' the the delivered the negative cloud region. This, however, does prevent the the appearance cloud region. This, however, does not not prevent appearance of of the the reflected reflected wave. At the first moment of time, time, the the capacitance capacitance still still preserves its charge charge and and wave. At the first moment of preserves its is the reflection reflection condition, to aa short-circuited is similar, similar, in in accordance accordance with with the condition, to short-circuited
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end of of the the line, line, which generates aa reflected reflected current current wave of the the same same sign sign and and end which generates wave of amplitude as as the incident one. one. As capacitance becomes discharged, the the amplitude the incident As the the capacitance becomes discharged, reflected wave wave amplitude amplitude decreases decreases and and then the sign sign is is reversed. The comcomthen the reversed. The reflected pletely pletely discharged discharged capacitance, capacitance, incapable incapable of of supporting supporting current, current, eventually eventually becomes to an numerical becomes equivalent equivalent to an open open line line end. end. It It is is clear clear even even without without aa numerical calculation how much the current changes changes at at the the earth earth after after the arrival of of aa how much the current the arrival calculation reflected of such such complexity. complexity. It It should should be be emphasized emphasized again again that that reflected wave wave of nothing been observed real lightning. nothing of of the the kind kind has has ever ever been observed in in real lightning. One try to by complicating boundary One can can also also try to rectify rectify the the situation situation by complicating the the boundary condition condition with with the the allowance allowance for for the the final final resistivity resistivity of of the the 'metallized' ‘metallized’ cloud region. The streamer streamer and and leader leader branches filling the the cloud cloud possess possess aa cloud region. The branches filling resistance moment of branch can resistance at at the the moment of their their generation. generation. A leader leader branch can hardly hardly be heated be heated as as much much as as aa single single descending descending leader. leader. The The resistance resistance of of the the 'metallized' R cb is high during ‘metallized’ cloud, cloud, R,,, is quite quite likely likely to to be be high during the the whole whole return return stroke this is be formally stroke stage. stage. If If this is so, so, the the boundary boundary condition condition should should be formally reprerepresented /dt -- R,, R cl di/dt. the boundary boundary sented as as i(H, i ( H ,t)t) = =C C,c dU dU/dt dildt. Strictly, Strictly, it it is is not not only only the condition that changes the case metallization, but but condition that changes in in this this case, case, like like in in the case of of ideal ideal metallization, also the set potential U no more more be also the set of of equations. equations. The The cloud cloud potential U.a can can no be considered considered as being constant re-written (4.24) should should be be re-written as being constant in in time. time. The The second second equation equation of of (4.24) as as
oi
aT
ax at'
T
= C j [U(x, t) - Ua(x, t)]
having taken into the change having taken into account account the change in in U U,a due due to to the the change change in in the the cloud cloud charge Qc' Q,. Then Then the the function function Ua(Qc) Uo(Q,) must must allow allow for for the the delay delay because of charge because of the problem becomes the finite finite rate rate of of the the electromagnetic electromagnetic field field propagation. propagation. The The problem becomes extremely registrations do extremely complicated. complicated. Although Although radar radar registrations do indicate indicate the the developdevelopment wide network branches in there has no investigation ment of of aa wide network of of branches in clouds, clouds, there has been been no investigation of reason for no doubt, of cloud cloud 'metallization'. ‘metallization’. The The reason for this, this, no doubt, is is the the lack lack of of initial initial data. data. One not discard two other unaccounted for by the One should should not discard two other factors factors unaccounted for by the numernumerical model. First, practically never never be be single. ical model. First, aa leader leader channel channel can can practically single. Owing Owing to to the branches of heights, the numerous numerous branches of different different lengths lengths developing developing at at different different heights, numerous reflected waves will arrive numerous reflected waves will arrive at at the the earth earth at at different different moments moments of of time, time, creating creating aa sort sort of of 'white ‘white noise' noise’ with with aa nearly nearly zero zero total total signal. signal. This This will the current bending which will deprive deprive the current of of its its characteristic characteristic bending which is is usually usually created created by aa single reflected wave wave at by single reflected at the the earth. earth. Second, Second, constant constant linear linear capacitance capacitance only of aa lightning lightning leader. leader. We We only approximately approximately describes describes the the real real re-charging re-charging of have mentioned above is have mentioned above that that the the cover cover charge charge around around aa leader leader channel channel is changed changed by by numerous numerous streamers streamers starting starting from from it. it. Their Their velocity velocity decreases decreases rapidly when rapidly when the the streamer streamer tips tips go go away away from from the the channel channel surface surface with with its its high radial field. the voltage the wave high radial field. So, So, when when the voltage at at the wave front front changes, changes, the the charge the channel charge near near the channel changes changes almost almost immediately, immediately, while while its its change change at at the external boundary occurs the external cover cover boundary occurs with with aa delay. delay. In In other other words, words, aa lightning lightning
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Physical processes in a lightning discharge
discharge proceed for discharge can can proceed for aa fairly fairly long long time. time. The The quasi-stationary quasi-stationary current current from periphery, having from the the discharge discharge of of the the cover cover periphery, having the the same same direction direction as as the the current current in in the the forward forward wave, wave, can can compensate compensate for for the the reverse reverse current current induced induced by wave reflected reflected by by the the earth. earth. by the the wave To To conclude, conclude, the the model model of of aa single single long long line line with with varying varying linear linear resisresistance return stroke but cannot tance allows allows elucidation elucidation of of many many aspects aspects of of the the return stroke but cannot claim reliable quantitative claim to to give give reliable quantitative description description of of all all characteristics characteristics of of this this phenomenon. The more simplified phenomenon. The much much more simplified models models of of return return stroke stroke are are usually usually used review of of used calculating calculating electromagnetic electromagnetic field field for for technical technical application. application. A review this model model is this is given given [30]. [30]. 4.4.6 4.4.6 The return stroke of a positive lightning lightning
Two be distinguished Two kinds kinds of of current current impulse impulse can can be distinguished in in oscillograms oscillograms taken taken at at the positive leader. the earth earth after after the the arrival arrival of of aa positive leader. Common Common impulses impulses are are similar similar to those those registered registered in in negative negative lightnings, lightnings, although although they they have slightly longer longer to have aa slightly duration be naturally and less less steep steep fronts. fronts. Such Such impulses impulses can can be naturally interpreted interpreted duration t pp and as as return return stroke stroke currents currents corresponding corresponding to to the the wave wave discharge discharge of of the the leader leader channel, channel, as as described described above. above. Sometimes, Sometimes, however, however, quite quite different different impulses impulses are registered registered with an order order longer longer duration duration and and an an amplitude amplitude as as large large as as are with an 200 kA. kA. A A closer closer examination examination shows shows that that impulses impulses with with an an 'anomalous' ‘anomalous’ 200 duration cannot cannot be interpreted as as resulting resulting from from aa grounded grounded leader leader disdisduration be interpreted charge. process, and charge. They They appear appear to to result result from from another another process, and we we shall shall offer offer some their nature nature in some suggestions suggestions concerning concerning their in section section 4.5. 4.5. Here, Here, only only common impulses impulses will will be discussed. common be discussed. It was was shown shown in in section section 4.4.2 4.4.2 that that the the stroke stroke current current front front is is unrelated to It unrelated to the wave wave discharge discharge process process in in the the channel channel but, but, rather, rather, is is associated associated with an the with an imperfect commutator commutator closing closing the the channel channel on on the the earth. earth. The The leader leader streamer streamer imperfect zone acting acting as as aa commutator commutator possesses possesses aa finite finite resistance resistance and and is is reduced reduced zone during aa finite finite period of time. time. The The front front steepness steepness is is determined determined by during period of by the the rate rate of resistance resistance reduction reduction in in this transient link link between the channel channel and and the the of this transient between the earth. But But the the streamer streamer zone zone length length of of aa positive leader at at the the same same voltage voltage earth. positive leader is about about twice twice as as large large as as that of aa negative leader and and takes takes more more time time to to is that of negative leader be that this this is why, with be reduced. reduced. It It is is quite quite likely likely that is the the main main reason reason why, with the the 50% probability, the current current front front duration duration in in positive positive lightnings, lightnings, 50% probability, the tr == 22lls, 22 ps, is is four four times times longer longer than than in in negative negative leaders leaders [1]. [l]. Approximately Approximately tf maximum pulse pulse steepness. the the same same proportion proportion is is characteristic characteristic of of the the maximum steepness. is primarily primarily determined determined by the stroke stroke The duration duration of of the the pulse pulse itself, itself, tpp,, is The by the 4.3 that that it it is is only only positive positive leaders leaders channel length. length. It It was was shown shown in in section section 4.3 channel which have real starting positive region starting from from the the top top positive region of of aa storm storm cloud cloud which have aa real chance to the earth. earth. This This region region is is twice twice as as high high as as the the negative negative chance to reach reach the cloud bottom. Hence, the channel length length of of aa positive positive descending descending leader leader is, is, cloud bottom. Hence, the channel at least, least, twice twice as as long. long. But But the the vertical vertical positive positive channel channel transverses transverses the the at negative Vi to to the the earth, earth, so so it it negative cloud cloud region, region, delivering delivering aa very very low low potential potential U,
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Anomalously large large current impulses ofpositive of positive lightnings
195 195
is stroke with is incapable incapable of of producing producing aa return return stroke with an an appreciably appreciably high high current current (section descending leaders (section 4.3.6). 4.3.6). Only Only those those lightnings, lightnings, whose whose positive positive descending leaders bypass cloud region curved path, bypass the the negative negative cloud region along along aa very very curved path, can can actually actually be be identified statistics shows shows that identified in in the the registrations. registrations. The The statistics that the the total total length length of of such aa leader, leader, including including the the path path bendings, bendings, is is 1.3-1.7 1.3-1.7 times times greater greater than than such that of of aa straight straight leader. leader. Therefore, Therefore, aa positive positive channel channel length length and and its its stroke stroke that pulse duration duration appear appear on on average average to to be be three three times times greater greater than than in in aa negative negative pulse leader. As As for for other other characteristics, characteristics, common common positive positive pulses pulses are are the the same same as as leader. the the negative negative ones ones described described above. above.
4.5
Anomalously large current impulses impulses of positive positive lightnings lightnings
Anomalous Anomalous impulses impulses of of aa positive positive lightning lightning have have the the duration duration t, tp M :::::; 1000 1000 ps llS of of the 0.5 the 0.5 amplitude amplitude level level and and the the rise rise time time tf q :::::;x 10011S. 100 ps. The The current current in in some some of of them them is is as as high high as as 100 100 kA kA or or more more [1]. [l]. Although Although such such lightnings lightnings are are rare, rare, their their effects effects on on industrial industrial objects objects are are so so hazardous hazardous that that they they should should not not be be underestimated. A A current current impulse impulse delivers delivers to to the the earth earth aa charge charge underestimated. Q be located Q :M: :; 100C; 1OOC; therefore, therefore, as as large large aa charge charge must must be located in in the the cloud cloud cell cell at the the boundary boundary of charged where the the lightning originated. The potential at where lightning originated. The potential of aa charged cloud region region of of radius, radius, say, km is is U :::::; 1000MV, with 1500MV cloud say, R R,c M:::::; 11 km U,, 1000 MV, with 1500 MV OR M at its its centre. centre. Any Any attempt to treat treat aa long long current current impulse impulse as return stroke at attempt to as return stroke current inevitably leads to contradictions. Indeed, in order to reduce the current inevitably leads to contradictions. Indeed, in order to reduce the near-earth current current by by half half of of its its maximum maximum value value for lO0011S, it it would would be be near-earth for lOOOps, l at = - 0.7 0.7 and and aa = =R R1/2L1 = 700s-’; hence, necessary to to assume assume in in (4.29) (4.29) atp necessary /2L = 700shence, ; 1 1 p. the the average average linear linear resistance resistance behind behind the the wave wave front front of of the the return return stroke stroke would would be be R Rl1 :::::; M 3.5xlO3 . 5 ~ 3 O/m. a / m . The The total total resistance resistance of of aa channel channel of of length length H = = 4000m 4000m would would be be R1H:::::; R1H M 140, 14R, i.e., i.e., 40 40 times times less less than than the the wave wave resistance. The to be be discharged resistance. The line line would would seem seem to discharged as as an an ideal ideal line, line, i.e., i.e., for for 20 2011S ps instead instead of of 1000 lO0011S, ps, with with the the velocity velocity of of an an electromagnetic electromagnetic signal. signal. Excessively smooth impulses are are sometimes in ascending ascending Excessively smooth impulses sometimes observed observed in leaders. A A positive positive impulse impulse 1IM :::::; 28 kA with t :::::; 80011S was registered leaders. M 28 kA with t, = 800 ps was registered M p during the the propagation propagation of of aa negative negative ascending ascending leader leader from 70-m tower tower during from aa 70-m Mount in in Switzerland fact in is of on the the San on San Salvatore Salvatore Mount Switzerland [31]. [31]. This This fact in itself itself is of interest, interest, but but its its analysis analysis may may offer offer an an explanation explanation of of 'anomalous' ‘anomalous’ currents currents of descending descending positive positive lightnings. lightnings. Note, Note, at at first, first, the the unusual unusual situation situation at at of the the start. start. Since Since the the negative negative charge charge of of the the dipole dipole is is located located at at the the cloud cloud bottom, bottom, the the ascending ascending leader leader is is to to be be positive positive rather rather than than negative. negative. Therefore, Therefore, the has either negative the dipole dipole axis axis has either deviated deviated from from the the normal normal or or the the bottom bottom negative charge say, an situation charge was was neutralized neutralized earlier earlier by, by, say, an intercloud intercloud discharge. discharge. This This situation occurs rarely but but it it is occurs rarely is possible. possible. We mentioned mentioned in in section 4.1 that that ascending lightnings have have no no return return We section 4.1 ascending lightnings strokes because their their channels are grounded from the the very very beginning. beginning. strokes because channels are grounded from
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Physical processes in a lightning discharge
However, when when the ascending leader leader penetrates the charged charged cloud cloud region region However, the ascending penetrates the (positive, potential difference (positive, in in this this case), case), aa large large potential difference arises arises between between the the front front so the the leader leader current current end of of its its grounded grounded channel channel and and the the space space around around it, it, so end has been has been found found from from many many registrations registrations to to rise rise to to several several kiloamperes. kiloamperes. This be triggered by the is This event event seems seems to to be triggered by the same same mechanism, mechanism, but but its its effect effect is greatly by the the leader the very greatly enhanced enhanced by leader hitting hitting the very centre centre of of aa large large cloud cloud Q, >=::: x 30C 30C and and radius radius R, M 500 500m, where the the potential potential is is as as charge of, of, say, say, Qe R e >=::: m, where charge high MV. At high as as V U,o >=::: M 500-800 500-800MV. At such such voltages, voltages, the the streamer streamer zone zone and and cover cover appear much much extended. extended. Negative streamers develop develop until the average average field field appear Negative streamers until the M 10kV/cm 10 kV/cm under normal conditions conditions (or (or in their their streamers streamers drops drops below in below E, E s >=::: under normal Streamers elongate elongate very very quickly quickly 1.5 times times less less at at aa 5-6 5-6km height [16]). Streamers 1.5 km height when the field very powerful powerful streamer when the field is is higher. higher. Therefore, Therefore, aa very streamer corona corona conconsisting sisting of of numerous numerous branched branched streamers streamers (they (they are are likely likely to to originate originate not not only the stem numerous branches, branches, too) will fill only from from the stem but but from from its its numerous too) will fill up up aa R M VolEs>=::: Uo/EsM R R,.e. The The negative negative charge charge of of the the streamer streamer zone zone space of of size size R>=::: space If the the streamers streamers will partly the positive charge of of the the cloud cloud cell. cell. If will partly neutralize neutralize the positive charge 6 have velocity mis, they region for have velocity VU,s >=::: x 10 106m/s, they will will fill fill the the charged charged cloud cloud region for 3 t x Relv Rc/vU, M s. Since the capacitance of the leader portion inside the t>=::: >=::: 10s. Since the capacitance of the leader portion inside the s cloud, C CL, is comparable comparable with that of of the charged cloud cloud region, region, Cel> Ccl, with that the charged cloud, L , is CL
>=:::
27rc o2R In(H I R) ,
Cel
>=:::
47rcoRe ,
CL
>=:::
0.5Cel
charge of of opposite opposite sign, sign, comparable comparable with with the the cloud cloud intrinsic intrinsic charge, charge, aa charge penetrates the cloud. cloud. The The resultant resultant effect effect is is such such that that most of the the cloud cloud penetrates the most of x Qol Q o /tt >=::: M 30 30 kA charge would would seem seem to down to the earth earth with with current current i >=::: charge to run run down to the kA 3 M 10lop3 s. Microscopically, Microscopically, the the cloud cloud medium medium remains remains non-conductive, non-conductive, for t >=::: for s. as before. before. Charges Charges do do not recombine but neutralize one one another another on on average. average. as not recombine but neutralize of current current organization organization reduces reduces to to the the neutralization neutralization of of the the The process The process of cloud rather than leader leader charge. charge. cloud rather than Returning to to long long current current impulses impulses after after the the positive leader arrival arrival at at the Returning positive leader the us imagine imagine that that the leader has developing along along aa vertical line earth, let let us earth, the leader has been been developing vertical line Q, >=::: x 100C lOOC or or somewhat away away from from the the axis axis of of aa powerful cloud dipole dipole with with Qe somewhat powerful cloud x 1km, 1 km, and and V U,,OR >=::: M 1000 1000 MV. MV. Suppose Suppose the the leader leader cover cover has no more, R, more, R e >=::: has no contact with with the the cloud cloud charge charge boundary boundary but is close close to to it. it. All All the the same, same, aa contact but is Q, arises arises in in the the vicinity vicinity of of huge, actually induced induced charge charge comparable comparable with huge, actually with Qe the cloud cloud charges. charges. Note the arrival arrival of of aa vertical vertical positive positive leader leader does does the Note that that the to zero because of not not reveal reveal itself itself in in any any way, way, since since its its potential potential is is close close to zero because of aa nearly complete symmetry symmetry of of charges charges induced induced in in the the lightning lightning channel. channel. nearly complete Suppose now now that that while while this this leader leader still still preserves preserves conductivity conductivity (this (this Suppose period of time time is is measured measured in in dozens dozens of of milliseconds milliseconds because of the the current current period of because of supply of of '" -100 A), an an intercloud intercloud discharge discharge occurs, occurs, connecting connecting the the lower lower 100 A), supply negative charge charge of of the the dipole dipole to to another another positive positive charge. charge. Intercloud Intercloud disdisnegative charges have have been been observed observed to to be much more more frequent frequent phenomenon charges be aa much phenomenon than than So our our suggestion suggestion is is not at all all improbable. improbable. The The cloud-earth discharges. discharges. So cloud-earth not at
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of a negative leader Stepwise behaviour of
197 197
charges of of opposite opposite signs signs connected connected by by intercloud intercloud leaders leaders will will gradually gradually charges neutralize each each other other via via the the same same mechanism mechanism as as the one underlying underlying an an neutralize the one ascending leader.t 1eader.t As the the neutralization goes on, on, the earlier induced induced but ascending neutralization goes the earlier but now positive charge now liberated liberated positive charge of of the the grounded grounded leader leader will will flow flow down down to to the the ground. will occur ground. This This will occur at at aa lower lower velocity velocity than than the the return return stroke stroke velocity, velocity, in accordance accordance with rate of of the the negative cloud charge. charge. in with the the neutralization neutralization rate negative cloud This is is aa likely likely explanation explanation for for long long powerful current impulses impulses in in positive This powerful current positive lightnings. lightnings.
4.6
Stepwise Stepwise behaviour behaviour of a negative negative leader
When negative leader When discussing discussing the the negative leader in in section section 4.3.2, 4.3.2, we we put put off off the the consideraconsideration of its its stepwise stepwise behaviour until the the reader reader has has become familiar with the tion of behaviour until become familiar with the return stroke, phenomenon is principal event return stroke, since since aa similar similar phenomenon is the the principal event occurring occurring in would be point, to to the the nature in each each step. step. It It would be reasonable, reasonable, at at this this point, to turn turn to nature and and effects effects of of the the stepwise stepwise leader leader behaviour. behaviour. But But we we should should like like to to warn warn the reader that why aa negative negative the reader that there there is is no no clear clear answer answer to to the the question question why leader positive one Nonetheless, leader has has aa stepwise stepwise structure structure while while aa positive one has has not. not. Nonetheless, some presented in some observations observations of of stepwise stepwise positive positive lightning lightning leaders leaders were were presented in [32]. phenomenon has been observed [32]. This This phenomenon has never never been observed in in laboratory laboratory conditions. conditions.
4.6.1 The step formation and parameters 4.6.1 The thing one nature ofleader The only only thing one can can rely rely on on today today in in discussing discussing the the nature of leader steps steps is is the results oflaboratory the results of laboratory experiments experiments with with long long negative negative sparks sparks (section (section 2.7). 2.7). Natural lightning Natural lightning observations observations are are not not informative, informative, except except for for the the step step lengths m [13,32-37] lengths 6x Ax,s :::::; x 5-100 5-100m [13,32-371 and and the the registrations registrations of of leader leader channel channel flashes photographs indicate flashes occurring occurring at at the the step step frequency. frequency. Streak Streak photographs indicate that that only bright flashes. only the the front front channel channel end end of of 1-2 1-2 steps steps in in length length shows shows bright flashes. But But weak flashes kilometre length weak flashes may may appear appear even even along along aa kilometre length (the (the vision vision field field of of aa photocamera photocamera does not always whole channel). does not always cover cover the the whole channel). Laboratory Laboratory streak streak pictures pictures show show that that aa step step originates originates from from two two secondsecondary the main ary twin twin leaders leaders at at the the front front end end of of the the streamer streamer zone zone in in the main negative negative leader the Russian leader (in (in the Russian literature, literature, these these are are termed termed bulk bulk leaders). leaders). The The positive positive leader moves towards leader moves towards the the main main leader leader tip tip while while the the negative negative one one develops develops along pause between between two the along the the latter latter (figure (figure 4.22). 4.22). During During the the pause two steps, steps, the secondary not have velocity, but but secondary negative negative and and the the main main leaders leaders do do not have aa high high velocity, the the distance the positive positive leader leader moves moves faster faster for for two two reasons. reasons. First, First, as as the distance to to the between the positive tip the main main leader leader tip tip becomes becomes shorter, shorter, the the difference difference between the positive tip t The that intercloud The fact fact that intercloud discharges discharges do do neutralize neutralize charged charged regions regions is is supported supported by by electric electric field field flow through lightning channels channels are are indicated indicated by peals of of measurements, and and high high currents currents that by peals measurements, that flow through lightning thunder. thunder.
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Physical processes in a lightning lightning discharge
u secondary leaders tnirial point
4L.------=-----f----------:: X
u
basic negatiw
v L
4/eadertip
x
u L
4'
4
Figure 4.22. The various stages stages of of aa step step formation. formation. The The main main The potential potential distribution distribution for for various potential U,: Ua: (top) (top) secondary secondary leaders leaders leader U ( x )is is counted counted from from the the external external potential leader potential potential U(x) the streamer streamer zone zone end; end; (centre) (centre) the the tip tip of of aa positive positive secsec11 and and 2 are are formed formed at at point point 33 at at the ondary reached tip tip 4 of of the the main main leader, leader, and and aa discharge discharge wave wave has has started started its its ondary leader leader has has reached travel leader channel channel (dashed (dashed line); line); (bottom) (bottom) the the main main leader leader tip tip travel along along the the secondary secondary leader is repeated. repeated. after has taken taken aa new new position, position, and and the the process process is after the the step step formation formation has
potential the external external (for (for the the tip) tip) potential potential U V(XI) at the the tip tip site site x1 Xl potential V U j1 and and the ( x l )at increases the streamer streamer zone zone field field of of the the main main increases (figure (figure 4.22 4.22 (top)). (top)). Second, Second, the leader, the generated generated negative negative streamers, streamers, leader, E E,s ~ % 10 10 kV/cm, kV/cm, which which must must support support the is required for for the the development development of of positive positive E,s ~ x 5 kV/cm kV/cm required is higher higher than than the the field field E leaders. For For this this reason, reason, the the streamers streamers generated leaders. generated by by the the secondary secondary positive positive leader tip tip develop develop in in aa fairly fairly strong leader strong field, field, become become accelerated accelerated and and all all reach reach the main main leader leader tip. Since the the tip. Since the long long channel channel of of the the main main leader leader has has aa capacitance that of of the the short short secondary secondary leader, leader, itit absorbs absorbs capacitance greatly greatly exceeding exceeding that completely carried by by the the positive positive streamers. streamers. In In other other words, words, completely all all charges charges carried the secondary secondary positive the positive leader leader develops develops in in the the final final jump jump mode. mode. We We know know from section section 2.4.3 2.4.3 that from that this this leads leads to to its its acceleration. acceleration. The The secondary secondary negative negative leader, on on the the contrary, contrary, develops develops in in aa decreasing leader, decreasing field field beyond beyond the the streamer streamer of the the main main leader, leader, whose whose streamers streamers stop zone of stop in in space, space, so so it it moves moves much much zone more slowly, slowly, similarly similarly to to the the main main leader. leader. more When the the tip tip of of the the secondary secondary positive positive leader leader comes comes in in contact contact with with the the When main channel, channel, the the positive positive leader leader experiences main experiences the the transition transition to to the the return return stroke. Charge Charge variation stroke. variation waves waves run run along along both both channels, channels, as as described described in in section 4.4, 4.4, and and their section their potentials potentials tend tend to to become become equalized equalized (figure (figure 4.22 4.22 (centre)). But But the the capacitance capacitance of of the the kilometre kilometre length length channel channel is is much much (centre)). higher than than that that of of the shorter secondary the shorter secondary channel, channel, so so their their fusion fusion results results in in higher
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Stepwise behaviour of of a negative leader
199
establishing establishing aa potential potential only only slightly slightly differing differing from from the the initial initial potential potential of of the the main leader leader tip, tip, U,. Vt . main The moment moment at at which which the the potential potential U1 VI is is taken taken off off the the secondary secondary leader leader The channel of the the step. step. channel and and the the latter latter joins joins the the main main leader, leader, manifests manifests the the end end of The The main main leader leader tip tip ‘jumps’ 'jumps' over over to to aa new new place, place, the the one one occupied occupied previously previously by the the tip tip of by Ut, of the the secondary secondary negative negative leader, leader, delivering delivering to to it it its its potential potential V (figure (figure 4.22 4.22 (bottom)). (bottom». The The tremendous tremendous potential potential difference difference that that arises arises in in the the vicinity of vicinity of the the newly newly formed formed tip tip at at this this moment moment produces produces aa flash flash of of aa powerful powerful negative streamer streamer corona, corona, which which transforms transforms to to the the novel novel streamer streamer zone zone of negative of the the main leader. leader. Then Then the the sequence sequence of main of events events is is repeated. repeated. The The combination combination of of the the charge of the the secondary secondary positive positive leader leader and and charge utilized utilized for for the the short short recharging recharging of for charging charging the the secondary secondary negative negative one, one, plus plus the the charge charge incorporated incorporated into into for the new new streamer streamer zone, zone, create create the the step step current current impulse. impulse. (Recall (Recall that that there there is is the always aa local local current current peak peak at at the the streamer streamer tip tip or or in in the the leader leader streamer streamer always zone, related related to to the the displacement displacement of of the the charge charge in in this this region; region; see see sections sections zone, 2.2.3 and and 2.3.2.) 2.3.2.) Part Part of of the the step impulse creates impulse in in the the 2.2.3 step impulse creates aa current current impulse main channel channel and the other other part part is is spent for the the formation of aa new new cover cover main and the spent for formation of Qs pumped pumped into into the the main main channel channel can can be be evaluated portion. The The charge charge Q, portion. evaluated in in terms of of the the mean mean velocity velocity of of the the stepwise leader, wL VL M ~ 3 the terms stepwise leader, 3 x 10mls, 10m/s, the of aa step ~xs M ~ 30m, 30 m, and and the the current ~ 100 A averaged averaged over over the the length of length step Ax, current iiL lOOA L x whole duration the process. process. Since the time time between between two two steps is whole duration of of the Since the steps is 2 C. 4 s, ~ts x ~ Ax,/wL ~xs/vL M ~ 10the charge is Q, Qs M ~ iLAt, iL~ts x ~ 10c. At, lop4 s, the charge is lop2 4.6.2 Energy effects in the leader channel The The energy energy pumped pumped by by the the charge charge pulse pulse Qs Q, into into the the channel channel can can be be evaluated evaluated if if the the effect effect is is assumed assumed to to be be similar similar to to that that observed observed when when the the small small capacicapacitance tance (of (of the the secondary secondary leader) leader) is is added added parallel parallel to to the the large large capacitance capacitance (of (of the the main main leader). leader). While While aa common common potential potential is is being being established, established, there there is is aa dissipation dissipation of of energy energy nearly nearly equal equal to to the the electrical electrical energy energy stored stored by by the the small small capacitance capacitance at at aa voltage voltage equal equal to to the the difference difference between between the the initial initial capacitance capacitance voltages voltages V U,t -- VI, U1, where where VI U1 is is the the potential potential of of secondary secondary leaders. leaders. It It was was pointed pointed out out in in section section 2.4.1 2.4.1 that that the the leader leader tip tip potential potential is is shared shared nearly nearly equally equally between between the the streamer streamer zone zone and and the the space space in in front front of of it. it. Secondary Secondary leaders leaders are are produced produced at at the the streamer streamer zone zone edge, edge, so so we we have have V U1j -- Va U, ~ M !(V ( Ut, -- Va); U,); hence, hence, V Ut, -- VI U1 = = !(V f ( Ut, -- Va). U,,). With With the the accepted accepted average average values values of of current current ii ~ xC C1 ( Ut, -- Va)VL Uo)vLand and of of velocity velocity VL wL and and assuming assuming j (V C C1 x lOpFjm, 10pF/m, we we find find U U,t -- Ua, ~ = 30MV 30MV and and V U,t -- U U1j ~ = 15MV. 15MV. Thus, Thus, the the I ~ step step energy energy is is
4
W
M
Q,( U, - U1)/2 x 7.5 x lo4J.
(4.39) (4.39)
Of Of course, course, not not all all of of this this energy energy is is released released in in the the main main channel. channel. During During the the disdissipation sipation of of charge charge Qs' Q,, the the channel channel potential potential rises rises appreciably. appreciably. This This leads leads to to the the radial of aa streamer streamer corona corona which which radial field field enhancement enhancement and and to to the the excitation excitation of
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Physical processes in a lightning discharge
pumps cover. This This process, process, the the cover cover pumps some some of of the the charge charge into into the the leader leader cover. if the the energy energy released released ionization But even even if ionization in in particular, particular, requires requires much much energy. energy. But in is still very large large energy. energy. W M 104 lo4 J, J, tthis h s is still aa very in the the channel channel is is assumed assumed to to be be W:::::: The current of of 100 A in in remote remote The power power required required to to support support an an average average current lOOA impulses are are averaged averaged and and channel channel portions, portions, where where the the effects effects of of current current impulses 5 smeared, removed from from the the channel channel by by heat heat smeared, should should be be PI P1 :::::: M 10 lo5W/m. Wjm. It It is is removed conduction parameters correspond correspond to to the the conduction and, and, partly, partly, by by radiation. radiation. These These parameters maximum field E E:::::: lOV/cm, and resisresismaximum channel channel temperature temperature T:::::: T M 10 10 000 000 K, K, field x 10 Vjcm, and D/m taken (section 2.5.2). 2.5.2). At At this this tance tance R) R1 :::::: M 10 10R/m taken for for the the above above estimations estimations (section per unit unit channel channel length length will will power, power, the the energy energy released released between between two two flashes flashes per 4 :::::: 10 J /m for the time tlt :::::: 10s. Therefore, the single pulse be be W Wl,, M 10J/m for the time At, M lOP4s. Therefore, the single pulse lav s energy km long. long. In In reality, reality, aa energy W W would would be be sufficient sufficient to to support support aa channell channel 1 km At aa distance distance of of about about step step pulse pulse is is damped damped at at aa much much shorter shorter length. length. At 1km, the energy released in in the the channel channel 1 km, the the step step effects effects are are smeared smeared and and the energy released from the the tip, tip, the the energy energy becomes becomes totally totally averaged. averaged. But But at at aa short short distance distance from by the the intensive intensive flash. flash. The The effect effect of of the the step step is is very very strong, strong, as as is is indicated indicated by [35], temperature was as as high high as as 30 30000 K [35], temperature registered registered in in some some measurements measurements was 000 K i.e., return stroke. stroke. i.e., the the same same as as at at the the wave wave front front in in aa return Let us us evaluate evaluate the the distance distance at at which which the Let the energy energy effect effect of of an an individual individual step is is still still essential. essential. When When aa short short step step joins joins aa long long channel, channel, charge charge Q, Qs is is step pumped into into the the channel channel for for aa short short time. time. Since Since we pumped we are are interested interested in in distance distance and of aa charge charge source, source, let let and time time much much larger larger than than the the real real length length and and duration duration of point-like, as as is is usually usually done done in in us us assume assume the the source source to to be be instantaneous instantaneous and and point-like, the initial initial point point of of the the line, line, x = = 0, 0, physics: physics: aa point point charge charge Qs Q, is is introduced introduced at at the at the the initial initial moment moment of of time time t = = O. 0. The The resistance resistance of at of not not very very short short channel channel fragments, Rlx, Rlx, is is higher higher than than the the wave wave resistance, resistance, so so the the inductance inductance effect effect fragments, will be be neglected. neglected. At At an an average average resistance resistance of 10R/m, of 10 D/m, this this distance distance is is just just will about the the step step length length tlx Ax,.s . At At shorter shorter distances, distances, the about the instantaneous instantaneous point point source model model is is invalid, invalid, since since it it implies implies an source an infinite infinite initial initial voltage voltage and and energy. They They drop drop to to realistic realistic values values only only if energy. if the the charge charge affects affects aa length length exceeding tlx Ax,, at which which the the source source was actually placed. exceeding was actually placed. Therefore, Therefore, with with s , at the neglect neglect of of inductance inductance (and (and the the precursor), precursor), the the line line charging charging to to potential potential the U,(x, above the the background background potential potential is described by Up(x, t)t ) above is described by equations equations (4.36). (4.36). On the the assumption assumption of of R) R1 = = const, const ,tt they they have have an On an exact exact solution solution correspondcorresponding to to heat heat flow flow from from an an instantaneous instantaneous lumped lumped source: ing source:
Up = .
lp = -
2Qs C) (471"Xt)
1/2
2/ t , exp ( -x 4x)
1 BUp
-R ~ = )
uX
QsX 1/2
(471"Xt)
(4.40) 2
t
exp( -x /2xt),
1
x= R)C)'
(4.41 )
The value value of of resistance resistance R 1Ito to be be taken taken for for evaluations evaluations may tI.The may be be smaller smaller than than that that in in the the leader, leader, of the the channel channel due due to to the step current. having in in mind mind aa transformation transformation of the step current. having
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Stepwise behaviour of of a negative leader
201 20 1
The step per The power power released released by by the the current current step per unit unit channel channel length length is is described as described as 2 2
PI
=
.2 lpR I
=
QsX 2
4JrX Cit
3
2/ ) exp (-x 2xt.
(4.42) (4.42)
2 At the the point point x, x, the the power power reaches reaches aa maximum maximum at at moment moment tt,m = = xx2/6x, and At /6x, and
(4.43) (4.43) For energy released length at at point For the the time time of of the the pulse pulse action, action, the the energy released per per unit unit length point x x is is W,
WI:::::; M
1J;oc ·2 o
Q;
Q2 W (~xs)2 Ax, W lpRldt=--2:::::;~ - ,, iiR, dt = 2x - JrClx L.l.X d 1 x 2 Ax,s xx
(
x > Ax,
(4.44) (4.44)
where where W W is is the the total total energy energy injected injected into into the the channel channel by by the the pulse, pulse, with with its its upper upper limit limit given given by by formula formula (4.39). (4.39). The The effective effective duration duration of of energy energy release release from from aa single single step step at at point point xx is is expressed expressed as as
W xX22 ~tw :::::; _ _ I :::::; 2.2t 2.2tm = Wl - -2-' -. At, - 2.7% P lmax .7X PImax N
(4:45) (4.45)
Consequently, the contribution contribution of charge injection injection to to the the energy release at at Consequently, the of charge energy release aa given given channel channel site decreases in the direction direction of perturbation propagation propagation as site decreases in the of perturbation as WI x--2 ~and independent of of R I1. . The The latter latter fact fact justifies justifies the the use use of of x x and is is independent Wl :::::; R I == const const without without reservations reservations concerning concerning the the resistance resistance variation variation during during R1 the the current current impulse impulse passage. passage. The The energy energy pulses pulses released released at at point point xx owing owing to to the the two two subsequent subsequent steps steps superimpose superimpose at at xx > > (2.7x~ts)1/2; (2.7xAts)ll2; this t h s critical critical distance distance follows follows from from the the condition condition ~tw At, :::::; M ~ts' At,. For For example, example, at at the the average average 4 resistance X= Is and s, resistance R II = = 10 10 O/m, R/m, with with x = 10 1010 m m2 /s and step step frequency frequency ~ts At, = = 10lop4 s, this km in ~ts/2.2 :::::; /-lS, after pulse this happens happens at at aa distance distance xx :::::; x 1.6 1.6 km in tm, :::::; M AtJ2.2 M 45 45 ps, after the the pulse arrival here. here. Thus, Thus, the the effects effects of energy release release from individual steps arrival of energy from individual steps are are detectable along an an extended lightning path, path, and and this this is is the the cause cause of of detectable even even along extended lightning observable flashes of almost almost the the whole whole channel. For aa flash to arise, arise, there there is observable flashes of channel. For flash to is no need need for for aa strong effect. A A short temperature rise rise of, of, say, above no strong energy energy effect. short temperature say, above 1000 Kover 000 K would would be be sufficient for aa flash to be be detected detected by by l000K over 10 l0000K sufficient for flash to modern modern photographic photographic equipment. equipment. The The channel channel energy energy is is affected affected by by the the temperature temperature rise rise above above the the average average background, background, rather rather than than by by the the time time separation separation of of the the energy energy pulses pulses between between two two subsequent subsequent steps. steps. In In this this respect, respect, the the impulse impulse effect effect on on the the channel channel during during the tip. The plasma the wave wave propagation propagation is is damped damped at at distances distances close close to to the the tip. The plasma large distances is temperature modulation modulation determining the flash intensity at temperature determining the flash intensity at large distances is due to to the the imbalance imbalance between between the the energy energy release release and and heat heat removal removal from from due the channel during the the pauses pauses between between the the steps. is no no imbalance imbalance the channel during steps. There There is at aa large large distance the tip tip after the channel channel development has been been at distance from from the after the development has
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202 202
Physical processes in a lightning discharge
established. plasma radiation established. At At T :::::: x 10000 10 000 K, K, the the losses losses for for air air plasma radiation are are not not parparheat of ticularly but become ticularly great great but become appreciable appreciable at at T :::::: M 12-14000 12- 14 000 K. K. The The Joule Joule heat of current primarily by current is is eliminated eliminated from from the the channel channel primarily by heat heat conduction. conduction. This This process occurs pressure when process occurs at at constant constant (atmospheric) (atmospheric) pressure when the the energy energy release release is hundreds of is moderate, moderate, as as is is the the case case for for distances distances of of hundreds of metres metres from from the the tip. tip. At T:::::: T M 10000K, l0000K, the the air air heat heat conductivity conductivity is is XA:::::: x 1.5 1 . x510~ 2 W/cm WjcmK and At K and 2 the thermal thermal conductivity conductivity at at pressure 1 atm is is XT == X/pcp x 180cm 180cm2/s, the pressure pp == 1atm A/pcp:::::: /s, where where p is is air air density density and and ccp is heat heat capacity. capacity. The The average average conductivity conductivity in in p is the channel channel 0" corresponds corresponds to to aa temperature lower than than the the maximum the temperature lower maximum temtemperature. perature. To To illustrate, illustrate, at at 0"c :::::: x 10 10 (rl (Cl . cm)-l cm)-’ corresponding corresponding to to T == 8000 8000 K, K, l 2 the effective effective radius radius of of aa conductive conductive channel channel is is r:::::: r M (71"O"R (mrR1)-”2 M 0.6cm 0.6cm / :::::: the 1 for R1 = 10 10R/m. The heat heat is is removed removed from from the the channel channel for for time time for R1 = rl/m. The r 2 /2xT '" 10- 3 s, s, an an order order of of magnitude magnitude longer longer than than the pause between between t '" ,2/2XT the pause the steps. The The repeated energy pulses pulses dissipate dissipate rather rather slowly, slowly, and and the the the steps. repeated energy temperature modulation modulation relative relative to to its its average average value value T:::::: T M 10000 10000K is not not temperature K is large at at long long distances distances x. x.Indeed, Indeed, the the energy energy released released in in the the remote remote channel channel large portions during during aa pause pause is is W Wla, xP PlavAt, x 10J/m 10 Jim at at an an average average power power portions lav :::::: lav !:1t s :::::: 5 Plav:::::: M 10 lo5 Wjm. Even Even if if we we assume assume that that all all energy energy of of aa step step is is released released in in Play W/m. the channel channel and and W W/Ax, x 2500 2500 J/m Jim in in (4.45), (4.45), the the excess excess of of the the pulse pulse release release the / !:1x s :::::: over the the average average heat heat removal, removal, which which is is equal equal to to the the average average energy energy release release over W Wla,, will be be small small at at xx > > !:1xs(WdWlav)I/2:::::: Ax,( Wl/Wlav)1’2x lO!:1x lOAx,s :::::: zz 500m. 500m. With With allowallowlav , will ance ance for for other other energy energy expenditures, expenditures, this this reduction reduction in in the the pulse pulse effect effect will will be be evident evident even even at at shorter shorter distances. distances. This This circumstance circumstance makes makes the the use use of of average average parameters reasonable reasonable in in the the consideration consideration of of the the evolution evolution of of long long stepwise stepwise parameters lightning leaders, leaders, ignoring ignoring the the stepwise stepwise behaviour behaviour effects. effects. In In any any case, case, laboralaboralightning tory experiments experiments show show that that there there is is no no appreciable appreciable difference difference between between aa tory positive continuous continuous and and aa negative negative stepwise stepwise spark spark discharge discharge as as for for the the positive velocity, average average leader leader current current or or breakdown breakdown voltage voltage in in superlong superlong gaps. gaps. velocity, However, even even aa small small excess excess of of the the average average temperature temperature over over its its averaverHowever, age value value may may be be sufficient sufficient for for aa flash flash to to be be registered registered optically. optically. As As for for chanchanage nel portions portions located located at at aa distance distance of of one one or or two two steps steps from from the the tip, tip, the the energy energy nel pulses and and outbursts outbursts of of temperature temperature and and brightness brightness are are found found to to be be very very pulses strong there. there. A A gas-dynamic gas-dynamic expansion expansion of of the the channel channel is is also also possible, possible, as as strong happens in in the the return return stroke stroke (section (section 4.4), 4.4), although although it it occurs occurs on on aa smaller smaller happens scale. scale. No No doubt, doubt, aa flash flash is is also also produced produced by by aa powerful powerful impulse impulse corona corona giving rise rise to to aa new new streamer streamer zone zone of of the the elongated elongated leader. leader. Photographs Photographs giving show that that the the transverse transverse dimension dimension of of aa step step flash flash is is about about 10 10m [38]. show m [38].
r
N
4.7
N
The subsequent subsequent components. components. The The M-component M-component The
The processes processes in in the the lightning lightning channel channel following following the the first first component component are are The known as as subsequent subsequent components. components. Of Of interest interest among among these these are are so-called so-called known M-components and and dart dart leaders. leaders. In In the the first first case, case, the the current current impulse impulse M-components
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The subsequent components. components. The M-component
203 203
registered registered at at the the earth earth has has aa very very smooth smooth front front (0.1-1 (0.1-1 ms), ms), aa similar similar duration duration and 1-22 kA. and an an amplitude amplitude of of several several hundreds hundreds of of amperes, amperes, sometimes sometimes of of 1kA. The The channel radiation radiation intensity intensity increases increases abruptly abruptly during during the the impulse, impulse, but but one one channel can hardly hardly identify identify in in the the photographs photographs aa structure structure similar similar to to the the impulse impulse can front. The The current current impulse impulse of of an an M-component M-component is is always always registered registered against against front. the 100A the background background of of about about 100 A continuous continuous current current of of the the interpulse interpulse pause. pause. For For aa dart dart leader leader to to arise, arise, this this current current must must necessarily necessarily be be cut cut off off [39,40]. [39,40]. A A few few microseconds microseconds after after the the cut-off, cut-off, aa short short high-intensity high-intensity region region - aa dart dart leader leader tip tip - runs runs down down to to the the earth earth along along the the previous previous channel channel 7 with of ",10 -107m/s. The contact with aa velocity velocity of m/s. The contact of of the the dart dart leader leader with with the the earth earth produces produces aa return return stroke stroke with with its its typical typical characteristics characteristics but but having having aa much much shorter ps or shorter impulse impulse front front than than in in the the first first component component (less (less than than 1IllS or even even 0.1 ps in It is is hard hard to to say say anything anything definite definite about about the the lower lower O.lIlS in some some impulses). impulses). It limit limit of of the the front front duration: duration: it it is is likely likely to to lie lie beyond beyond the the time time resolution resolution of of the measuring equipment. equipment. the measuring The papers published published almost The papers almost simultaneously simultaneously [41, [41, 42] 421 interpret interpret the the subsubsequent component component qualitatively qualitatively as as representing representing the the discharge, discharge, into into the the sequent earth, of of an an intercloud intercloud leader leader after after its its contact contact with with the the upper upper end end of of the the precedprecedearth, ing grounded grounded but but still still conductive conductive channel. channel. Here Here we we describe describe the the evolution evolution of of an an ing M-component M-component in in terms terms of of aa numerical numerical simulation. simulation. The underlying the The model model underlying the simulation simulation is is as as follows follows (figure (figure 4.23). 4.23). Initially, Initially, there potential, which which there is is aa grounded grounded plasma plasma channel channel of of length length HI with with zero zero potential, by the the preceding preceding lightning lightning component. 0, aa leader was left left behind behind by was component. At At time time t == 0, leader channel channel of of length length H 22 and and potential potential Ui Vi joins joins it it in in the the clouds clouds (the (the voltage voltage drop drop is neglected). neglected). from the the leader leader current current and and from from the the intercomponent intercomponent current current is from The the channel the streamer The short short process process of of the channel commutation commutation through through the streamer zone zone
(a)
-u t=0 r-----I t1
-a
I
I
(b) X x Figure Figure 4.23. 4.23. The The formation formation of of aa subsequent subsequent component: component: (a) (a) the the grounded grounded channel channel of of
the the previous previous component component and and an an intercloud intercloud leader; leader; (b) (b) channel channel charging-discharging charging-discharging waves. waves.
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204 204
Physical processes in a lightning discharge
of moment of of the the intercloud intercloud leader leader is is ignored. ignored. At At the the moment of closure, closure, the the leader leader channel possesses typical resistance resistance R RIL % 10O/m. The resistance R I, of of i"::j 100/m. The resistance Ric channel possesses aa typical IL the previous channel the duration the intercomponent the previous channel depends depends on on the duration of of the intercomponent pause. After the the return return stroke stroke current current impulse impulse of of the the previous previous component component pause. After
25001 2000
d
g 1 500 C
f
-
E
61000500-
0-
Simulation of of the the M-component M-component on on closing closing an an intercloud intercloud leader leader 2km 2 km in in Figure 4.24. Simulation length and and 10 lOMV potential on on aa 4-km 4-km grounded grounded channel. channel. The The initial initial linear linear resistances resistances length MV potential 10I1jm n/m and and the the steady steady field field E EL = 10 10Vjcm. Vjcm. The The waves waves of of potential potential of the the channels channels R 1I== 10 of L = (this page, page, top), top), current current (this (this page, page, bottom), bottom), the the power power of of Joule Joule losses losses (opposite, (opposite, top) top) (this and the the current current impulse impulse at at the the grounded grounded end end of of the the channel channel (opposite, (opposite, bottom); bottom); for for and is also also given given for for R jI== 20 20 I1jm G/m and and E EL = 20 20 Vjcm V/cm (curve (curve B). B). comparison, the the latter latter is comparison, L =
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T h e subsequent components. components. The The M-component M-component The
205
is is damped, damped, the the channel channel resistance resistance increases increases gradually gradually due due to to the the gas gas cooling. cooling. But But if if the the intercomponent intercomponent current current is is comparable comparable with with the the leader leader current, current, as as is usually the case case by the moment moment the the M-component M-component arises, arises, the the increased increased is usually the by the resistance of resistance of the the grounded grounded channel channel may may be be suggested suggested to to be be limited limited by by the the value of of Ric:::::: R I , M riL' r l L . The The reactive reactive parameters parameters of of both both lines, lines, which which are are not not value very very sensitive sensitive to to the the channel channel plasma plasma state, state, can can be be taken taken to to be be identical identical to to those of of the the leader: leader: C C1 :::::: M lOpF/m 10pF/m and and L L1 :::::: M 2.7IlH/m. 2.7 pH/m. those l l During During the the intercloud intercloud leader leader discharge discharge into into the the earth earth via via the the preceding preceding channel path, path, the the channel channel resistances resistances change, change, as as in in the the return return stroke stroke channel
16 14 12
.
.gE 3 ~
10 10
::;E 2 88 .... b 7 -
~
g6
Il)
~
6
0
~ & 44
22 0
800
0
i
22
x, kIn x, km
33
4
-
<
'€m 600600 c, -5 <'0
a 'l.l, a,
~ 400400
4-3
It,E
u 200200
B
o+-h....;...................,..-.........,--..-.................-....-~ 200 300 300 400 500 600 700 0o 100 200 Time, Time,
4.24. Continued. Continued. Figure 4.24.
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ps ~s
206 206
Physical processes in in a lightning discharge
(section (section 4.4.4). 4.4.4).Suppose Suppose that that these these changes changes follow follow the the relaxation relaxation law law expressed expressed by (4.38). This This formula formula describes describes adequately adequately the the qualitative qualitative tendentendenby formula formula (4.38). cies; cies; there there are are no no quantitative quantitative results results to to compare compare them them with. with. This This process process is is described described by by the the long long line line equations equations of of (4.24) (4.24) with with the the following following initial initial and and boundary boundary conditions: conditions: V(x,O) U ( x , O )== 00
at at 00 ~ < xx ~< HHI,I ,
+ i(H i(H1 + H Hz, r ) == O.0. 2 , t)
< <
V(x,O) at HI H I ~ xx ~ HI HI+H H 22, , U ( x , O )== Vi U, at i(x,O) i ( x ,0) == 0, 0;
V(O, U ( 0 ,t)t ) == 0, 0,
(4.46) (4.46)
j
At At the the site site of of contact contact of of the the two two lines, lines, their their potentials potentials are are identical identical at at tr > > O. 0. After After the the contact contact of of the the intercloud intercloud leader leader with with the the grounded grounded channel, channel, current current and and voltage voltage waves waves start start running running along along both both lines lines away away from from the the point point of of contact. contact. As aa result, result, the the grounded grounded channel channel becomes becomes charged charged while while the the leader leader channel channel is is discharged. discharged. If If the the initial initial parameters parameters of of the the lines lines are are identical, identical, the the initial initial current current at at the the point point of of contact contact is is i = = V;/2Z, Ui/2Z, where where Z is is the the wave wave resistance resistance of of the the lines. lines. Rapidly Rapidly attenuated attenuated precursors precursors run run in in both both directions directions at at velocities velocities of of electromagnetic electromagnetic signal, signal, while while the the main main current current and and voltage voltage waves waves propagate propagate via via the the diffusion diffusion mechanism mechanism (figure (figure 4.24). 4.24).These These waves waves spread spread much much stronger stronger than than in in the the return return stroke, stroke, since since the the current current and and voltage voltage are are lower lower here here (the (the more more so so that that the the initial initial voltage voltage amplitude amplitude is is half half the the value value of of Vi)' Ui). The The channel channel resistance resistance decreases decreases more more slowly slowly and and the the wave wave fronts fronts become become smooth smooth instead instead of of becoming becoming steeper. steeper. The The initial initial voltage voltage Vi Ui in in the the subsequent subsequent components components seems seems to to be be lower lower on on average average than than in in the the stepwise stepwise leader leader of of the the first first component component because because this this process process involves involves the the increasingly increasingly less less mature mature cloud cloud cells cells with with lower lower charges. charges. If If we we ignore ignore the the weak weak displacement displacement current current induced induced by by the the changing changing charges charges of of the the recharged recharged channels, channels, the the current current impulse impulse at at the the earth earth can can be be registered registered only only after after the the diffusion wave front has reached the earth. By that moment, the wave diffusion wave front has reached the earth. By that moment, the wave has has become very be very become very diffuse, diffuse, so so the the current current impulse impulse front front appears appears to to be very smooth power smooth (figure (figure 4.24 4.24 (bottom, (bottom, page page 205».t 205)).t The The more more or or less less uniform uniform power distribution uniform enhancement distribution along along the the channel channel is is to to look look as as aa uniform enhancement of of its its radiation of this this distribution distribution and and such such current current radiation intensity. intensity. The The calculations calculations of impulse impulse characteristics characteristics as as the the front front steepness, steepness, duration, duration, and and amplitude amplitude are are similar to their similar to their observations observations in in M-components. M-components. A still still better better agreement agreement with be attained with the the measurements measurements can can be attained by by varying varying the the parameters parameters in in the the calcalculations, in fully transformed L Land and quasistationary quasistationary field field E EL in fully transformed culations, primarily primarily R IL L channels. channels. These These arguments arguments favour favour the the above above suggestions suggestions concerning concerning the the nature of nature of lightning lightning M-components. M-components.
t The The current current impulse impulse of of aa return return stroke stroke is is of of a different different form. form. The The current current amplitude amplitude is is registered registered
right process when right after after the the short-term short-term commutation commutation process when the the leader leader contacts contacts the the earth earth via via its its reducing reducing streamer streamer zone, zone, which which takes takes aa few few microseconds. microseconds.
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of a dart leader Subsequent components. The problem of
4.8
207
Subsequent components. components. The problem problem of a dart leader Subsequent
There There is is still still no no clear clear understanding understanding of of the the nature nature of of aa dart dart leader, leader, so so we we shall shall discuss data available available and and suggest suggest aa hypothesis hypothesis based based discuss the the scarce scarce experimental experimental data on shall consider consider some some possible on them. them. Then Then we we shall possible consequences consequences of of this this hypothhypothesis and dart leader esis and the the difficulties difficulties that that may may arise. arise. The The dart leader problem problem remains remains unsolved of the unsolved but but it it cannot cannot be be put put aside aside because because of of the the importance importance of the dart dart leader process. process. leader 4.8.1
'waveguide'? A streamer in a ‘waveguide’?
There are are no no grounds grounds to to believe believe that that the the mechanism mechanism of of dart dart leader leader initiation initiation There in the the clouds clouds is is essentially essentially different different from from that that of of aa M-component. M-component. Rather, Rather, in both processes processes result result from from the the closure closure of of an an intercloud intercloud leader leader on on aa grounded grounded both channel remaining remaining after after the the passage passage of of the the return return stroke stroke in in the the previous previous lightlightchannel ning component. component. But But the the potential potential wave wave running running along along this this track track to to the the ning earth, known as as aa dart dart leader, leader, differs differs radically radically from that of of an an M-component. M-component. earth, known from that It has aa well-defined well-defined front front identifiable identifiable by by the the intense intense radiation radiation of of dart leader It has dart leader tip travelling travelling to to the the earth earth with with velocity velocity ?&L VdL "" mis, which which is is at least by by an an tip N 10 lo77 m/s, at least order of of magnitude magnitude hgher higher than than the the typical typical velocities velocities of of the the first first stepwise order stepwise leader. The The ability ability of of aa dart dart leader leader to to travel travel so is especially especially remarkable remarkable leader. so fast fast is because its its potential potential is is most most likely likely to to be be lower lower than than that that of of aa stepwise because stepwise leader. This This is is indicated by the the return return stroke which are are on average leader. indicated by stroke currents, currents, which on average 2-2.5 times times lower lower (1M :::::: U;/Z). The potential potential drop drop from from the the dart leader tip tip 2-2.5 (I, M U , / Z ) . The dart leader in the the previous, previous, still untransformed channel channel towards towards the the earth must occur occur in still untransformed earth must very very quickly. quickly. This This is is indicated indicated by by aa very very fast fast front front rise rise of of the the return return current current impulse, impulse, tj' tf. To To gain gain the the full full current current ,1ZM :::::: M U //Z, Z , where where U is is the the potential potential carried carried by by the the dart dart leader, leader, the the return return wave wave must must run run along along aa leader leader section section with with aa rising rising potential potential bx Sx and and reach reach the the totally totally charged charged portion portion of of the the we have have ~x return wave channel. channel. Therefore, Therefore, we Ax "" vrtj, ‘U,?,and and if if the the return wave velocity velocity is is V m/s and 0.1 ps, IlS, the the length length of of the the region region with with an an abrupt potenw,r M:::::: 10 10’8 m/s and tj M:::::: 0.1 abrupt potendrop in in the the dart leader front front is is Sx bx:::::: m. It is quite quite possible possible that that this this tial drop tial dart leader M 10 10m. It is value is is actually because the the return return wave wave cannot first gain gain the the value actually smaller smaller because cannot at at first 8 full return return stroke stroke velocity velocity VU,r M :::::: 10 m/s. On On the the other other hand, hand, the the potential potential full lo’,/,. drop region region should not be be shorter than Ax ~x :::::: the cross cross drop should not shorter than M vdLtj vdL?f :::::: x 1m, 1 m, since since the section section of of aa channel channel with with total total potential potential U approaches approaches the the earth earth at at velocity velocity VdL' m is ‘udL. Such Such aa steep steep front front of of 1-10 1-10m is unattainable unattainable not not only only by by aa diffusion diffusion wave wave with with its its potential potential varying varying along along many many hundreds hundreds of of metres metres (figure (figure 4.24) 4.24) but but even even by by an an 'ordinary' ‘ordinary’ leader leader of of the the first first lightning lightning component, component, in in which which ~x Ax is is determined determined by by the the streamer streamer zone zone length. length. At At the the moment moment of of contact contact with with the the earth, earth, the the latter latter is is measured measured in in dozens dozens of of metres metres at at the the tip tip MV. This potential of 20-30 20-30MV. This is is the the reason reason why why the the time time necessary necessary for for the the potential of return of times times return wave wave current current to to grow grow to to its its maximum maximum value value is is dozens dozens of longer longer than than that that for for the the dart dart leader. leader. N
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208 208
Physical processes in a lightning discharge
It that aa dart dart leader leader travels travels as as It follows follows from from the the foregoing foregoing and and the the fact fact that fast has no no streamer streamer zone zone which which fast as as aa very very fast fast streamer streamer that that the the former former has would leader mechanism. mechanism. It It appears appears would serve serve as as the the primary primary prerequisite prerequisite for for aa leader not aa leader, leader, although although that that the the dart dart leader, leader, contrary contrary to to its its name, name, is is essentially essentially not itit has to acquire under somewhat somewhat different different has aa charge charge cover, cover, which which it it has has to acquire under circumstances like aa diffusion diffusion wave wave of of the the MMcircumstances (see (see below). below). Nor Nor does does it it look look like magnitude higher higher velocity velocity and and aa type. type. The The latter latter would would have have an an order order of of magnitude very very diffuse diffuse front. front. of the the known known types types of of propapropaA dart dart leader leader looks looks more more like like the the oldest oldest of gating head represents represents an an ionization ionization gating plasma plasma channel channel -- aa streamer, streamer, whose whose head wave. to that that of of aa high high voltage voltage streamer. streamer. wave. The The velocity velocity of of aa dart dart leader leader is is close close to in air air - aa rapid rapid The being non-viable non-viable in The principal principal reason reason for for aa streamer streamer channel channel being is very very weak weak in in this this case. case. A dart dart loss loss of of conductivity conductivity by by the the cold cold plasma plasma -- is leader preceding component, component, so so that that the the leader follows follows the the track track heated heated by by the the preceding still-hot track serves as a kind of waveguide to the leader. The high gas still-hot track serves as a kind of waveguide to the leader. The high gas temperature Therefore, the the possibility possibility for for the the temperature greatly greatly retards retards electron electron losses. losses. Therefore, region temperatures increases increases considerably. considerably. region behind behind the the tip tip to to be be heated heated to to arc arc temperatures in aa ‘classical’ 'classical' leader. leader. This This provides provides aa stable stable higWy highly conductive conductive state state inherent inherent in The probably more more important, important, funcfuncThe preheated preheated air air pipe pipe serves serves another, another, probably tion. Its Its hot hot and and rarefied rarefied air air is is surrounded surrounded laterally laterally by by cold cold dense dense air, air. Since Since tion. the rate rate of of ionization ionization due due to to the the field field is is described described by by the the E / N ratio, ratio, the the the behind the the streamer streamer tip tip is is abruptly abruptly radial radial expansion expansion of of the the channel channel region region behind motion of of the the ionization ionization wave. wave. So So retarded retarded as as compared compared with with the the forward forward motion the is reduced, reduced, permitting permitting the the channel channel the air air mass mass to to be be heated heated by by the the current current is gas cold air air restricts restricts the the channel channel gas to to be be heated heated to to aa higher higher temperature. temperature. The The cold by the the streamer streamer zone zone expansion produced by expansion because because it it acts acts as as aa charge charge cover cover produced of the the leader. leader. of One should should not not think think that that the the channel channel does does not One not expand expand through through the the ionization mechanism mechanism at at all. all. This This process is just ionization process is just much much slower slower than than the the forward motion motion of of an an ionization ionization wave wave towards towards the the earth, earth, so so most most of of the the forward Joule heat heat is is released released into into the the yet yet unexpanded unexpanded channel channel having having aa smaller smaller Joule radius. The The radial radial field field leads leads to to the the channel channel expansion expansion only only at at the the beginning, beginning, radius. as is is the the case case with with common common streamers streamers (section (section 2.2.2). 2.2.2). When as When the the radial radial field field is is somewhat reduced, the channel becomes the source of a radial streamer somewhat reduced, the channel becomes the source of a radial streamer corona which which does does not not require require aa high high field. field. Radial Radial streamers streamers rapidly rapidly lose lose corona their conductivity conductivity in in cold cold air, air, and and their their immobile immobile charges of their charges form form aa cover cover of the type type that that surrounds surrounds aa common common leader leader channel. channel. Now, the Now, though though with with some some delay, the the mechanism mechanism of of radial radial field field attenuation attenuation and and hot hot channel channel stabilization stabilization delay, comes into into action. action. Thus, Thus, aa dart dart leader, leader, being comes being essentially essentially aa streamer streamer (i.e., (i.e., an an ionization wave wave having having no no streamer streamer zone zone in in front front of of the the tip), tip), must must possess possess ionization charge cover, cover, as as aa leader. leader. Unlike Unlike the the case case with with aa common common leader, leader, the the aa charge cover is is not not inherited inherited from from aa streamer streamer zone zone but cover but is is formed formed entirely entirely behind behind the tip, tip, which which is is the the seat seat of of the the principal principal processes processes driving driving the the dart dart leader. leader. the (In aa classical classical leader, leader, the the cover cover formation formation partly (In partly continues continues behind behind the the tip, tip,
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The problem of of a dart leader Subsequent components. The
209 209
as as described described in in section section 2.2.4.) 2.2.4.) The streamer mechanism of the dart leader leader development development due due to to impact impact The streamer mechanism of the dart ionization of of the the gas gas in in the the strong strong field field of of the the tip tip can can manifest itself only only if if the the manifest itself ionization conductivity in in the the channel channel of of the the previous component has has dropped dropped below conductivity previous component below aa critical value value by the time time the the next next lightning lightning component component is is to to arise. arise. This This is is critical by the unambiguously supported supported by the following following observations. observations. The The M-component M-component unambiguously by the is produced against the the background of aa continuous continuous current current of of the the interinteris produced against background of pause, whereas whereas the the dart dart leader leader arises arises some some time time after after this current is is cut cut off. off. pause, this current As long as as the medium preserves preserves aa high high conductivity, conductivity, the the diffuse diffuse penetration penetration As long the medium of the the field field and and current current prevents the ionization ionization wave wave propagation. propagation. The The diffudiffuof prevents the sion wave wave has has practically no ionization ionization due due to to aa direct direct action action of of the the low low field. field. sion practically no The medium medium pre-ionization does not not stimulate stimulate but rather hampers hampers the the propapropabut rather The pre-ionization does gation of of the the ionization ionization wave. wave. The The latter latter requires strong field, field, but the high high gation requires aa strong but the conductivity of of the the medium medium in in front front of of the the wave wave leads leads to to the the field field dissipation. dissipation. conductivity In order order to to focus focus the the potential potential drop drop to to aa narrow narrow region, region, one one must must stop stop the the In charge flux flux (electric (electric current) current) by by concentrating concentrating charge charge in in aa narrow narrow region region to to charge produce aa strong strong field. field. The The charge charge flux flux can can be be 'locked ‘locked in' in’ only only by by creating creating produce resistance to to it it if if one one places places aa poor poor conductor conductor in in front front of of the the well-conducting well-conducting resistance portion of of the the channel. channel. portion 4.8.2
The non-linear non-linear diffusion diffusion wave front front
At this this point, point, we we have have to to make make aa short short digression digression to to discuss discuss the the structure structure of of At the the near-front near-front region region of of aa diffusion diffusion potential potential wave. wave. One One will will see see later later that that this this is is directly directly related related to to the the ionization ionization wave wave problem. problem. The The diffusion diffusion wave wave velocity velocity vw is is determined determined by by the the propagation propagation process process along along the the whole whole wave wave length. length. Its Its variation variation along along the the path path from from the the cloud cloud to to the the earth earth is is illustrated illustrated in in figure figure 4.24. 4.24. By By order order of of magnitude, magnitude, the the velocity velocity of of aa non-linear non-linear wave wave is is FZ Xav/Xj, x,,/-xf, where where Xj -xf is is its its total total length length from from the the source source to to the the initial initial front front vU ~ point and and Xav xavisis an an averaged averaged diffusion diffusion coefficient coefficient in in the the transformed transformed channel channel point behind the the front, front, which which better better fits fits the the final final linear linear resistance resistance of of the the channel channel behind than than to to its its initial initial resistance. resistance. If If constant constant potential potential U Uii is is applied applied to to the the initial initial channel channel end, end, the the value value of of Xav xavdoes does not not change change much. much. The The velocity velocity changes changes appreciably appreciably over over the the time, time, during during which which the the wave wave covers covers aa distance distance comparcomparable with with that that between between the the cloud cloud and and the the earth. earth. But But its its change change is is relatively relatively able small small over over the the time time the the wave wave covers covers aa distance distance of of the the order order of of its its front front width width where where the the potential potential U(x) U(x) rises rises steeply. steeply. This This means means that that we we have have U(x, U(x, t)t ) ~ M U(x U(x -- vt) wt)in in the the wave wave front, front, and and the the distributions distributions of of all all parameters parameters along the the x-axis x-axis are are quasistationary quasistationary in in the the coordinate coordinate system system related related to to the the along moving [12]). With With this this circumstance circumstance in in moving front front (as (as in in aa non-linear non-linear heat heat wave wave [12]). mind, mind, we we can can rewrite rewrite the the potential potential diffusion diffusion equation equation (4.35) (4.35) as as dU d dU -V---Xdx-dx dx'
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(4.47) (4.47)
210 210
Physical processes in a lightning discharge
Taking E = = -dU/dx -dU jdx = = 00 and and U U= = 00 in in front front of of the the wave wave at at Taking into into account account E x -> of this this equation equation is is -+ 00, m, the the integral integral of
vU =
dU
E
.
-x dx '
1=
R = C I Uv =
(4.48) (4.48)
TV.
I
The valid at at every every point point of of the the quasistationary quasistationary The familiar familiar relation relation ii = = TV rv is is valid of the the current current cut-off cut-off in in front front of of the the wave wave portion portion but but not not only only at at the the site site of streamer streamer or or leader leader tip. tip. of the the quasistationary quasistationary channel channel is is The W1 per per unit unit length length of The energy energy WI described described as as dWj
dt
=
_v dWI
dx
=
2
i2RI
2
(4.49) (4.49)
= v Cj U X
and through the the local local potential. potential. Indeed, Indeed, reducing reducing the the and is is expressed expressed directly directly through rank (4.48) and and (4.49) (4.49) by by dividing dividing them them by by one one rank of of the the set set of of equations equations (4.48) another, another, we we find find
w, - w,o = c1u2/2
(4.50) (4.50)
the channel channel far far out out the the wave wave front. front. Thus Thus the the where where W Wlo is the the initial initial energy energy in in the lO is statement that the the energy energy dissipated dissipated in in the the statement repeatedly repeatedly used used in in evaluations evaluations that in its its capacitance capacitance is is valid valid exactly exactly channel channel is is of of the the same same order order as as that that stored stored in in in the the stationary stationary case. case.tt We when the the gas gas is is heated heated at at constant constant We shall shall consider consider moderate moderate waves, waves, when 2 pressure, released at at constant constant mass mass m = = r7rrr 22 Pp = = mopo 7rr6Po pressure, and and the the Joule Joule heat heat is is released per the initial initial radius radius and and gas gas density density in in per unit unit channel channel length length (ro (yo and and pP are are the is the the specific specific gas gas front front of of the the wave). wave). Then Then we we have have WI W1 = = mh, mh, where where h is enthalpy. equilibrium ionizaionizaenthalpy. Assume Assume for for simplicity simplicity that that thermodynamically thermodynamically equilibrium that conductivity conductivity 0 CJ and and tion tion is is established established at at every every point point of of the the wave, wave, so so that X= I) -I are Then X is is = mCJ(pC ma(pC1)-’ are the the functions functions of of temperature temperature T or or hh((T). T ) . Then unambiguously With (4.48)-(4.50), (4.48)-(4.50), unambiguously related related to to U through through formula formula (4.50).t (4.50).$ With finding the the distributions distributions along along the the wave finding wave reduces reduces to to the the quadrature quadrature
x
x
1J
X(h)dh __ 2vx, h - h -= -2vx.
2m(h - h,)
]
1’2
_ W m
(4.51) (4.51) o Let us us calculate calculate the the integral integral by p XE ::::: oCJ /j P Let by approximating approximating the the relationship relationship x n by the the power power function function X x == Ah Ahn in the the temperature temperature range by in range typical typical of of the the wave. wave. =0 0 is is taken taken at at an an arbitrary arbitrary point The coordinate coordinate origin origin xx = The point of of the the wave wave front front U=[
cl
.
lO ho=-. mlo h0 -w '
tt This the channel is not not created created anew anew but but This isis quite quite natural natural because because under under the the problem problem conditions conditions the channel is exists from from the the very very beginning beginning with with its its linear linear capacitance capacitance C j, ,. Then exists Then every every channel channel portion portion is is (2.17) in section 2.2.4). charged as as lumped lumped capacitance capacitance (cf, (cf. the the comment comment on o n formula formula (2,17) charged in section 2,2,4), sections 4.7 4.7 and and 4,4, 4.4, the the quantity quantity X x was was related related to electrical electrical parameters t$In In sections parameters through through relation relation (4.38)which which refers refers to to strong strong waves waves with with aa high high energy energy release, release. If If desired, desired, one (4,38) one can can use use this this relation relation aG/ar by by -vdC/dx -U dC/dx and and doing doing the the above above operations, operations. after substituting substituting aGio! after
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components. The The problem of a dart leader leader Subsequent components.
211 21 1
start, in in front front of of which which (x> (x > 0) 0) the the channel channel transforms transforms very very slightly, slightly, so so that that X x start, increases slightly, slightly, followed followed by by (x (x < < 0) 0) where where itit changes changes noticeably. noticeably. The The increases parameters of of the the initial initial front front point point will will be be marked marked by by the the subindex subindex I,1, assumassumparameters x1== 2Xo, 2x0, where where Xo xo corresponds corresponds to to the the initial initial channel channel ing for for definiteness definiteness XI ing 1 n conductivity. Then Then we we have have hi h l -- hho Sho, where 8S == 22l/" 1. An An exponenexponen/ -- 1. conductivity. o == 8h o, where tially damping damping tail tail of of the the electric electric field field and and current current extends extends forward forward along along the the tially wave where where the the diffusion diffusion is is 'linear': 'linear': wave U E (x ) U = E = exp - .6.x' I
I
Xo .6.x = -;'
E I = _UI = vU. .6.x Xo
(4.52)
Within the the front, front, where where h exceeds exceeds hhoo considerably considerably or, or, asymptotically asymptotically at at Within + -00, -CO, we we have have xx ----t U
X::::: 2nv( -x),
E
= 2n( -x)' (4.53)
By matching matching the the approximate approximate solutions solutions asymptotically asymptotically valid valid at at xx By
+xand and
+ +00 ----t
-+ -00, -CO, the the parameters of the the front front start start and and the the matching matching point coordixx ----t parameters of point coordi-
nate x1 can be be found found as as XI can nate
U1
[($)r5(9)1'n]1'2,U1 '
Ax x1 = -. 2n
El =2nxl
(4.54) (4.54)
position of This point is This point is closer closer to to the the a priori position of the the front front start start xx == 00 than than .6.x, Ax, which which justifies justifies the the approximations. approximations. Let us us illustrate illustrate this this situation situation numerically numerically with with reference reference to to the the conditions conditions Let of the M-component (figure 4.25). Suppose the diffusion wave has has typical typical of the M-component (figure 4.25). Suppose the diffusion wave 8 velocity v = 10 mjs running along a channel with the initial radius ro = I em, velocity v = 10' mjs running along a channel with the initial radius = 1 cm, 5 3 temperature To To = = 5900 5900 K K (h, (h o = = 14.8 kJjg) and and po Po = = 5 5 xX 10gjcm which which temperature 14.8 kJ/g) lop5 g/cm3 4 is by by aa factor factor of of 25 25 less less than than the the normal; normal; m m= = 1.54 gjcm; 1.54 xx 1010-4g/cm; is 10 2 3, the initial linear resistance R = 10 [ljm, Xo = 10 10 14 n ::::: 1.8x 10 cmm /s js nee 1 . 8 10'4cm-3, ~ the initial linear resistance R1I = 10R/m, xo = 10 m and C1 CI = = 10pF/m. IOpFjm. For For the the temperature temperature range range T T::::: 6-10000K in in air air at at and x 6-10000K 3 1 we have have aa'/p::::: 17h 3 (where a[([l.cmr ], p[g/cm3], p[gjcm ], and and h[kJ/g]). h[kJjgJ). (where .[(a 1latm, atm, we l p = 17h3 cm)-'], 2 6 Hence, A = = 2.7 2.7 xX lo6 10 (m*/~)(kJ/g)-~ (m js)(kJjg)-3 and and S8 = = 0.25. 0.25. From From formulas formulas (4.51)(4.51)Hence, (4.53), kV/cm, and we find find for for the the initial initial front front point point U1 UI = = 3.5 3.5 MV, MV, El EI = = 2.2 2.2kVjcm, and (4.53), we a
-e -x/Ax x, 0 A x
x ..
Figure Figure 4.25. 4.25. Schematic Schematic diagram diagram of of the the nonlinear non-linear diffusion diffusion wave wave front. front.
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212 212
Physical processes in a lightning discharge
the m. The the effective effective field field length length before before the the wave wave front front ~x Ax == 100 100m. The point point behind behind the U == 10MV lOMV (h (h ~ M 50kJ/g, 50 kJ/g, TT ~ M 10000K) 100OOK) lies lies at at aa distance distance the wave wave with with V 2 xx == 500 m from lOll m /s, the 500m from the the front. front. There, There, X x ~M 3x 3x10" m2/s, the resistance resistance is is by by aa factor 30 lower lower than than before before the the front, front, and and the the field field drops drops to to 33 33 V/cm. V/cm. The The factor of of 30 field field maximum maximum lies lies near near the the initial initial front front point. point. The The qualitative qualitative picture picture presented presented in 4.24(a). in figure figure 4.25 4.25 agrees agrees with with the the numerical numerical results results of of figure figure 4.24(a). 4.8.3
The possibility possibility of diffusion-to-ionization diffusion-to-ionizationwave wave transformation transformation The
Let Let us us define define the the conditions, conditions, under under which which aa diffusion diffusion wave wave can can transform transform to to an an ionization ionization wave wave which which is is supposed supposed to to be be aa dart dart leader. leader. Consider Consider aa simple simple situation. U, is is applied applied to to the the upper upper end end of of aa situation. It It is is suggested suggested that that potential potential Vi grounded conductive conductive channel channel of of the the previous previous lightning lightning component. component. It It grounded begins to to diffuse diffuse into into the the channel. channel. It It is is assumed assumed that that there there is is no no transfortransforbegins mation and and the the initial initial conductivity conductivity corresponding corresponding to to the the diffusion diffusion coefficient coefficient mation xo isis preserved. preserved. The The diffusion diffusion is is 'linear' 'linear' in in this this case. case. The The potential potential and and field field Xo vary as as vary
V = Vi [1 - erf(
iR1 = E = iR I =
Vi
1/2
(..xot) lI2 (71"Xot)
x (4Xot)
1/2)]'
(- A). 4x0 x2
exp (_ 4 exp
Xott
(4.55) (4.55)
).
At every every point point x, x, the the field field first first rises rises with with time time but but then then falls falls after after the the At 2/2Xo. The E,,max == 2(71"e)-1/2 2(7re)-li2U,/x at moment moment t == xx2/2x0. The point point E E,,max maximum E maximum Vi / X at moves at at velocity velocity vwg = Xo/ xo/x, and the the potential potential at at this this point point is is V Um, == 0.33 0.33U1. moves x, and Vi' g = An ionization ionization wave wave can can be be formed formed if if the the maximum maximum field field is is sufficiently sufficiently An E,. The ionization ionization wave wave is is assumed assumed high and and exceeds exceeds aa certain certain critical critical value value E high j • The supposed to to be be equal equal to to that that of of aa dart dart leader. leader. Since Since to propagate propagate at at velocity velocity Vw,s supposed to I 2 / , the E E,,max '" x-I", x-l ttC1I2, the ionization ionization wave wave could could principally principally arise arise at at earlier earlier E,,max > E E,, but if if its its velocity velocity is is VU,s < < vwg, is immediately immediately overcome overcome times when when E times i , but g , is by aa diffusion diffusion wave. wave. This This will will not not happen happen if if vwg drops below below VU,s while while E E,,max is is by g drops E,, i.e., i.e., if if the the conditions conditions Vvs 3 vwg, E,,max ?: 2E E,i are are fulfilled fulfilled still higher higher than than E;, still s ?: g, E together. together. For For this this to to happen, happen, the the diffusion diffusion coefficient coefficient must must be be smaller smaller and and the the linear linear channel channel resistance resistance larger: larger: N
N
1 R 1cr = - - ' C 1XOcr
(4.56) (4.56)
For this, this, the the gas gas temperature temperature in in the the initial initial channel channel should should not not be be high. high. On On For the the other other hand, hand, for for the the 'waveguide' 'waveguide' properties properties to to manifest manifest themselves, themselves, the the temperature temperature must must be be as as high high as as possible possible to to make make the the air air rarefied. rarefied. Because Because of of the the very very sharp sharp temperature temperature dependence dependence of of conductivity conductivity (when (when it it is is low), these these conditions conditions are are met met only only in in aa very very short short temperature temperature range, range, low), T T~ M 3000-4000 3000-4000K, K, where where the the air air density density is is by by aa factor factor of of 10-15 10-15 lower lower than than
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of a dart leader Subsequent components. The problem of
213 213
normal. normal. For For the the estimations, estimations, we we take take Ei Ei = = 33 kV/cm kVjcm corresponding corresponding to to aa field field characteristic of initial initial air air ionization, ionization, 30-40 30-40 kV/cm kV jcm under under normal normal conditions. conditions. characteristic of Vs = = 107m/s, 107 mjs, Ui Uj = = 5MV 5 MV (such (such potential potential usually usually provides provides current current Suppose w, Suppose 1 ~ lOkA for the next component at Z = 500n during the return stroke), IM M lOkA for the next component at 2 = 500R during the return stroke), M and C, = = 10pFjm. We find XOcr x ~ l.lx10sm2/s l.lx10 8 m 2 js and and Rlcr R lcr == 880njm. The 10pFjm. We find xOcr 880R/m. The and C1 resistance is is two two orders magnitude higher higher than than that that supposed to resistance orders of of magnitude supposed to precede the the M-component. M-component. For For this this reason, reason, aa dart dart leader appear only precede leader can can appear only after the current cut-off during the interpause partial cooling the after the current cut-off during the interpause and and aa partial cooling of of the channel. channel. In reality, reality, the the channel channel undergoes undergoes transformation transformation due due to to the the diffusion diffusion In wave, its its conductivity conductivity rises, rises, and and the the field field dissipates dissipates faster faster than than what what is is wave, expected from from the the second second formula formula of of (4.55). (4.55). To To provide provide for for the the critical critical conconexpected ditions, ditions, the the initial initial conductivity conductivity may may seem seem to to be be lower lower than than the the estimated estimated value. So we we should should consider consider the the other other extremal extremal case case when when the the diffusion diffusion value. of air air and and an an equilibrium equilibrium ionization ionization is is estabestabwave heats heats aa limited limited amount amount of wave lished. The diffusion wave is now non-linear. Its maximum field is near the lished. The diffusion wave is now non-linear. Its maximum field is near the initial point of of the the wave wave front, front, and use, instead instead of of (4.56), the last initial point and one one should should use, (4.56), the last relation of of (4.52) similar to to it with E, One should keep in in mind mind that that of relation (4.52) similar it with El == E Ei. should keep of j • One interest are the the temperatures temperatures 3000-6000 K, at which 17 varies with with T much much interest are 3000-6000K, at which 0 varies 6 9 more strongly: ~ 1.8 h (the dimensionalities are are the the same as in in more strongly: 171P alp M 1.8 xX 1O10-6h9 (the dimensionalities same as 7 4.8.2). Now Now we we have have nn == 9; at U, Vs = mls and and the illustration illustration of the of section section 4.8.2). 9; at = 10 107m/s 7 2 ro = = 1cm, we have have U, = 1.2MV, Xo = = 4x 4xl0 m js, and and channel radius radius yo 1 cm, we U 1= 1.2MV, xo lo7m2/s, channel R 'Icr, M~ 2500 2500 anjm. The field extends before the wave front only for 6.x = 4 m. / m . The field extends before the wave front only for Ax = 4m. The electron density in the initial channel under critical conditions The electron density in the initial channel under critical conditions is is ne x ~ 6x 6xl10l2 012 ~ cmm-3,- corresponding corresponding to its its temperature temperature 4000 4000 K. K. ne ~, to 4.8.4
The ionization ionization wave in a conductive conductive medium
The The values values obtained obtained in in section section 4.8.3 4.8.3 on on two two extremal extremal assumptions assumptions do do not not differ to be be reasonable. reasonable. The problem of the conditions differ much much and and seem seem to The problem of the conditions necessary for to arise arise may may seem to have have been been solved. necessary for aa dart dart leader leader to seem to solved. This This optimism will, will, however, however, disappear one evaluates evaluates the the parameters parameters optimism disappear as as soon soon as as one of an an ionization wave when when it it propagates propagates through through aa medium medium with with critical critical of ionization wave conductivity. conductivity. Consider aa wave wave in in the the front-related was done Consider front-related coordinate coordinate system, system, as as was done in in section section 4.8.2. 4.8.2. The The equation equation for for field field (4.48) (4.48) will will be be supplemented supplemented by by an an ionizaionization X because tion kinetics kinetics equation equation written written directly directly for for x because X x '" 17CT '" nne: e: N
-U-
dX = uix, dx
vj Vi
= Nf(EIN). Nf(E/N). =
N
(4.57) (4.57)
This This equation equation describes describes aa new new law law for for the the channel channel transformation. transformation. Owing Owing to to (4.48), 4 turns turns to to the the potential potential function. function. Then, Then, by by (4.48), the the ionization ionization frequency frequency Vi X(U) dividing U) dividing (4.48) (4.48) and and (4.57), (4.57), the the problem problem is is reduced reduced to to the the equation equation for for x( and and the the quadrature, quadrature, as as in in section section 4.8.2. 4.8.2.
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214 214
processes in in a lightning discharge Physical processes
To To advance advance further, further, one one should should choose choose the the functionf(EIN) functionf(E/N) in in aa way way suitable suitable for for integration. integration.1t But But difficulties difficulties and and doubts doubts arise arise immediately immediately k , as here. vi by by the the power power function function Vi vi == bE bEk, as in in the the here. The The approximation approximation of of Vi streamer theory theory when when this this approximation approximation with with kk = = 2.5 2.5 provided provided fairly fairly streamer good results, results, does does not not work work in in this this case. case. The The ionization ionization wave wave propagating propagating good through aa conductive conductive medium medium appears appears absolutely absolutely diffuse, diffuse, as as in in any any other other through channel channel transformation transformation law: law: (4.38) (4.38) or or on on the the assumption assumption of of equilibrium equilibrium ionization ionization (section (section 4.8.2). 4.8.2). Let Let us us impart impart aa threshold threshold nature nature to to the the function function 4 = 00 at at E E < E* E* and and Vi 4= = const const at at E > E*. E*.This This is is vi(E) in aa simple simple way way -- Vi Vi (E) in what of [43] [43] when when solving solving aa similar similar problem problem for for what was was done done by by the the authors authors of of (4.48) (4.48) and and the laboratory laboratory ionization ionization wave wave in in aa tube. tube. The The integration integration of the (4.57) by by the the above above method method yields yields the the following following result. result. The The change change in in the the (4.57) x inin the the ionization ionization wave wave is is defined defined by by the the ratio ratio of of electron density density and and X electron potentials U22 and and U U11 at at the the points points of of ionization ionization outset outset and and onset, onset, where where potentials U E == E*. E*. A potential potential 'tongue' 'tongue' of of effective effective length length ~x Ax == Xolv xo/uextends extends in in front front E of the the ionization ionization wave, wave, as as in in other other diffusion diffusion modes. modes. The The potential potential at at the the of front ~x. The = E* E*Ax. The parameter parameter ratios ratios at at the the wave wave boundaries boundaries are are front is is U 11= ne2 X2 U2 neo = Xo = U1 ::::; exp
v (ViXO2) .
(4.58) (4.58)
This relation relation can can be be regarded regarded as as the the dependence dependence of of the the wave wave velocity velocity on on an an This U2 applied to to its its back. back. On On the the other other hand, hand, the the velocity velocity 'external potential' potential' U 'external 2 applied is expressed expressed by by aa formula formula similar similar to to (2.2) (2.2) for for the the streamer: streamer: is V
=
l/~~ 1
1.
In(X2Ixo) .
~
~Xi = 2A' vi I..l.X
-
~
ViXO
>1
(4.59)
Axi is is the the extension extension of of the the ionization ionization region region from from the the initial initial to to the the final final where ~Xi where point of of the the wave. wave. For For aa wave wave to to survive, survive, its its parameters parameters must must meet meet the the last last point of (4.59). (4.59). Otherwise, Otherwise, the the field field within within the the wave wave will will be be unable unable to to inequality of inequality E*, so so no no ionization ionization will will occur. occur. exceed E*, exceed The capabilities capabilities of of an an ionization ionization wave wave are are limited, limited, and and this this limit limit increases increases The with increasing increasing initial initial conductivity conductivity of of the the medium. medium. For For example, example, if if the the initial initial with 12 7 2 ne0 M 10 lOl2cmP3 and x0 x 10 107m2/s were even even lower lower than than the the parameters neo::::; parameters cm- 3 and Xo::::; m Is were kV/cm, critical values values found found in in section section 4.8.3 4.8.3 and and if if the the threshold threshold field field was was 33 kV/cm, critical 6 it would would be be necessary necessary to to have have the the ionization ionization frequency frequency Vi 4 == 2.1 2.1 Xx 10 lo6 s-' S-1 it and potential potential U U2 = 300MV in order to increase ne and x by three orders of and = 300 MV in order to increase n and X by three orders of 2 e (Ax = 1m, 1 m, U I1 = 0.3 0.3 MY) MV) and and U U2 30 MV by two orders. The magnitude (~x magnitude = 30 MV by two orders. The 2 wave width width in in this t h s case case is is ~Xi Axi ::::; E 22 22m, i.e., it it is is very very extended. extended. Only Only when when m, i.e., wave Sometimes, it it seems seems better better to to describe describe !lj vi the the function function of of E and, and, on on the the contrary, contrary, to to remove remove U t Sometimes, from (4.48) (4.48)and and (4.57). (4.57). Instead Instead of of (4.48), (4.48), we we then then get get from
~~=E(;-~),
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!lj=!lj(E)
Subsequent components. of a dart leader components. The The problem of
215
the lower (nCo (neo = = 10" 1011 cmP3, cm -3, the initial initial conductivity conductivity is is still still an an order order of of magnitude magnitude lower 6 2 5 Xo Is, and wave width width begins begins to to approach approach what what xo == 10 lo6m m2/s, and R R= = 10 lo5Dim), fl/m), the the wave the same medium at at the the same same Uv would would be be desired desired for for aa dart dart leader. leader. In In the same medium and the ratio ratio xX2/XO = lo3 103 would would be be and E*, E*, the the parameters parameters necessary necessary for for the 2 / x 0= 7 IIi = 30 MV, D.x = 10 cm, D.xi = 5 m, and U\ = 30 kV. vi = = lAx 1 . 410 lo7 ~ S-I, sP1, U U2 = 30 MV, Ax = 10 cm, Axi = 5 m, and U , = 30 kV. 2 A rise would would be be obtained obtained at at aa still still A still still narrower narrower region region of of the the potential potential rise the applicability applicability limits limits of of lower lower initial initial conductivity. conductivity. But But then then we we approach approach the the propagation in in aa conductive conductive the basic basic concepts concepts of of the the theory theory of of perturbation perturbation propagation we are probably coming coming closer closer to to the the medium medium and and of of the the long long line line theory, theory, and and we are probably production. understanding understanding of of criteria criteria for for the the dart dart leader leader production. 4.8.5 4.8.5 The dart dart leader as a streamer in a 'nonconductive waveguide'
The channel, or or in in aa long long line, line, is is The diffusion diffusion mechanism mechanism of of field field evolution evolution in in aa channel, incompatible hence, with with strong strong fields. fields. If If incompatible with with abrupt abrupt potential potential changes changes and, and, hence, abrupt by diffusion. diffusion. We We believe believe abrupt changes changes do do arise, arise, they they are are rapidly rapidly smeared smeared by for wave nor nor aa dart dart leader leader can can for this this reason reason that that neither neither aa narrow narrow ionization ionization wave be find the the conditions, conditions, in in which which aa be formed formed in in aa well-conducting well-conducting channel. channel. To To find remind ourselves ourselves of of the the prerequiprerequivery very strong strong field field can can be be induced, induced, we we should should remind sites for for the the long long line line equations. equations. sites The geometry has has the the form: form: The electrostatics electrostatics equation equation for for cylindrical cylindrical geometry
-+--rE dEx 1 d -P aEx+~~fEr=.f!... (4.60) (4.60) rax r or EO dx r dr EO where p is is space space charge charge density. density. By By integrating, integrating, in where in the the cross cross section, section, aa conducconductor of of radius radius ro ro and and neglecting neglecting the the dependence dependence of of the the longitudinal longitudinal field field E, Ex on on tor we obtain obtain r,r, we
a::
1;
2 8Ex m-0 -+ + 2rrroEro:a 27rroEr, = - , rrr~ 7-
(4.61) r= = LO 27rrpdr T 2rrrp dr (4.61) dX EO where E Er0 is the the radial radial field field on on the the surface surface of of aa conductor where conductor of of length length II > »> yo: ro: ro is E
U
ro ~ ro In(l/ro) ,
C
2rrEo
1
= In(l/ro) .
(4.62) (4.62)
If the the longitudinal longitudinal field field varies varies along along the the channel channel so so slowly If slowly that that the the axial axial diverdivergence can can be be neglected neglected (the (the characteristic characteristic length length for of E, Ex is is gence for the the variation variation of Ax » >> fO), ro), we we arrive arrive at at one one of of the the basic basic conceptions conceptions of D.x of the the long long line line theory, theory, ~ ( x= =) C\ C1U(x), U(x), whose whose implication implication is is the the potential T(X) potential diffusion diffusion mechanism. mechanism. It It is suggested suggested implicitly implicitly that that the the resistance resistance varies is varies very very slowly slowly along along the the channel, so so this this variation variation cannot cannot be be an an obstacle obstacle to to aa charge charge flux, flux, making making channel, the flux flux velocity velocity decrease decrease abruptly abruptly and and create create aa space space charge charge due due to to its its the local accumulation accumulation (a (a long long line line has has no no 'jams'). 'jams'). local However, space space charge charge does does accumulate accumulate at at aa sharp However, sharp boundary boundary between between aa poorly- and and aa well-conducting well-conducting channel channel portion. is formed formed at at poorlyportion. A charged charged tip tip is the end end of of an an ideal ideal (or (or non-ideal) non-ideal) conductor, conductor, the the potential potential in in front front of of it it the
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Physical processes in a lightning discharge
drops drops abruptly, abruptly, at at distances distances about about equal equal to to ro, ro, inducing inducing there there aa strong strong field field capable capable of of sustaining sustaining an an ionization ionization wave. wave. This This is is what what happens happens in in aa common medium. It what is common streamer streamer in in aa non-conductive non-conductive medium. It is is then then clear clear what is necessary to the ionization necessary to support support aa sharp sharp potential potential drop drop at at the ionization wave wave front front for for aa long the perspective perspective trajectory long time. time. The The conductivity conductivity along along the trajectory must must drop drop to to aa value be unable unable to the value low low enough enough for for the the diffusion diffusion field field tongue tongue to to be to smear smear the sharp potential drop. sharp potential drop. Therefore, Therefore, the the tongue tongue length length must must become become comparable comparable with the the channel channel radius radius 6.x Ax ~ x xo/v N roo yo. Because Because of of the the strong strong temperature temperature with Xolv '" dependence dependence of of the the degree degree of of equilibrium equilibrium ionization ionization in in air air at at low low temperatures, temperatures, aa drop K would would be be sufficient. drop to to T ~ x 3000 3000K sufficient. The The equilibrium equilibrium electron electron density density 10 3 established for for the the long long zero-current zero-current pause pause will will be be neo'" neo 10 10'o-lO1l cmP3; _10 11 cmestablished ; 5 2 6 5 6 hence, RIO'" _10 ll/m, and Xo '" 10 _10 m Is. But Rlo 10 105-1060jm, and xo 106-105m2/s. But the the air air density density in in hence, the cooled cooled channel channel of of the the previous previous component component at at T ~ x 3000 3000K is by by an an order order the K is of that the of magnitude magnitude lower lower than than that that of of cold cold air, air, so so that the conductivity conductivity drop drop will will not interfere properties of not interfere with with the the 'waveguide' 'waveguide' properties of the the track. track. The The velocity velocity of of aa dart dart leader leader as as an an ionization ionization wave wave is is defined, defined, in in order order of the streamer of magnitude, magnitude, by by the the same same formula formula (2.2) (2.2) as as the streamer velocity. velocity. But But the the 10 'pre-ionization' this case _10 11 cm -3) is 'pre-ionization' in in this case (neo (nee '" 10 10'o-lO'l cmP3) is considerable, considerable, and and aa x 5) 5 ) is is to much smaller smaller number number of of electron electron generations generations (In(n (1n(ne2/neo) to be be much e2In eo) ~ produced in the the wave. wave. With With the the account account of of the similarity law law for for Vi vi at at produced in the similarity an an order order of of magnitude magnitude lower lower gas gas density, density, the the ionization ionization frequency frequency is is Vi '" 1010 we obtain vi 10" S-l sC1 and and ro ro '" 1cm; 1 cm; then then we obtain aa correct correct order order of of the the velocity velocity qro/ In(ne2/ne0) io77 m/s. mjs. vU = ~ viral In(n e2Ineo) '" 10 One cannot cannot say say that that all all the the details details of of the the dart dart leader leader behaviour have been been One behaviour have clarified by the the dart clarified by the above above considerations. considerations. For For the dart leader leader channel channel to to be be well well conductive, the electron density density in in it it must must be at least least 5-6 5-6 orders orders of of magnitude magnitude conductive, the electron be at higher than than the initial value for the the track. track. But But the the capabilities capabilities of of the the ionizaionizahigher the initial value for tion wave wave to to produce more electrons electrons are are limited. limited. The The maximum maximum conductivity conductivity tion produce more of an an ionization ionization wave wave propagating propagating through through aa non-conductive non-conductive medium medium is is of defined, in in order order of of magnitude, magnitude, by the relation relation umaxl ~ F co~ '" ~Vi vi (section (section ~ / E 2.2.2), 2.2.2), ~ defined, by the because space charge charge of of the the streamer streamer tip, tip, providing strong ionization ionization because the the space providing aa strong field is is dissipated dissipated with the Maxwellian Maxwellian time time TM == col Eo/cF.t After the the wave wave with the u. t After field
-
-
-
-
-
--
-
It also also determines determines the the rate rate at at which the linear linear charge charge 7T == C C1 U is is established, established, if if it it is, is, in in the the t It which the j U channel. Let Let us us integrate integrate the relation for for charge charge conservation conservation in in the conductor cross cross section. section. channel. the relation the conductor along the the channel channel length, length, we we obtain obtain Neglecting, for simplicity, simplicity, the variation in in a0 along Neglecting, for the variation op
oE _ 0
at + aa;x - ,
07
2
oE,
at + 1rroaa; = O.
Using (4.61) we arrive relation between between 7T and Using (4.61) and and (4.62), (4.62), we arrive at at aa refined refined equation equation for for the the relation and U: U: co 07
--; at + 7
= C j U.
The postulate of the the long long line line theory, theory, 7T = = C j, U, U, is is valid valid if if the changes in in the the system, system, which which also also The postulate of the changes ~ ( t )occur occur , slower than than with T ,= = ~ co/a. co/u. When When applied applied to to the wave front front moving moving at at define 7(t), slower with 7M the wave define in aa line line with with conductivity conductivity aD, uo,this happens at at aD 00 » >> vco/ro u i o / r o and and xo >> vro. vr0. velocity velocity vU in this happens Xo »
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Experimental checkup of of subsequent component theory
217
has and ionized, ionized, but but both both proprohas passed, passed, the the channel channel still still needs needs to to be be heated heated and cesses as in in aa classical classical leader leader channel. channel. cesses are are to to occur occur in in aa moderate moderate electric electric field, field, as Besides, radial field field makes makes the the channel channel Besides, this this must must take take place place before before aa strong strong radial become enveloped enveloped by by expand expand beyond beyond the the hot hot gas gas tube, tube, or or if if it it has has already already become aa stabilizing stabilizing charge charge cover cover (section (section 4.8.1). 4.8.1). There the processes processes in in aa dart dart leader leader that that There are are still still many many questions questions about about the remain quantitative theory theory is is also also aa remain to to be be answered; answered; the the development development of of its its quantitative task task of of further further research. research. To features of of aa current current To conclude, conclude, it it is is worth worth noting noting some some specific specific features components. Generally, Generally, the the impulse impulse in in the the return return stroke stroke of of subsequent subsequent components. takes the the return return stroke stroke to to run run impulse impulse duration duration is is related related to to the the time time it it takes components, this this time time must must along along the the whole whole channel. channel. For For the the subsequent subsequent components, be to the the attached attached intercloud intercloud be longer longer than than for for the the first first component component due due to the subsequent components is is about about leader. leader. But But the the impulse impulse duration duration in in the subsequent components twice wave velocities velocities are are generally generally the the same. same. twice as as short, short, although although the the return return wave The to be be the the absence absence of of branches branches in in aa The reason reason for for this this difference difference is is likely likely to relatively slow slow process process of of their their reredart dart leader. leader. It It is is quite quite possible possible that that the the relatively charging elongates the current impulse tail of the first component. The charging elongates the current impulse tail of the first component. The do not not reverse reverse the the sign, sign, similarly similarly impulses impulses of of the the subsequent subsequent components components do to branches, the the action action of of the the reflected reflected to those those of of the the first first one. one. In In the the absence absence of of branches, the randomly randomly reflected reflected waves waves of of the the wave wave can can no no longer longer be be screened screened by by the of ‘white 'white noise’ noise' should, should, numerous hypothesis of numerous branches branches (section (section 4.4.5). 4.4.5). The The hypothesis probably, This problem, problem, like like the the others others probably, be be discarded discarded as as being being inadequate. inadequate. This above, awaits awaits its its solution. solution. above,
4.9
Experimental component theory Experimental checkup checkup of subsequent subsequent component
The theoretical theoretical treatment treatment of of processes processes occurring occurring in The in the the channel channel of of the the previous component component has has been been reduced reduced to previous to the the various various wave wave propagation propagation modes -- the the diffusion diffusion mode mode in in the the M-component M-component and modes and the the ionization ionization wave wave mode in in the the dart dart leader. leader. The The former former has has aa strongly strongly elongated mode elongated front front with with aa slowly varying varying potential, potential, and and the the latter latter must must possess slowly possess aa tip tip with with aa concentrated concentrated charge, producing producing an an abrupt abrupt potential potential change. change. Indirect charge, Indirect evidence evidence for for the the significant difference in the field distribution is the registrations of current significant difference in the field distribution is the registrations of current impulses at at the the earth. earth. The The impulse impulse front front durations durations are impulses are found found to to differ differ by by 2-4 orders of magnitude between an M-component and a dart 2-4 orders of magnitude between an M-component and a dart leader. leader. There is is aa possibility possibility for for aa direct direct experimental experimental evaluation There evaluation of of the the potential potential distribution in in aa wave wave approaching approaching the the earth. earth. This be done done by by measuring measuring distribution This can can be the electric electric field field gain gain at at the the earth earth during during the the wave wave motion. motion. If If the the potential potential the slowly rises rises along along the the whole whole wave wave length, length, as slowly as in in an an M-component M-component (figure (figure 4.24(a)), the the distributions distributions of of the the potential potential and and linear 4.24(a», linear charge charge from from the the initial initial h, to to the the cloud cloud can front point, point, located located at at height height h, can be be considered considered to to be be front linear, T(X) ~ ( x= = ) aq(x a,(x -- h) h ) (x (x isis counted counted from from the 2 h). linear, the earth earth and and xx> h). For For the the
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218 218
Physical processes in a lightning discharge
field at distance distance r from from aa vertical vertical channel. channel, we we find find field at
[
aq
AE(r)= 27rEO
+
+r 2 y 2
]
lnH ( H 2 H-h h (h2 + r2)1'2 ( H 2 r2)1/2
+
+
(4.63 (4.63))
where of the at least, several times where H is is the the height height of the grounded grounded channel. channel. If If H is, is, at least, several times r, the of field A E on on the larger than larger than 1', the dependence dependence of field 6.E the distance distance between between the the registration point and the channel be only registration point and the channel line line will will be only logarithmic. logarithmic. The The same same is is true true of of the the front front duration duration of of aa field field pulse. pulse. The situation situation must must be be quite quite different different for for aa dart dart leader leader with with the the abrupt abrupt The potential potential drop drop at at the the wave wave front, front, since since the the first first approximation approximation in in the the field field calculation calculation may may assume assume aa uniform uniform potential potential along along the the channel channel and and ~ ( x= = ) const const at at xx > h. h. This This gives gives formulae formulae (3.6) (3.6) and and (3.7), (3.7), which which yield yield the the T(X) maximum (r) rv r~]. Such a large difference in the field variation AE,,,(r) r ' . Such a large difference in the field variation maximum value value 6.E max is easily easily detectable detectable experimentally, experimentally, especially especially if if we we remember remember that that it it concerns concerns is not not only only the the field field pulse pulse amplitude amplitude but but also also its its front front rise rise time. time. To To see see that that this this is so, it it is is sufficient sufficient to to introduce introduce into into (4.63) (4.63) and and (3.6) (3.6) the the h-coordinate h-coordinate for for the the is so, wave ut. wave front, front, expressed expressed through through the the respective respective velocities: velocities: h == H -- vt. Triggered lightning lightning is is aa perfect perfect source source for for such such measurements. measurements. A triggered triggered Triggered lightning is is initiated initiated by by launching launching aa small small rocket rocket raising raising aa very very thin thin wire wire which which lightning evaporates during during the the development development of of the the first first component. component. The The point point of of evaporates so itit is is easy easy to to contact of of the the lightning lightning with with the the earth earth is is strictly strictly defined, defined, so contact position current current detectors detectors at at the the necessary necessary distances. distances. Besides, Besides, the the channel channel at at position the earth earth follows follows the the wire wire track track and and is is strictly strictly vertical, vertical, as as is is implied implied in in the the numerical numerical formulae. formulae. Such Such measurements measurements have have been been partly partly made made [44-45]. [44-451. the A E at at distances distances In section section 3.5, 3.5, we we discussed discussed the the measurements measurements of of field field 6.E In r1 == 30 30m and r2 r2 == 500 500m from the the channel channel during during the the dart dart leader leader developdeveloprj m and m from ment. These These measurements measurements were were not not synchronized. synchronized. However, However, the the ratio ratio ment. 17.4 for for approximately approximately equal equal currents currents is is nearly nearly the the AE(30)/AE(500) == 17.4 6.E(30)/6.E(500) r2/r1 == 16.7. 16.7. same as as r2lr] same The field field measurements measurements for for M-components M-components have have been been reported reported only only for for The = 30m 30m [42]. [42]. The The oscillogram oscillogram of of 6.E(t) A E ( t ) is is accompanied accompanied by by aa simultaneous simultaneous I'r = 800A and the the front front registration of of aa current current impulse impulse with with the the amplitude amplitude of of 800 registration A and ps. The The duration duration of of the the impulse impulse front front 6.E A E is is approximately approximately rise time time rv -100 rise 100 /-lS. the same, same, but but the the field field reaches reaches its its maximum maximum of of l350Vjm 1350 V/m earlier, earlier, when when the the the current has has reached reached half half of of its its maximum maximum amplitude amplitude (until (until the the potential potential current wave arrives, arrives, the the current current at at the the earth earth is is zero, zero, whereas whereas the the field field begins begins to to wave rise and rise since since its its start). start). Figure Figure 4.26 4.26 shows shows the the calculated calculated functions functions i(t) and 6.E(t) m and m. The A E ( t ) at at the the observation observation points points with with r = 30 30m and 500 500m. The long long line line C1 = lOpFjm, 10pF/m, model described described in in section section 4.7.1 4.7.1 was was used used with with the the same same C model j = L1j == 2.7 2.7 /-lH/m, pH/m, and and R jI(0) (0) == 10 10 [lIm. njm. The The length length of of the the grounded grounded channel channel L was 4000 4000m and that that of of the the intercloud intercloud leader leader contacting contacting itit was was 2000 2000m. The was m and m. The of 800 800 A was was reproduced reproduced in in the the calculation calculation experimentally observed observed current current of experimentally at the the leader leader potential potential U U,i = = 9.7 9.7MV. Under these these conditions, conditions, the the field field MV. Under at N
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References
219 219
~PO)
1.5
800
,-
,
I
\
• 600 is
.§
.,
\,
>
1.0
,, ,
-0
!.;: (,,)
,,
,,
400
~
1)
,, ,,
~:::l
U
.,,
'.6 ,,
(,,) ~
,,
,,
200
0.5 ~ ,,
' ..........
o+---,-.........---r-.,.....--,-~-"T"""--+-
o
100
200 300 Time, IlS
0.0
400
Calculated variations variations of of the the electric electric field field at at the earth’s surface surface due due to to the Figure 4.26. Calculated the earth's the M-component under under the conditions of of figure figure 4.24. The dashed dashed curve curve shows shows the the M-component the conditions 4.24. The current impulse impulse II.. current amplitude of of 1500 1500V V/m at the the point point r == 30 30 m m is is close close to to the measured value. value. It It amplitude 1m at the measured 4.26 that that the the temporal temporal parameters of the the current current impulse impulse follows from from figure figure 4.26 follows parameters of = 500 500m, the are also also consistent consistent with the measurements. At the the point point r = are with the measurements. At m, the calculated field field amplitude amplitude is is aa factor factor of of three smaller and and the for the the calculated three smaller the time time for maximum amplitude m. Both 30m. Both parameters parameters maximum amplitude is is nearly nearly the the same same as as for for r == 30 would differ by an order order of of magnitude magnitude in in aa dart dart leader leader with with this this increase increase in in would differ by an r.r . Therefore, Therefore, the the diffusion diffusion model model of of the the M-component M-component reproduces reproduces fairly fairly well available observations. observations. It It would, would, certainly, certainly, be most desirable desirable to well the the available be most to make simultaneous field field registrations registrations at at different different distances distances from from aa grounded grounded make simultaneous lightning channel. channel. lightning
References References [I] [l] Berger Berger K, K, Anderson Anderson R R Band B and Kroninger Kroninger H H 1975 1975 Electra 41 41 23 23 [2] Idone Idone V VP P and and Orville Orville R RE E 1985 1985 J. J. Geophys. Geophys. Res. 90 6159 6159 [2] [3] Antsurov Antsurov K K V, V, Vereschagin Vereschagin II P, P, Makalsky Makalsky L LM M et al1992 a1 1992 Proc. 9th Intern. Con! Con$ [3] on Atmosph. Electricity 11 (St Voeikov Main (St Peterburg: Peterburg: A I Voeikov Main Geophys. Geophys. Observ.) Observ.) 360 360 [4] A, Makalsky Izvestiya. [4] Vereschagin Vereschagin II P, P, Koshelev Koshelev M M A, Makalsky L LM M and and Sysoev Sysoev V VS S 1989 1989 Izvestiya. 100 Akad. Nauk SSSR, S S S R , Energetika i transport 4 100 [5] Simpson Simpson G GC C and and Robinson Robinson G GD D 1941 1941 Proc. R 117 281 [5] R.. Soc. Soc. London A 117281 [6] Res. 65 1873 [6] Kasemir Kasemir H HW W 1960 1960 J. J . Geophys. Geophys. Res. 1873
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Physical processes in a lightning discharge Physical
[7] [7] [8] [8] [9] [9] [lo] [10] [ll] [11] [12] [12]
Gorin Nand Gorin B BN and Shkilev Shkilev A AV V 1976 1976 Elektrichestvo 6 31 31 Geophys. Res. 76 1078 1078 Proctor D DA A 1971 1971 J. Geophys. Proctor Mazur V, Gerlach Gerlach JJ C C and and Rust Rust W WD D 1984 1984 Geophys. Geophys. Res. Lett. 11 11 61 61 Mazur V, V, Rust Rust W WD D and and Gerlach Gerlach JJ C C 1986 1986 J. J. Geophys. Geophys. Res. 91 91 8690 8690 Mazur V, Mazur Raizer Yu P 1991 1991 Gas Discharge Physics (Berlin: (Berlin: Springer) Springer) p 449 449 Raizer Yu P of Shock Waves Waves and HighZel’dovich Ya Ya Band B and Raizer Raizer Yu Yu P P 1968 1968 Physics of Zel'dovich (New York: York: Academic Academic Press) Press) p 916 916 Temperature Hydrodynamic Phenomena (New [13] [13] Schon1and Schonland B B 1956 1956 The The Lightning Discharge. Handbuch der Physik 22 (Berlin: (Berlin: Springer) 576 576 Springer) E 1999 1999 J. Geophys. Res. 104 104 [14] Orvill Orvill R RE [14] J. Geophys. [15] Gorin Gorin B BN and Shkilev Shkilev A AV V 1974 1974 Elektrichestvo 2 29 29 [15] Nand EM M and and Raizer Raizer Yu Yu P P 1997 1997 Spark Discharge (Boca (Boca Raton: Raton: CRC CRC Press) Press) [16] Bazelyan Bazelyan E [16] pp 294 294 Gegechkori N M, Drabkina Drabkina S II and and Mande1'shtam Mandel’shtam S L L 1947 1947 Zh. Zh. [17] Abramson I S, Gegechkori [17] Abramson IS, N M, Eksper. i Teor. Teor. Fiz. 17862 17 862 [18] Drabkina Drabkina S II 1951 1951 Zh. Zh. Eksper. ii Teor. Teor. Fiz. Fiz. 21 473 473 [18] [19] Dolgov Dolgov G GG G and and Mandel'shtam Mandel’shtam S L L 1953 1953 Zh. Zh. Eksper. Eksper. ii Teor. Teor. Fiz. Fiz. 24691 24 691 [19] [20] N 1958 1958 Soviet Phys. JETP 7 (Eng. (Eng. Trans.) Trans.) 1068 1068 [20] Braginsky Braginsky S N [21] Zhivyuk Zhivyuk Yu Yu Nand N and Mandel'shtam Mandel’shtam S L L 1961 1961 Soviet Phys. Phys. JETP 13 13 (Eng. (Eng. Trans.) Trans.) [21] 338 338 14 2111 and and 2124 2124 [22] P100ster Plooster M MN N 1971 1971 Phys. Fluids 142111 [22] [23] Paxton Paxton A A N, N, Gardner Gardner R RL L and and Baker Baker L L 1986 1986 Phys. Fluids 29 2736 2736 [23] [24] [24] Sneider Sneider M MN N 1997 1997 Unpublished report [25] Gorin Gorin B BN N and and Markin Markin V V II 1975 1975 in in Research of of Lightning and High-Voltage [25] (Moscow: Krzhizhanovsky Krzhizhanovsky Power Power Engineering Engineering Inst.) Inst.) pp 114 114 (in (in Discharge (Moscow: Russian) Russian) [26] Baze1yan Bazelyan E E M, M, Gorin Gorin B B Nand N and Levitov Levitov V V II 1978 1978 Physical and Engineering [26] of Lightning Protection (Leningrad: (Leningrad: Gidrometeoizdat) Gidrometeoizdat) pp 223 223 (in (in Fundamentals of Russian) Russian) [27] 10 [27] Gorin Gorin B BN N 1985 1985 Elektrichestvo 4 10 Proc. 9th Intern. Intern. Con! Conf. on Atmosph. Electricity 1 (St (St Peterburg: Peterburg: [28] Gorin Gorin B BN N 1992 1992 Proc. [28] Voeikov Main Main Geophys. Geophys. Observ.) Observ.) 206 206 A I Voeikov A J. Geophys. Geophys. Res. Res. 88 88 6555 6555 [29] Jordan Jordan D DM M and and Uman Uman M MA A 1983 1983 J. [29] [30] Rakov Rakov V VA A and and Uman Uman M M A 1998 1998 IEEE Trans. Trans. on EM E M Compatibility 40 40 403 403 [30] in Lightning, vol. vol. 1, 1, Physics Lightning (R (R Golde Golde (ed) (ed) New New York: York: [31] Berger Berger K K 1977 1977 in [31] Academic Academic Press) Press) pp 119 119 E 1966 1966 Bull. Bull. SEV S E V 57(13) 57(13) 11 [32] Berger Berger K K and and Vogrlsanger Vogrlsanger E [32] [33] Schonland Schonland B, B, Malan Malan D D and and Collens Collens H H 1935 1935 Proc. Proc. Roy. Roy. Soc. Soc. London Ser A 152 152 [33] 595 595 [34] [34] Schon1and Schonland B, B, Malan Malan D D and and Collens Collens H H 1938 1938 Proc. Proc. Roy. Roy. Soc. Soc. London Ser A 168 168 455 455 [35] [35] Orvill Orvill R RE E 1968 1968 J. J . Geophys. Geophys. Res. Res. 736999 73 6999 RE E and and Idone Idone V V PP 1982 1982 J. J . Geophys. Geophys. Res. Res. 87 87 11177 11177 [36] Orvill Orvill R [36] E PP 1974 1974 J. J. Geophys. Geophys. Res. Res. 794542 79 4542 [37] Krider Krider E [37] [38] (New York: York: McGraw McGraw Book Book Company) Company) pp 300 300 [38] Uman Uman M MA A 1969 1969 Lightning (New A, Uman Uman M M A et al1993 a1 1993 J. J . Geophys. Geophys. Res. 9822887 98 22887 [39] Fisher Fisher R R G, G, Rakov Rakov V V A, [39] V A, A, Uman Uman M MA A et al1992 a1 1992 Proc. Proc. 9th Intern. Con! Con$ on on Atmosph. [40] Fisher Fisher R R G, G, Rakov Rakov V [40] (St Petersburg: Petersburg: A A II Voeikov Voeikov Main Main Geophys. Geophys. Observ.) Observ.) pp 873 873 Electricity 3 (St
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[41] Phys. Rep. 21 21 [41] Bazelyan Bazelyan E EM M 1995 1995 Fiz. Plazmy 21497 21 497 (Engl. (Engl. transl.: transl.: 1995 1995 Plasma Phys. 470) 470) [42] Rakov Rakov V V A, A, Thottappillil Thottappillil Rand R and Uman Uman M MA A 1995 1995 J. J . Geophys. Res. 100 100 25701 25701 [42] [43] N 1999 transl.: 1999 [43] Sinkevich Sinkevich 00 A A and and Gerasimov Gerasimov D DN 1999 Fiz. Plazmy 25376 25 376 (Engl. (Engl. transl.: 1999 Plasma Phys. Rep. 25 339) 339) [44] Uman M A et al1995 [44] Rubinstein Rubinstein M, M, Rachidi Rachidi F, F, Uman MA a1 1995 J. J . Geophys. Geophys. Res. 100 100 8863 8863 [45] Uman M J. Geophys. [45] Rakov Rakov V V A, A, Uman M A, A, Rambo Rambo K K JJ et al1998 a1 1998 J. Geophys. Res. 103 14117 14117
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Chapter 5
Lightning Lightning attraction by objects
In this this chapter, chapter, we we shall shall describe describe the the way lightning channel channel chooses chooses aa point In way aa lightning point to terrestrial or to strike strike (a (a terrestrial or aa flying flying body). body). This This is is the the principal principal issue issue for for lightning lightning protection technology. technology. In In any any case, case, aa direct direct stroke stroke is is more more hazardous hazardous than than aa protection remote lightning lightning effect effect via the electromagnetic electromagnetic field field or or shock shock wave wave in in the air. remote via the the air. Historically, direct direct lightning lightning strokes strokes were observed earlier earlier than indirect ones, ones, Historically, were observed than indirect and research into protection problems was associated with and the the first first research into lightning lightning protection problems was associated with direct strokes. strokes. direct Everyday experience experience and and scientific scientific observations, observations, including including those those made Everyday made as far far back back as as the the 18th 18th century, century, indicate indicate that that lightning lightning most most often often strikes strikes as individual the earth. towers, individual structures structures elevated elevated above above the earth. These These may may be be towers, churches, high open just high high trees. this list churches, houses houses on on high open hills, hills, and and just trees. Today, Today, this list is is much longer longer and and includes includes power lines, transmitting transmitting and and much power transmission transmission lines, receiving antennas, receiving antennas, and and the the like. like. The The experience experience in in maintaining maintaining such such structures that the the frequency with the structures indicates indicates that frequency of of strokes strokes increases increases with the object's height. This was used used as basis for the most most common object’s height. This observation observation was as aa basis for the common lightning protection protection techniques. techniques. A grounded the object grounded rod rod higher higher than than the object to to lightning lightning rod rod -- put up in in the the vicinity of the object is is be protected -- aa lightning be protected put up vicinity of the object supposed the object. supposed to to attract attract most most strokes, strokes, thus thus protecting protecting the object. The The underlying underlying principle has not not changed was principle of of this this approach approach has changed since since the the first first lightning lightning rod rod was constructed has changed constructed two two and and aa half half centuries centuries ago. ago. What What has changed is is the the requirerequirement the protection become extremely ment for for the protection reliability, reliability, which which have have become extremely stringent. stringent. For reason, the with exceptions rather than than the For this this reason, the specialists specialists have have to to deal deal with exceptions rather the rules, focusing focusing on on the the rare cases of of lightning lightning breakthroughs to the the object object rules, rare cases breakthroughs to being they lead lead to to emergencies emergencies and and sometimes sometimes to being protected, protected, because because they to catastrophes. catastrophes. The timeThe study study of of lightning lightning attraction attraction mechanisms mechanisms is is extremely extremely timeconsuming the number number of consuming and and expensive. expensive. Even Even aa simple simple measurement measurement of of the of lightning various heights heights is very hard hard to lightning strokes strokes at at objects objects of of various is very to arrange. arrange. Most houses and Most apartment apartment houses and industrial industrial premises premises in in Europe Europe are are less less than than 222 Copyright © 2000 IOP Publishing Ltd.
The The equidistance principle
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50 m high. high. On On the the average, average, aa lightning lightning strokes strokes aa 50m 50 m building building once in five five 50m once in years. attracts years. Every Every kilometre kilometre of of aa power power transmission transmission line line 30m 30 m high high attracts approximately lightning discharge of approximately one one lightning discharge per per year. year. Long-term Long-term observations observations of aa large and multi-kilometre large number number of of buildings buildings and multi-kilometre transmission transmission lines lines are are necessary to to accumulate accumulate aa representative representative statistics. statistics. The The difficulties difficulties increase increase necessary many-fold when when one one needs needs to to extract extract information information on on the the protection protection reliabilreliabilmany-fold ity from from the the observational observational statistics. statistics. To To illustrate, illustrate, 10-20 10-20 years years of of continuous continuous ity observations of of aa 50 50 m m building building would would be be required required to to obtain obtain information information on on observations the lightning lightning discharge discharge frequency, frequency, and and at at least least 1000 1000 years years would would be be necessary necessary the to check whether its lightning lightning rod can really really provide provide aa '99% ‘99% protection’ to check whether its rod can protection' promised by by the the rod rod producers. producers. promised In situation like evaluations, In aa situation like this, this, one one has has to to resort resort to to theoretical theoretical evaluations, and this one reason attraction theory focal and this is is one reason why why lightning lightning attraction theory has has been been the the focal point of research specialists. Here, research for for many many lightning lightning specialists. Here, as as in in many many other other point of lightning there is an acute lack of of factual factual data. data. The The available available lightning problems, problems, there is an acute lack evidence evidence obtained obtained from from laboratory laboratory investigations investigations on on long long sparks sparks does does not not always always provide provide an an unambiguous unambiguous interpretation, interpretation, and and this this makes makes one one treat treat with with caution caution many, many, even even generally generally accepted, accepted, concepts. concepts. We We shall shall focus focus on on the the most most advanced advanced approaches, approaches, discussing, discussing, where where necessary, necessary, alternative alternative hypotheses. hypotheses.
5.1 5.1
The equidistance principle principle
This approach is oldest oldest and in its theoretical formulation. formulation. This approach is and clearly clearly correct correct in its theoretical Suppose is located located on on aa flat flat earth's earth’s Suppose that that an an object object of of small small area area and and height height h is surface surface (it (it is is aa rod rod electrode electrode in in laboratory laboratory simulations). simulations). Let Let us us assume assume further further that that aa lightning lightning channel channel is is shifted shifted from from it it horizontally horizontally at at aa distance distance r, r, and and the the channel channel tip tip is is at at an an altitude altitude H Hoo (figure (figure 5.1). 5.1). In In order order to to predict predict whether whether the the object the earth, we shall the lightning lightning will will strike strike the object or or the earth, we shall take take into into account account the breakdown breakdown voltage voltage measurements measurements of long air with aa sharply nonthe of long air gaps gaps with sharply nonuniform electric They show show that that the the longer longer the the gap, gap, the the higher higher the the uniform electric field. field. They average voltage required required for its breakdown breakdown and and the the longer longer the the time time necessary necessary average voltage for its for the discharge discharge formation. This means means that that the the shortest shortest gap gap has has the the best best for the formation. This chance to experience experience aa breakdown, breakdown, provided provided that that the the same voltage is is applied applied chance to same voltage simultaneously simultaneously to to several several gaps. gaps. Let Let us us keep keep in in mind mind that that the the distance distance from from the the 2]1/2, is h)2 + r shorter than that to the lightning [(Ho h)’ r2I1/’, is shorter than that to the lightning tip tip to to the the object, object, [(H o earth's earth’s surface surface H Hoo at at (5.1 )
The distance distance Re, R eq is is known known as as the the equivalent equivalent attraction radius for for an an object of The attraction radius object of height h. h. It indicates the the surface from which which lightning lightning discharges that height It indicates surface area, area, from discharges that
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Lightning attraction by objects
Estimating the the equivalent equivalent attraction attraction radius. radius. Figure 5.1. Estimating
Hoo are are attracted attracted by by the the object. object. For For aa compact compact have descended descended to to the the altitude altitude H have is aa circle circle of of area area Seq Seq~ M 7rR~q; T R ; ~for for ; an an object of of small small cross cross section, section, this this is object >> h and and width width b « << h (e.g., (e.g., aa power power transmission transmission extended object object oflength of length L » extended line), this this is is aa stripe stripe of of area area Seq Seq ~ x 2R 2ReqL. The average average number number of of strokes strokes per per line), eq L. The Seqas as storm season season is is evaluated evaluated from from Seq storm N NIj == njSeq nlSeq
(5.2) (5.4
where nj nl is is the the year year density density of of lightning lightning discharges discharges into into the the earth earth at at the the where object’s site. site. Global Global and and regional regional maps maps of of storm storm intensity intensity are are made made from from object's 2 The nj n1 data data are are usually usually given given per per 1km 1 km2 meteorological survey survey data data [1,2]. [l, 21. The meteorological per of storm storm days days or or per year. year. Quite Quite often, often, the the maps maps indicate indicate the the number number of nl. hours, together together with with empirical empirical formulae formulae to to relate relate this this parameter parameter to to nj. hours, The equidistance equidistance principle, principle, simple simple and and clear clear as as it it may may seem, seem, is is of of little little The use, because because one one can can employ employ formulas formulas (5.1) (5.1) and and (5.2) (5.2) to to advantage advantage only only if if use, Hoo (the (the attraction attraction altitude), altitude), at at which which aa descending descending one knows knows the the altitude altitude H one lightning leader leader begins begins to to show show its its preference preference and and selects selects the the point point to to lightning on it it strike. The The condition condition of of the the earth's earth’s surface surface and and the the objects objects located located on strike. cannot influence influence the the lightning lightning behaviour behaviour high high up up in in the the clouds. clouds. A lightning lightning cannot approaches the the earth, earth, the the develops by by changing changing its its path path randomly. randomly. As itit approaches develops field perturbation perturbation by by charges charges induced induced by by terrestrial terrestrial objects objects become become field increasingly comparable comparable with with random random field field fluctuations. fluctuations. Eventually, Eventually, the the increasingly perturbation begins begins to to play play the the dominant dominant role, role, determining determining the the channel channel perturbation Ho at which which this this happens happens path more more or or less less rigorously. rigorously. The The average average altitude altitude H path o at is is known known as as the the attraction attraction altitude. altitude. is unlikely unlikely that that the the altitude altitude H Ho should be be determined determined only only by by the the It is It o should h. It It must must also also depend depend on on the the leader leader field field varying varying terrestrial object's object’s height height h. terrestrial statistically with with the the lightning lightning due due to to the the variations variations in in the the storm storm cloud cloud statistically
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The equidistance principle
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charge, the starting point of the the descending descending leader, leader, its its path, number of of charge, the starting point of path, number branches, etc. This This diversity diversity of of lightning lightning conditions conditions is is uncontrollable. uncontrollable. The The branches, etc. only parameter parameter that that can, can, to to some some extent, extent, depend depend on on observations observations is is the the only attraction attraction altitude altitude averaged averaged over over all all descending descending discharges. discharges. It It deserves deserves attention because the attention because the averaging averaging will will require require only only the the statistics statistics of of descending descending lightnings which have lightnings which have struck struck objects objects of of various various heights. heights. These These statistics statistics cannot be be said said to to be be reliable reliable but but it it provides provides some some factual factual information information cannot important for for lightning lightning protection important protection practice. practice. Before use the the stroke treatment, we Before we we use stroke statistics statistics in in aa theoretical theoretical treatment, we think think it it worthwhile worthwhile defining defining the the range range of of object object heights. heights. Unfortunately, Unfortunately, one one has has to to discard high constructions. discard the the stroke stroke data data concerning concerning high constructions. Ascending Ascending discharges discharges become h > 150m. 150m. Data Data on on such such strokes strokes cannot cannot be be become dominant dominant at at heights heights h> included without reservations, included in in the the statistics statistics without reservations, even even though though they they were were obtained obtained from from well-arranged well-arranged observations, observations, in in which which every every discharge discharge was was identified unambiguously. The that ascending partly identified unambiguously. The point point is is that ascending lightnings lightnings partly discharge the clouds, number of discharge the clouds, reducing reducing the the number of descending descending discharges. discharges. This This interference so appreciable appreciable that that aa further further interference into into the the storm storm cloud cloud activity activity is is so increase increase of of h above above 200 200 m m does does not not practically practically change change the the stroke stroke frequency frequency of of an an object object by by descending descending lightnings. lightnings. Of Of little little use use are are the the data data on on low low structures this case structures (l0-15m). (10-15m). The The number number of of strokes strokes in in this case is is greatly greatly affected affected by the the nearest the local be by nearest neighbours neighbours and and the local topography. topography. Account Account should should be taken of buildings, but taken of the the statistics statistics for for low low buildings, but such such observations observations are are scarce. scarce. too large The The overall overall data data have have aa too large spread. spread. The The authors authors of of [3] [3] selected selected the the most most reliable reliable data data and, and, by by averaging averaging many observations, the relationship relationship between between the many observations, derived derived the the number number of of descenddescending Figure 5.2 ing strokes strokes and and the the terrestrial terrestrial object object height. height. Figure 5.2 shows shows individual individual the results representative representative values values to to demonstrate demonstrate the the data data spread. spread. All All of of the results are are normalized to the storm normalized to the the intensity intensity of of the storm cloud cloud activity, activity, which which is is 25 25 storm storm days per year. 9 150 150 m m considered, considered, we we can can admit, admit, with with some some days per year. In In the the range range h ::::; height dependence number reservations, the the existence reservations, existence of of aa quadratic quadratic height dependence of of the the number of of lightning lightning strokes strokes for for concentrated concentrated objects objects and and aa linear linear dependence dependence for for const. R eq / h ~ const. extended extended ones. ones. Both Both dependencies dependencies mean mean Re,/h Figure that the R eq = used for Figure 5.2 5.2 shows shows that the expression expression Re, = 3h, 3h, sometimes sometimes used for rough estimations rough estimations of of the the expected expected number number of of strokes, strokes, agrees agrees fairly fairly well well with the R eq derived derived from from observations. observations. The The substitution substitution with the averaged averaged values values of of Re, of R eq = = 3h 3h into into (5.1) (5.1) yields yields for for the the average average attraction attraction altitude altitude for for descending descending of Re, leaders leaders Ho = = 5h. 5h. Ho
(5.3) (5.3)
This be aa large the This does does not not seem seem to to be large height. height. A A lightning lightning is is insensitive insensitive to to the earth's most of path and earth’s surface surface along along most of its its path and it it is is only only its its last last 50-500 50-500mm which which are the mechanism mechanism of are predetermined. predetermined. Below, Below, we we shall shall discuss discuss the of the the more more or or less behaviour by by aa particular less rigid rigid determination determination of of the the leader leader behaviour particular site site on on the the earth (section (section 5.6). 5.6). earth
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226
Lightning attraction by objects
N ,
1.0
".
~;.,,'"
I- - - 1 - - -
~
---
•
--x
I
/
I
I
,I "I ./I
I
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0. 0.1
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,
. I
'.;
!
h,m 240
120
360
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N, kIn-I /
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Ig o
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./
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-- -- h,m
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Figure 5.2. The The average average number of strokes strokes per for compact compact (top) (top) and and extended extended Figure number of per year year for of height height h. h. The The dashed dashed curves curves bound spread zones zones in in observation observation (bottom) objects objects of (bottom) bound spread data. Solid Solid curves curves are are plotted using the the equivalent equivalent attraction attraction radius. data. plotted using radius.
5.2
The electrogeometric electrogeometric method method
Popular among among some some lightning lightning specialists, specialists, this this method of calculating calculating the the Popular method of number of of lightning lightning discharges discharges into into aa grounded grounded structure structure [4-8] [4-81 should should be number be modification of of the the equidistance equidistance principle. The main main considered as as aa modification considered principle. The
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The electrogeometric method
227
Figure 5.3. 5.3. Lightning Lightning capture capture regions. regions.
calculation parameter parameter in in this this method method is is the the striking striking distance distance rr,.s . Surfaces Surfaces calculation located upper points from the the upper points of of aa structure structure (roof), (roof), the the located at at aa distance distance rr,s from adjacent buildings, and adjacent buildings, and from from the the earth's earth’s surface surface define define by by means means of of their their interception regions (figure 5.3). interception lines lines the the lightning lightning capture capture regions (figure 5.3). The the capture The further further path path of of aa lightning lightning channel channel which which has has reached reached the capture region is considered considered unambiguously predetermined. The The leader leader will will move move unambiguously predetermined. region is to to the the object object (or (or to to the the earth), earth), whose whose capture capture surface surface it it has has intercepted. intercepted. With number of N,I With these these initial initial assumptions, assumptions, the the calculation calculation of of the the number of strokes strokes N reduces to reduces to geometrical geometrical constructions, constructions, since since the the lightning lightning density density nj nl at at an an is considered considered to to be and the value of of N can be be altitude z > h + rr,s is be uniform, uniform, and the value NIj can altitude if one one knows knows the the area area 5S,s of of the the capture capture surface surface projection on to calculated if calculated projection on to NI == n ytlSs. the earth’s plane, plane, N, the earth's l 5s' The witnessed only The long long history history of of the the electrogeometric electrogeometric method method has has witnessed only one one improvement that of the selection principles concerning the striking improvement - that of the selection principles concerning the striking distance rr,s [1,3]. [l, 31. Discarding Discarding inessential inessential details, details, the the quantity quantity rY,s is is found found from from distance an average electric field E between the object's top (or the earth's surface) an average electric field E,a between the object’s top (or the earth’s surface) and region. Usually, and the the lightning lightning leader leader tip tip which which has has reached reached the the capture capture region. Usually, the be equal breakdown strengths E,a were were taken taken to to be equal to to the the average average breakdown strengths the values values of of E of the laboratory laboratory study study of of increasingly increasingly longer longer of the the longest longest laboratory laboratory gaps. gaps. As the sparks E a introduced introduced into into the the calculation calculation method method sparks progressed, progressed, the the values values of of E, decreased this to 2kV/cm, 2 kV/cm, entailing entailing larger larger striking striking distances. distances. In In this decreased from from 66 to approach, height is independent independent of of the the grounded grounded object's object’s height approach, the the parameter parameter rr,s is U,t (r (Y,s ::::: zz Uti U,/&). However for for applicaapplicabut is is sensitive sensitive to to the the leader leader tip tip potential potential U E a ). However but the tions, the relation of the tions, attempts attempts are are made made to to find find the relation to to the the current current amplitude amplitude 1Z,Mw of return using simulation models of the kind discussed in , return stroke, stroke, rather rather than than U U,, using simulation models of the kind discussed in t 4.4. If If the the function function rY,s =!(IM) =f(ZM) and, and, hence, are known, the section 4.4. section hence, 5S,(ZAw) (IM) are known, the s number of of lightning lightning strokes strokes at at an an object object with with current current 1,M > hIo Zlwo is is found found as as number
+
Nj({',.fo) = nj
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J:,
5 s(I)r..p(I) dl
(5.4)
228
Lightning attraction by objects
500l-----+----+----+-~+_f_+_t
400 l-------+---+--~~I'_+~f_+-+l 300 l----+------:~-+--,'---i--:---r-I_+_r_+___+_7'--i
200 J-----+-..".4-f---'--++----r---t-+-+-+--r-t----t-t-t-i
1m, 50
kA
150
100
Figure 5.4. The The dependence dependence of of striking striking distance distance on on lightning lightning currents. currents. The The lower lower curve using [5] curve is is plotted plotted using using [4] [4] data, data, the the upper upper using [5] data. data. The The spread spread region region isis hatched. hatched.
where where 'P(I) p(Z) is is the the probability probability density density of of current current of of amplitude amplitude IZ found found from from natural measurements. To total number natural measurements. To find find the the total number of of strokes, strokes, the the lower lower limit limit of (5.4) should should be be taken taken to to be be zero. zero. of the the integral integral in in (5.4) The in current current The generally generally correct correct idea idea of of differentiating differentiating distances distances rr,s in amplitude to refine refine the There is is no no factual factual amplitude actually actually fails fails to the calculation calculation of of N 1I•. There ( Z M ) experimentally, experimentally, while while information to to determine determine the the function function rrss == ff(IM) information so that that theoretical evaluations suffer suffer from from an an unacceptably unacceptably large large spread, spread, so theoretical evaluations the values times (figure the values obtained obtained by by different different authors authors differ differ several several times (figure 5.4). 5.4). The heights could, The stroke stroke statistics statistics for for objects objects of of various various heights could, to to some some extent, extent, but it it proves proves be used used for for fitting fitting the the calculated calculated total total number of strokes strokes N bI , but be number of unsuitable for finding finding the the function function rr,s ==f(Z,w). unsuitable for f(IM)' The procedures in The calculation calculation procedures in the the electrogeometric electrogeometric approach approach do do not not involve involve aa strong strong dependence dependence of of stroke stroke frequency frequency on on an an object's object's height. height. Indeed, for for aa single single construction, construction, like like aa tower, tower, the the capture capture region region projection projection Indeed, on on to to the the earth's earth's plane plane is is aa circle circle of of radius radius 2)1/2 at R == (2r (2r,h h2)'I2 at sh -- h
R == rr,s
at at
rrss ~ 2h rr,s
(5.5) (5.5)
and width 2R. and for for an an extended extended object, object, like like aa transmission transmission line, line, it it is is aa stripe stripe of of width 2R. Therefore, number of Therefore, the the number of strokes strokes of of low low power power lightnings lightnings with with aa small small stroke stroke (rss < h) h ) will will be be entirely entirely independent independent of of the the object's object's height, while the distance (r distance height, while the must increase increase with with height frequency of of powerful discharges with with rr,s > h must frequency powerful discharges height for compact compact objects objects and and as as h l/2 for for extended extended ones. ones. Actually Actually slower than than h for slower both of both of these these dependencies dependencies are are steeper. steeper.
5.3
The probability probability approach approach to finding finding the stroke stroke point point
choice of of the the discharge discharge path path through through an an air air gap gap contradicts contradicts A predetermined predetermined choice the Neither aa spark the experience experience gained gained from from long long spark spark investigations. investigations. Neither spark nor nor aa
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The finding the stroke point The probability approach to Jinding
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lightning path. When voltage is is simultaneously simultaneously lightning travel travel along along the the shortest shortest path. When voltage various lengths, lengths, it it is is the the longest longest applied applied in in parallel parallel to to several several air air gaps gaps of of various gap This is is supported supported by by the the large large gap that that is is sometimes sometimes closed closed by by aa spark. spark. This deviation u a for for multi-metre multi-metre spread spread of of breakdown breakdown voltages: voltages: the the standard standard deviation of the the average average breakdown breakdown gaps gaps with with aa sharply sharply non-uniform non-uniform field field is is 5-10% 5-10% of voltage. voltage. If the distributions distributions of of breakdown breakdown If two two gaps, gaps, tested tested individually, individually, possess possess the (U), then, provided provided that that the the voltage voltage of of the the probability probability densities densities 'PI cpl (U) ( U ) and and 'P2 p2( U ) ,then, the breakdown breakdown probprobvoltage voltage of of aa common common source source is is applied applied simultaneously, simultaneously, the is described described as as ability ability for for one one of of the the gaps, gaps, say, say, the the first first one, one, is (5.6) the probability probability of of the the gap gap breakbreakwhere where
