Relativistic-microwave theory of ball lightning

Abstract

Ball lightning, a fireball sometimes observed during lightnings, has remained unexplained. Here we present a comprehensive theory for the phenomenon: At the tip of a lightning stroke reaching the ground, a relativistic electron bunch can be produced, which in turn excites intense microwave radiation. The latter ionizes the local air and the radiation pressure evacuates the resulting plasma, forming a spherical plasma bubble that stably traps the radiation. This mechanism is verified by particle simulations. The many known properties of ball lightning, such as the occurrence site, relation to the lightning channels, appearance in aircraft, its shape, size, sound, spark, spectrum, motion, as well as the resulting injuries and damages, are also explained. Our theory suggests that ball lighting can be created in the laboratory or triggered during thunderstorms. Our results should be useful for lightning protection and aviation safety, as well as stimulate research interest in the relativistic regime of microwave physics.

Relativistic-microwave theory of ball lightning

H.-C.Wu,

Since Arago1 rst extensively discussed ball lightning in 1838, this rare natural phenomenon still remains a riddle. Ball lightning25 exhibits very diverse characteristics, such as close association with ordinary lightning, globate structure with steady glow for 15 seconds, and mostly horizontal motion. Ball lightning can be formed even inside aircra and closed rooms, permeate glass plates, decay explosively or silently, and produce sound and acrid odours. Many models of ball lightning have been proposed, but none have been fully accepted6. In particular, these theories do not explain the inscrutable appearance of ball lightning inside fully-screened aircra7. Here, we propose a theory for ball lightning formation, which can explain its appearing in aircra and many other properties.

Lodge8 considered that ball lightning might be excited by a standing electrical wave from lightning. Kapitza9 argued that ball lightning could be formed through air ionization at antinodes of electromagnetic standing waves in the microwave regime. Dawson and Jones10 proposed that ball lightning could be a microwave bubble conned inside a globate plasma shell. Continuous air ionization by the trapped microwave maintains the plasma shell11.

By dimensional analysis, we have pointed out12 that a microwave bubble can be formed similarly as light solitons observed in the laser-plasma interaction. Such a microwave bubble contains a half-cycle standing-wave mode and is sketched in Fig.1a. The microwave-type model of ball lightning can explain its permeation through glass plates. However, the origin of microwave emission from lightning was never found.

In this article, we propose a mechanism for microwave generation from lightning. As shown in Fig.1b, we assume that in a ball lightning event a relativistic electron bunch is generated by lightning. When this bunch strikes the ground or passes various media, powerful microwaves are emitted by coherent transition radiation (Fig.1c). We further verify that this specic microwave in air plasmas naturally evolves into a microwave bubble. These results are demonstrated by particle-in-cell (PIC) simulation using the code JPIC12.

Results

Relativistic electron bunch. The assumption of isolated relativistic electron bunches in ball lightning events is based on high-energy phenomena13,14 discovered in cloud-to-ground lightning. A lightning ash5 starts with a negative leader propagating downward in a stepping process with each step tens of metres. This stepped leader has a corona 110m in width. Moore et al.15 rst detected >1MeV radiation from a stepped leader. It was then observed that each step emits an x-ray burst16, which intensies when the leader approaches the ground. Recent data17 shows that the last step or the so-called leader burst closest to the ground produces the strongest x-rays. Electrons accelerated by the stepped leader account for these detected x-rays, so that the electron acceleration is the most violent in the last step.

Institute for Fusion Theory and Simulation (IFTS) and Department of Physics, Zhejiang University, Hangzhou

R A

P

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Figure 1. Ball lightning model. (a) Microwave bubble model. (b) Relativistic electron bunch generation. In the last leader step, a bunch of runaway electrons emerges from the leader tip, accelerates by electric elds between the leader and ground, and undergoes an avalanche. (c) Coherent transition radiation (CTR) of the electron bunch striking the ground or passing through aircra skins. is the relativistic factor of electrons.

Friction force of electron motion in air is maximum at an energy of 100 eV, which denes a critical electric eld Ec 30 MV/m14. Fields above Ec at the leader tip can accelerate thermal electrons to several keV18. This thermal runaway process19 can produce ~1011 electrons. The hot electrons can be further accelerated by the electric eld between the leader tip and the ground, and serve as seed electrons to undergo avalanche in air20. The electron ux quickly rises as exp(z/L), where L is the avalanche length. The electron energy follows a Boltzmann distribution exp(ke/7.3 MeV), such that the average energy is 7.3 MeV. Latest data analysis21 shows that colli-mating relativistic electrons are required to explain observed x-rays from the stepped leader, and should be either Boltzmann-distributed at 7MeV or monoenergetic from 1 to 10MeV.

The isolated x-ray bursts from the stepped leader are much shorter than 1s16. On the other hand, metre-scale laboratory sparks in air22 can emit very similar x-rays as in natural lightning. Duration of x-ray bursts from laboratory sparks is generally sub-10ns23 and can be as short as 1ns24. Accelerated electrons are expected to have the same temporal structure as the x-rays from lightning or sparks.

Accordingly, it can be expected that the last leader step generates a spatially well-dened relativistic electron bunch in a ball lightning event (see Fig.1b). For simplicity, we assume that this bunch has a density prole nb=nb0exp(r2/22), where nb0 is the peak density, and is the characteristic radius. We take a bunch size (4)

of tens of cm, i.e. ~1ns in duration. As discussed later, a bunch with total electron number Nb=(2)3/2nb031014

will lead to a microwave bubble. In the avalanche mechanism, this would need an avalanche path of 7L, corresponding to a multiplication rate of exp(7)103. The avalanche length L is 730cm near the ground14 and should support the rapid amplication of these nanosecond bunches. Another mechanism25 predicts that the leader can directly generate ~1016 energetic electrons on a timescale of 1ns without avalanche.

Microwave generation. Transition radiation is generated from medium surfaces when an electron enters or emerges26 and can be coherent for an isolated electron bunch27. As the electron bunch reaches relativistic energies, its self-elds are predominantly transverse i.e. EbcBb28, which is very close to an electromagnetic wave. In this case, coherent transition radiation can be considered as the reected wave of the bunch eld from the medium surface29. Therefore, we can write the radiation energy as

CTR b f

,

= +

( 1)/( 1) 2

* is the Fresnel reection formula, Wb,f refers to the total bunch eld energy, and is the medium permittivity. The radiation is strongest for a metal or perfect conductor where and R1 in microwave region. In addition, a Boltzmann-distributed electron bunch turns out to produce almost the same transition radiation pulse as a monoenergetic one30.

The lemost panel of Fig.2 shows the transverse eld Eb,x of a monoenergetic 7 MeV electron bunch with = 4cm, which is normalized to the peak eld .

where

b010 3 . The bunch eld is a unipolar wave with the same prole exp(z2/22) as the electron density along the direction of motion. Using JPIC12, we simulate the coherent transition radiation from a perfect conductor in Fig.2. The radiation eld Ex is initially opposite to Eb,x due to the conductor boundary, diracts transversely, and quickly evolves into a bipolar pulse. This radiation has a central wavelength 7.5= 30cm (i.e. 1GHz). The rapid eld evolution into the bipolar shape is due to diffraction losses of longer wavelength components in an unipolar pulse31. At normal incidence in Fig.2, the radiation field is radially polarised with a ring-like intensity distribution. Oblique incidence32 can enhance the radiation production and lead to an asymmetric intensity pattern. Considering surface fluctuations and non-axisymmetric bunches, the actual radiation could contain only one high-intensity emission spot, which is linearly-polarised and will make bubble formation more easily.

E 3 2 MV/m

b

0 10

Microwave bubble formation. Laser solitons have been observed in both PIC simulations33,34 and experiments3537 on relativistic laser-plasma interaction. The laser needs to exceed the relativistic field threshold

= * ( ) , (1)

n

cm

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Figure 2. PIC results of microwave generation. Distribution of the initial bunch eld and microwave elds at times 0.8ns, 1.5ns and 2ns. The eld is normalized to the bunch peak eld Eb0. In the lemost panel, the bunch is le-going to the plasma surface at z= 0. The white circle marks the bunch region with a density of 0.5nb0.

The radiation is a reection of the bunch eld and propagates along z. Arrows point to the eld propagation direction. Parameters are given in the text and Methods.

Er=mc/e38 and is typically multi-cycle. The plasma is underdense with an initial density n0<nc, where nc=0m2/e2 is the critical density39. During the laser propagation in the plasma, the self-phase modulation eect40 leads to a dramatic spectral broadening, which makes some part of laser energy to shi even below the background plasma frequency. Hence this part gets trapped in a plasma cavity with a half-cycle standing wave mode. The cavity is spherical and formed by evacuating electrons through the relativistic ponderomotive force41. The entire formation process takes tens of light cycles.

Here, we discuss the bubble formation for a mono-cycle microwave in Fig.2. The microwave must get trapped within a few cycles before it is diracted. In contrast to the mechanism discussed above, we nd that the initial plasma must be overdense with n0 nc, where nc 1.2 1010cm3 at /2 = 1 GHz. The existence of such a bubble-formation regime for single-cycle waves indicates self-consistency of our theory. The collisional eect is included by embedding air friction14,18 into JPIC. We launch microwave pulses with wavelength = 30cm into a uniform plasma. The simulation shows that the threshold eld required for bubble formation is

.

E E E

max (4 , ) (2)

bl c r

At 1 GHz, we have Er 10.7 MV/m and Ebl 11Er 120 MV/m, which is highly relativistic. Equation(2) clearly shows that the eld needs to be greater than Ec to efficiently accelerate electrons, and reach the relativistic regime to completely expel electrons by the relativistic pondermotive force. Surprisingly, Er matches with Ec to make the bubble formation possible. Here, we check the bunch parameters for giving the threshold eld Ebl. For the case in Fig.2, we get nb03.71011cm3 and Nb3.71014.

In Fig.3, we take n0=4nc and a microwave eld of 310MV/m, and let t= 0 when the eld touches the plasma. Snapshots of microwave eld and plasma density from t= 1ns to 11ns illustrate the entire process of microwave self-trapping and bubble formation. The radiation pressure of microwave rst pushes electrons to pile up into a semicircular shell at t= 1ns and leaves a low-density region at the rear. As the eld is reected by the front shell, peripheric electrons return to the low-density region and close up the cavity at t 3ns. The eld gets trapped and then evolves into a standing-wave mode. At t = 11 ns, a motionless electron cavity forms about 45cm deep into the plasma, and then it becomes circular and keeps stable aer t 15ns. Meantime, heavy ions are slowly pulled out by the charge separation eld.

In Fig.4a,b, snapshots of the stable bubble at t= 19ns show that the elds take on a half-cycle standing wave pattern, electrons have been almost emptied, and ions are partially evacuated. The electrostatic force between electrons and ions is balanced by the radiation pressure 0E2/4 64 kPa, whereE = 170 MV/m is the standing wave amplitude. The periodic conversion between electric and magnetic energies in Fig.4c conrms the standing wave mode. The conned eld oscillates at a longer period of 1.6ns. This redshi is caused by the Doppler eect and self-phase modulation. The cavity diameter is about 24cm, half of the wavelength of the trapped eld. For a ball shape, the conned eld energy in Fig.4b is about 800J. Tuning the microwave eld, the trapped eld energy in the bubble ranges from 200J to 1500J.

Three-dimensional eld structure of microwave bubbles can be close to that of the light solitons observed in PIC simulation34. With energy loss of microwave by collisional absorption, the bubble is expected to convert into an electromagnetic cavity resonator. The fundamental mode at the lowest eigenfrequency in a spherical resonator26

is similar to that in a cylindrical cavity28, which resembles that shown in Fig.4a.

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Figure 3. PIC results of microwave self-trapping and bubble formation. Snapshots of the microwave electric eld E=|Ex|, magnetic eld = +

B B B

y z

Figure 4. PIC results of stable microwave bubble. (a) Snapshots of the microwave electric eld E, magnetic eld B, electron density ne, and ion density ni at t= 19ns. White arrows mark the magnetic eld direction. (b)

Field energy density and plasma density ne,i verses y across the bubble centre. (c) Evolution of the electric eld, electric eld energy We, and magnetic eld energy Wm in the bubble. Parameters are the same as Fig.3.

2 2 , electron density ne, and ion density ni from t= 1ns to 11ns. Vertical dashed line marks the plasma surface. Parameters are given in the text and Methods.

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The properties of ball lightning25 are summarized from about 5000 published sighting reports.

Site of occurrence. As shown in Fig.2, a planar surface is necessary for microwave generation at least with a size of ball lightning, which can be easily fullled in reality. Microwave emission is also aected by the ground reectivity *. The soil permittivity increases with its moisture ms42. At 1GHz, we get . .

= i

8 4 2 2

m 60%

s ,

which correspond to *25% and 56%, respectively. Rainfall can lead to ms>60%43 and thus is favorable for the ball lightning formation. As stated by Stenho4, more than 50% of reports show that medium or heavy rainfall happens before the observation. Moreover, there is *65% for either pure or sea water44. Indeed, there are 18 reports at sea2 and a few reports over rivers2,4. Certainly, metal holds the highest chance of ball formation due to *1.

Relation to lightning channels. The lightning channel refers to the bright return stroke occurring aer the stepped leader attaches with a positive leader rising from the ground. The starting place of this positive leader would be the lightning strike point. We show that ball lightning is caused by the stepped leader, which is invisible with the naked eye. The stepped leader and its mirror charge underground establish a dark channel for electron acceleration and avalanche. Obviously, the ball formation site is unrelated to the lightning strike point. Their separation should be within one step length of tens of metres typically. This successfully explains the reports where ball lightning does not form near the lightning channel or strike point4.

Appearance in aircra. First, the avalanche electron energy 7.3 MeV is independent of the air density13, i.e. altitude. When lightning strikes an aircra, the same bunch is presumably produced and enters the aircra with an energy loss of ~2MeV due to the ~0.6cm aluminium skin45. Second, transition radiation26 is not sensitive to the energy of the relativistic electrons, and its efficiency from the electron emerging surface of the medium is almost the same as the reection side discussed above. Therefore, the same intense microwave will arise inside the aircra and form ball lightning there. In the same manner, ball lightning can appear in enclosed rooms.

Permeation through glass plates. Ball lightning is observed to enter rooms by passing through closed glass windows. In interference experiments of low-power microwave in metal cavity46, generated reballs in air are observed to pass through a 3 mm ceramic plate intact. This is a direct result of the ability of microwave passage across dielectrics. The microwave bubble resembles a laser cavity. According to laser theory47, the internal standing wave will not be disturbed if a glass plate (~5mm) is much thinner than the wavelength of microwave.

Shape. From dimensional analysis12, the microwave bubble of Fig.4 in reality should be ball-shaped as its micrometre-scale counterpart in laser-plasma experiments3537. The full trapping of the eld in Fig.2 can account for the 62 ring-shaped ball lightning reports2.

Size. Ball lightning has a common diameter of 2050 cm4. Our theory shows that the diameter of microwave bubbles approximately equals the electron bunch length in the direction of motion. The bunch length of tens of cm is supported by x-ray duration measured from lightning and laboratory sparks, which can be as short as 1ns.

Sound. Hissing, buzzing or uttering sounds from ball lightning have been reported, which can be perfectly explained by the microwave hearing eect48,49. At 0.1 mJ/cm2, a microwave pulse (microsecond or shorter) at 0.23 GHz can induce an audible sound wave. The sound can only be heard by persons whose heads are irradiated by the microwave, and has been described as a hiss, buzz or knocking. Therefore, ball lightning can be silent during its lifetime. In Jennisons sighting50, he was only 0.5m from a cruising ball, and did not report any noise.

Spark. Ball lightning sometimes emits sparks, which can be caused by the ejection of charged particles along the electric eld. Especially, the sparks are toward opposite directions in two reports2, which agrees with the linear polarisation of standing wave in the bubble.

Spectrum. Recently, Cen et al.51 recorded an optical spectrum of ball lightning. The spectrum contains emission lines of atoms in air and soil. Interestingly, the spectral intensities of O and N atoms oscillate at 100Hz, twice the frequency of the adjacent power lines (35kV, 50Hz). The latter is only 20m from the ball and can produce a 50Hz electric eld of ~1V/cm52 at the ball. This eld can induce electron dri on the ball surface by tens cm (see Methods). This dri motion can perturb the spectral emission in the plasma shell. The spectral intensity should be independent of dri direction and varies at 100Hz. The ball is attached to the soil on a hillside, where electrons cannot feel the oscillating eld due to the screening eect. Thus, Si, Fe and Ca in soil glow steadily51.

Odour. Ionized air can produce O3 and NO25,53, both of which have an acrid smell.

Decay. The microwave bubble decays silently once the internal radiation is exhausted. When it is strongly disturbed or pierced by a conductor, the leaking radiation can launch a shock wave like an explosion.

Injury and damage. Most reported injuries and damages can readily be attributed to ordinary lightning2,4.

However, Stenho4 noticed that some supercial burns are difficult to explain. In the Smethwick event4,54, the female witness did not get an electric shock but felt a burning heat all over. Wooding55 estimated that she received 250J whole-body ionizing radiation, which can be due to the electrons from the stepped leader and also be responsible for the redness on her hand and legs. She heard a knocking-like sound (rattle) from the

s and

48 10

= i

m 20%

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microwave hearing eect. Her legs were numbed, which can be due to nerve damage by the microwave at 0.1J/ cm2 56. When she brushed the ball away with her hand, the ring was burning into her nger. Wooding calculated that this rapid heating would need a resonant microwave at 1 GHz with an eld of ~1 MV/m, which agrees well with our model. Others57 reported skin redness, vomiting and loss of hair, which are typical results of ionizing radiation58. As reported by X. Zhang and Q. Yan in Shanxi Daily (8 Aug. 2014), during a thunderstorm on 5 Aug. 2014, a red ball of re 40 cm in diameter was witnessed entering an office through an open window at the local Water Conservancy Bureau in Xinjiang, Shanxi, China. The ball lasted for less than one second and then exploded loudly. Five computers in the room were damaged, which is a direct result of high-power microwaves56.

Motion. Near the ground, ball lightning moves mostly horizontally at about 2m/s2 and usually travels with the wind3. A light breeze typically at 1.53m/s59 can account for this motion speed. However, air convection will raise the ball if the background air is heated up by the ionized plasmas. Assuming a constant heat power of 100W, we obtain a convection speed 23cm/s for the ball of size 30cm (see Methods). Thus, the upward motion is not notable compared with the horizontal motion. Several models2,4 speculate that the ball could take a positive charge due to the greater mobility of electrons compared with ions. The charged ball can further resist the buoyancy or air convection by an attractive force from its mirror charge underground. Moreover, like a charged particle self-accelerating into an open waveguide60, the ball can enter rooms through chimneys.

Lifetime. The typical lifetime of ball lightning is 15 seconds. Statistical analysis61 shows that increase in humidity decreases the lifetime of the ball, which can be due to microwave absorption by vapour. Experiments62 show that reballs in air produced by a 5kW, 2.45GHz microwave can last for ~0.5s aer the source is turned o. Our self-organized microwave bubble can have the same potential to persist for a scale of seconds. Zheng11 calculated that hundreds of joule microwaves can maintain the plasma shell of the bubble for a few seconds. Air plasmas continuously depleted by recombination are relled by microwave heating. Non-neutral plasmas shown in Fig.4b can further resists the recombination loss.

Discussion

Experiments are required to verify our theory. First, forming a microwave bubble in laboratory will need hundreds of gigawatt microwave, which is one order of magnitude higher than the manmade sources. As stated in ref. 56, it is technically feasible to enhance current microwave devices to 100 GW. Alternatively, one can adopt a high-power electron beam63 to directly simulate the mechanism proposed in Fig.1. Second, on the lightning research, we suggest to detect microwave radiation at GHz near a lightning strike point. We already show that trans-ionospheric pulse pairs from lightning are caused by the same radiation mechanism64, which supplies a physical evidence of our theory. On attempts to create ball lightning by rocket-triggered lightning65, we propose to use ungrounded wires5 rather than grounded ones because ball lightning is thought to be only related to the stepped leader. Perhaps intense lasers can trigger lightnings by producing an ungrounded plasma channel near thunderclouds66. For in situ investigation of ball lightning, we suggest to look for evidence of high-ux energetic electrons. Finally, we note that relativistic terahertz waves could be produced from laser-accelerated hot electrons emerging from solid foils by coherent transition radiation67 or laser-driven plasma waves in gas target68. In particular, the former scenario is very close to the scheme in Fig.1 and may lead to a millimetre-scale terahertz radiation bubble.

Conclusion

In conclusion, based on a reasonable assumption on the electron bunch, we have constructed a self-consistent theory on the microwave generation and ball lightning formation. The theory successfully explains many properties of ball lightning. For the rst time, we revel that ball lightning is an alarm signal of the existence of ultrastrong microwaves and abundantly hazardous electrons near the ground or aircra. This result is of great signicance for lightning protection and aviation safety. Moreover, it is hoped that our work will stimulate research activities in relativistic microwave physics and technology, an unexplored area before.

Dimensional analysis. The interaction of relativistic electromagnetic wave and collisionless plasmas is governed by the Maxwells equations and relativistic Lorentz equation of electrons and ions. When time and space are normalized to the cycle and wavelength of electromagnetic wave respectively, the whole system only depends on two dimensionless quantities =

n n n

/ c

0 0 and

=

E eE mc

/

0 0 , where n0 is the initial plasma density, nc=0m2/e2 is the critical density, E0 is the initial eld amplitude, e is the fundamental charge, m is the electron mass, c is the light speed, is the angular frequency, and 0 is the vacuum permittivity. If

n0 and

E0 are same for any systems with dierent wavelengths = 2c/, the physical process should be identical in these systems12. By the way, =

E 1

0 denes the relativistic eld threshold Er=mc/e. For a microwave at =30cm, we have Er=10.7MV/m (Ir=1.5107W/cm2) and nc=1.21010cm3.

All simulations are done with the JPIC code12, which self-consistently solves the Maxwells equations and relativistic Lorentz equations for electrons and ions in a two-dimensional space. JPIC applies a eld solver free of numerical dispersion in the propagation axis and can accurately simulate the dynamics of half-cycle electromagnetic waves31. The simulation of transition radiation in Fig.2 is performed in the xz plane. An overdense plasma is used to represent the conductor and its density has a negligible eect on the results. In Fig.2, we take a density n0=50nc and resolution of 100 and 80 grids per wavelength (=30cm) along thez- and x-axes, respectively. The simulation of bubble formation in Figs3 and 4 is done in the yz-plane. Since the collision

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frequency in air is ~1012Hz, i.e., thousands of collisions per cycle, which makes the resolution of individual collisions unrealistic in the present work. For the simulation, we embed the eective air friction force within an electron energy range [1eV, 1GeV]14,18 into the Lorentz equation of electrons. The microwave eld Ex is perpendicular to the simulation plane and propagates from a vacuum to a uniform plasma along the z-axis. The microwave pulse has the form

=

E E y R t z c t z c

exp( / )sin [ ( / )/ ] sin [ ( / )]

x 0

V dt E f

/( )

72, where

g=9.8m/s2 is the gravitational acceleration, D is the bubble size and T0 is the initial air temperature. If thermal energy is transferred primarily by the air convection, one has H=Cp0SuT, where H is the total heat power of bubble, Cp is the specic heat capacity of air, 0 is the air density, and S = (D/2)2 is the cross sectional area of bubble. From these relations, we obtain

u g

T T g C

At the room temperature T0 = 293 K, we have 0 = 1.2 kg/m3 and Cp = 1 103J/K/kg73. For a bubble with D=30cm andH= 100W, the convection speed is u 23cm/s and temperature increase is T 5K. We also get the Reynolds, Peclet, and Rayleigh numbers of this system as 4.5103, 3.2103, and 1.4107 respectively. These dimensionless numbers conrm that convection is the dominant mechanism of heat transport72.

References

1. Arago, F. Sur le tonnerre. Annuaire au Roi par le Bureau des Longitudes. Notices Scientiques p. 221 (1838).2. Singer, S. The nature of ball lightning (Plenum, New York, 1971).3. Barry, J. D. Ball lightning and bead lightning: Extreme forms of atmospheric electricity (Plenum, New York, 1980).4. Stenho, M. Ball lightning: An unsolved problem in atmospheric physics (Kluwer Academic and Plenum Publishers, New York, 1999).

5. Rakov, V. A. & Uman, M. A. Lightning: Physics and eects (Cambridge Univeristy Press, Cambridge, 2003).6. Ball, P. First spectrum of ball lightning. Physics 7, 5 (2014).7. Dijkhuis, G. C. A model of ball lightning. Nature 284, 150151 (1980).8. Lodge, O. J. Lightning conductors and lightning guards (Whittaker and Co. and Bell and Sons, London, 1892).9. Kapitza, P. L. On the nature of ball lightning. Doklady Acad. 101, 245248 (1955).10. Dawson, G. A. & Jones, R. C. Ball lightning as a radiation bubble. Pure Appl. Geophys. 75, 247262 (1969).11. Zheng, X.-H. Quantitative analysis for ball lightning. Phys. Lett. A 148, 463469 (1990).12. Wu, H.-C. JPIC & How to make a PIC code. Preprint at http://arxiv.org/abs/1104.3163 (2011).13. Dwyer, J. R. & Uman, M. A. The physics of lightning. Phys. Rep. 534, 147241 (2014).14. Dwyer, J. R., Smith, D. M. & Cummer, S. A. High-energy atmospheric physics: Terrestrial gamma-ray flashes and related phenomena. Space Sci. Rev. 173, 133196 (2012).

15. Moore, C. B. et al. Energetic radiation associated with lightning stepped-leaders. Geophys. Res. Lett. 28, 21412144 (2001).16. Dwyer, J. R. et al. X-ray bursts associated with leader steps in cloud-to-ground lightning. Geophys. Res. Lett. 32, L01803 (2005).17. Howard, J. et al. RF and X-ray source locations during the lightning attachment process. J. Geophys. Res. 115, D06204 (2010).

2 2 2 , where E0= 310MV/m is the eld amplitude, R= 9cm is the spot size, = 2ns is the duration, and /2= 1GHz is the central frequency. The full width at half maximum of the eld envelope is /2 = 1 ns. There are 80 and 64 grids per wavelength along the z and y axes respectively. Air molecules take an average molecular weight 28.97 and charge state Z= 1. In Fig.3, to clearly recognize the bubble structure, color bars are based on specic values at each moment, and therefore no quantitative relation exists among the dierent panels.

Microwave can penetrate deeply into the tissue and cause an inuence by thermal eects. Microwave hearing48,49,56 is the lowest power eect on humans and occurs when the absorbed energy in the brain tissue reaches 10J/g for a 10 s pulse. For a typical adult brain with 14 cm in diameter and 1.4kg in weight, we get an energy ux threshold of 0.1mJ/cm2. Experiments48,49 show this hearing eect induced by 0.23GHz microwave pulses with 1100s in duration. Theoretical analysis reveals that rapid (~s) temperature rise (~106 degree) leads to a thermoelastic expansion of tissue, which launches an acoustic wave travelling by the skull to the inner ear. The audio frequency is located at an audible high-frequency band of 715kHz, which is responsible for the sounds of hiss, buzz, knocking or clicking48. Although rather resistant to ionizing radiation69, sensory nerves in the peripheral nervous system are found to be particularly sensitive to the microwave70. Occurring at 0.1 J/cm2 56, nerve damage can lead to a numbness in the limbs71. In our theory, the microwave reaches ~1 J/cm2 for the ball formation, which is enough to induce both microwave hearing and nerve damage on witnesses.

In an electric field Ef sin(2f t) with frequency f, electrons in air gain a drift velocity Ve = eEf sin(2ft)13, where e is the electron mobility. The amplitude of electron drift is then

f

= =

(2 ) 1 . Taking Ef= 1V/cm, f= 50Hz and e= 0.6m2/V/s18, we have 38cm. Ions have 0.1mm due to its small mobility.

The microwave bubble will heat up the initially uniform air by electron-molecule collisions. When the temperature rises, air will expand and be lifted up by buoyancy, which leads to air convection. Assuming the temperature change T is small, the convection speed is given by =

u gD T T

/ 0

e e f

0

=

H

D

4

C T

p

0 0

1/3

(3)

1/3

=

0 0

2 5

16( ) (4)

H

D

2

p

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18. Moss, G. D. et al. Monte Carlo model for analysis of thermal runaway electrons in streamer tips in transient luminous events and streamer zones of lightning leaders. J. Geophys. Res. 111, A02307 (2006).

19. Gurevich, A. V. On the theory of runaway electrons. Sov. Phys. JETP 12, 904912 (1961).20. Gurevich, A. V., Milikh, G. M. & Roussel-Dupre, R. A. Runaway electron mechanism of air breakdown and preconditioning during a thunderstorm. Phys. Lett. A 165, 463468 (1992).

21. Babich, L. P. et al. Analysis of the experiment on registration of X-rays from the stepped leader of a cloud-to-ground lightning discharge. J. Geophys. Res. 118, 2573 (2013).

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This work was supported by the Thousand Youth Talents Plan, NSFC (No. 11374262), and Fundamental Research Funds for the Central Universities. We thank W. M. Wang and S. M. Weng for discussions on collisional PIC, Z. H. Wang for instruction in laser cavities, and M. Y. Yu for helpful comments.

Competing nancial interests: The author declares no competing nancial interests.

How to cite this article: Wu, H.-C. Relativistic-microwave theory of ball lightning. Sci. Rep. 6, 28263; doi: 10.1038/srep28263 (2016).

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