Relativistic-microwave theory of ball lightning

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.

Microwave generation. Transition radiation is generated from medium surfaces when an electron enters or emerges 26 and can be coherent for an isolated electron bunch 27 . As the electron bunch reaches relativistic energies, its self-fields are predominantly transverse i.e. E b ≃ cB b 28 , which is very close to an electromagnetic wave. In this case, coherent transition radiation can be considered as the reflected wave of the bunch field from the medium surface 29 . Therefore, we can write the radiation energy as is the Fresnel reflection formula, W b,f refers to the total bunch field energy, and ε is the medium permittivity. The radiation is strongest for a metal or perfect conductor where ε → ∞ and R ≈ 1 in microwave region. In addition, a Boltzmann-distributed electron bunch turns out to produce almost the same transition radiation pulse as a monoenergetic one 30 .
The leftmost panel of Fig. 2 shows the transverse field E b,x of a monoenergetic 7 MeV electron bunch with σ = 4 cm, which is normalized to the peak field ≈ . . The bunch field is a unipolar wave with the same profile exp(− z 2 /2σ 2 ) as the electron density along the direction of motion. Using JPIC 12 , we simulate the coherent transition radiation from a perfect conductor in Fig. 2. The radiation field E x is initially opposite to E b,x due to the conductor boundary, diffracts transversely, and quickly evolves into a bipolar pulse. This radiation has a central wavelength λ ≈ 7.5σ = 30 cm (i.e. 1 GHz). The rapid field evolution into the bipolar shape is due to diffraction losses of longer wavelength components in an unipolar pulse 31 . At normal incidence in Fig. 2, the radiation field is radially polarised with a ring-like intensity distribution. Oblique incidence 32 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.
Microwave bubble formation. Laser solitons have been observed in both PIC simulations 33,34 and experiments 35-37 on relativistic laser-plasma interaction. The laser needs to exceed the relativistic field threshold E r = mcω/e 38 and is typically multi-cycle. The plasma is underdense with an initial density n 0 < n c , where n c = ε 0 mω 2 /e 2 is the critical density 39 . During the laser propagation in the plasma, the self-phase modulation effect 40 leads to a dramatic spectral broadening, which makes some part of laser energy to shift 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 force 41 . 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 diffracted. In contrast to the mechanism discussed above, we find that the initial plasma must be overdense with n 0 ≥ n c , where n c ≈ 1.2 × 10 10 cm −3 at ω/2π = 1 GHz. The existence of such a bubble-formation regime for single-cycle waves indicates self-consistency of our theory. The collisional effect is included by embedding air friction 14,18 into JPIC. We launch microwave pulses with wavelength λ = 30 cm into a uniform plasma. The simulation shows that the threshold field required for bubble formation is bl c r At 1 GHz, we have E r ≈ 10.7 MV/m and E bl ≈ 11E r ≈ 120 MV/m, which is highly relativistic. Equation (2) clearly shows that the field needs to be greater than E c to efficiently accelerate electrons, and reach the relativistic regime to completely expel electrons by the relativistic pondermotive force. Surprisingly, E r matches with E c to make the bubble formation possible. Here, we check the bunch parameters for giving the threshold field E bl . For the case in Fig. 2, we get n b0 ≈ 3.7 × 10 11 cm −3 and N b ≈ 3.7 × 10 14 .
In Fig. 3, we take n 0 = 4n c and a microwave field of 310 MV/m, and let t = 0 when the field touches the plasma. Snapshots of microwave field and plasma density from t = 1 ns to 11 ns illustrate the entire process of microwave self-trapping and bubble formation. The radiation pressure of microwave first pushes electrons to pile up into a semicircular shell at t = 1 ns and leaves a low-density region at the rear. As the field is reflected by the front shell, peripheric electrons return to the low-density region and close up the cavity at t ≈ 3 ns. The field gets trapped and then evolves into a standing-wave mode. At t = 11 ns, a motionless electron cavity forms about 45 cm deep into the plasma, and then it becomes circular and keeps stable after t ≈ 15 ns. Meantime, heavy ions are slowly pulled out by the charge separation field.
In Fig. 4a,b, snapshots of the stable bubble at t = 19 ns show that the fields 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 ε 0 E 2 /4 ≈ 64 kPa, where E = 170 MV/m is the standing wave amplitude. The periodic conversion between electric and magnetic energies in Fig. 4c confirms the standing wave mode. The confined field oscillates at a longer period of 1.6 ns. This redshift is caused by the Doppler effect and self-phase modulation. The cavity diameter is about 24 cm, half of the wavelength of the trapped field. For a ball shape, the confined field energy in Fig. 4b is about 800J. Tuning the microwave field, the trapped field energy in the bubble ranges from 200J to 1500J.
Three-dimensional field structure of microwave bubbles can be close to that of the light solitons observed in PIC simulation 34 . 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 resonator 26 is similar to that in a cylindrical cavity 28 , which resembles that shown in Fig. 4a.   Explanation of the diverse properties. The properties of ball lightning [2][3][4][5] are summarized from about 5000 published sighting reports. Fig. 2, a planar surface is necessary for microwave generation at least with a size of ball lightning, which can be easily fulfilled in reality. Microwave emission is also affected by the ground reflectivity . The soil permittivity ε increases with its moisture m s 42 . At 1 GHz, we get ε ≈ . − .

Site of occurrence. As shown in
which correspond to  ≈ 25% and 56%, respectively. Rainfall can lead to m s > 60% 43 and thus is favorable for the ball lightning formation. As stated by Stenhoff 4 , more than 50% of reports show that medium or heavy rainfall happens before the observation. Moreover, there is  ≈ 65% for either pure or sea water 44 . Indeed, there are 18 reports at sea 2 and a few reports over rivers 2,4 . 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 after 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 point 4 .
Appearance in aircraft. First, the avalanche electron energy 7.3 MeV is independent of the air density 13 , i.e. altitude. When lightning strikes an aircraft, the same bunch is presumably produced and enters the aircraft with an energy loss of ~2 MeV due to the ~0.6 cm aluminium skin 45 . Second, transition radiation 26 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 reflection side discussed above. Therefore, the same intense microwave will arise inside the aircraft 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 cavity 46 , generated fireballs 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 theory 47 , the internal standing wave will not be disturbed if a glass plate (~5 mm) is much thinner than the wavelength of microwave.
Shape. From dimensional analysis 12 , the microwave bubble of Fig. 4 in reality should be ball-shaped as its micrometre-scale counterpart in laser-plasma experiments [35][36][37] . The full trapping of the field in Fig. 2 can account for the 62 ring-shaped ball lightning reports 2 .
Size. Ball lightning has a common diameter of 20-50 cm 4 . 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 1 ns.
Sound. Hissing, buzzing or fluttering sounds from ball lightning have been reported, which can be perfectly explained by the microwave hearing effect 48,49 . At 0.1 mJ/cm 2 , a microwave pulse (microsecond or shorter) at 0.2-3 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 Jennison's sighting 50 , he was only 0.5 m 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 field. Especially, the sparks are toward opposite directions in two reports 2 , which agrees with the linear polarisation of standing wave in the bubble. 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 lightning 2,4 . However, Stenhoff 4 noticed that some superficial burns are difficult to explain. In the Smethwick event 4,54 , the female witness did not get an electric shock but felt a burning heat all over. Wooding 55 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 Scientific RepoRts | 6:28263 | DOI: 10.1038/srep28263 microwave hearing effect. Her legs were numbed, which can be due to nerve damage by the microwave at 0.1J/ cm 2 56 . When she brushed the ball away with her hand, the ring was burning into her finger. Wooding calculated that this rapid heating would need a resonant microwave at 1 GHz with an field of ~1 MV/m, which agrees well with our model. Others 57 reported skin redness, vomiting and loss of hair, which are typical results of ionizing radiation 58 . 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 fire 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 microwaves 56 .
Motion. Near the ground, ball lightning moves mostly horizontally at about 2 m/s 2 and usually travels with the wind 3 . A light breeze typically at 1.5-3 m/s 59 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 100 W, we obtain a convection speed 23 cm/s for the ball of size 30 cm (see Methods). Thus, the upward motion is not notable compared with the horizontal motion. Several models 2,4 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 waveguide 60 , the ball can enter rooms through chimneys.
Lifetime. The typical lifetime of ball lightning is 1-5 seconds. Statistical analysis 61 shows that increase in humidity decreases the lifetime of the ball, which can be due to microwave absorption by vapour. Experiments 62 show that fireballs in air produced by a 5 kW, 2.45 GHz microwave can last for ~0.5 s after the source is turned off. Our self-organized microwave bubble can have the same potential to persist for a scale of seconds. Zheng 11 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 refilled 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 beam 63 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 mechanism 64 , which supplies a physical evidence of our theory. On attempts to create ball lightning by rocket-triggered lightning 65 , we propose to use ungrounded wires 5 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 thunderclouds 66 . For in situ investigation of ball lightning, we suggest to look for evidence of high-flux energetic electrons. Finally, we note that relativistic terahertz waves could be produced from laser-accelerated hot electrons emerging from solid foils by coherent transition radiation 67 or laser-driven plasma waves in gas target 68 . 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 first time, we revel that ball lightning is an alarm signal of the existence of ultrastrong microwaves and abundantly hazardous electrons near the ground or aircraft. This result is of great significance 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. , where n 0 is the initial plasma density, n c = ε 0 mω 2 /e 2 is the critical density, E 0 is the initial field amplitude, e is the fundamental charge, m is the electron mass, c is the light speed, ω is the angular frequency, and ε 0 is the vacuum permittivity. If ′ n 0 and ′ E 0 are same for any systems with different wavelengths λ = 2πc/ω, the physical process should be identical in these systems 12 . By the way, ′ = E 1 0 defines the relativistic field threshold E r = mcω/e. For a microwave at λ = 30 cm, we have E r = 10.7 MV/m (I r = 1.5 × 10 7 W/cm 2 ) and n c = 1.2 × 10 10 cm −3 .

Methods
PIC simulation. All simulations are done with the JPIC code 12 , which self-consistently solves the Maxwell's equations and relativistic Lorentz equations for electrons and ions in a two-dimensional space. JPIC applies a field solver free of numerical dispersion in the propagation axis and can accurately simulate the dynamics of half-cycle electromagnetic waves 31 . 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 effect on the results. In Fig. 2, we take a density n 0 = 50n c and resolution of 100 and 80 grids per wavelength (λ = 30 cm) along the z-and x-axes, respectively. The simulation of bubble formation in Figs 3 and 4 is done in the yz-plane. Since the collision frequency in air is ~10 12 Hz, 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 effective air friction force within an electron energy range [1 eV, 1 GeV] 14 , where E 0 = 310 MV/m is the field amplitude, R = 9 cm is the spot size, τ = 2 ns is the duration, and ω/2π = 1 GHz is the central frequency. The full width at half maximum of the field envelope is τ/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 specific values at each moment, and therefore no quantitative relation exists among the different panels.
Microwave effects on humans. Microwave can penetrate deeply into the tissue and cause an influence by thermal effects. Microwave hearing 48,49,56 is the lowest power effect on humans and occurs when the absorbed energy in the brain tissue reaches 10 μJ/g for a 10 μs pulse. For a typical adult brain with 14 cm in diameter and 1.4 kg in weight, we get an energy flux threshold of 0.1 mJ/cm 2 . Experiments 48,49 show this hearing effect induced by 0.2-3 GHz microwave pulses with 1− 100 μs in duration. Theoretical analysis reveals that rapid (~μs) temperature rise (~10 −6 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 7-15 kHz, which is responsible for the sounds of hiss, buzz, knocking or clicking 48 . Although rather resistant to ionizing radiation 69 , sensory nerves in the peripheral nervous system are found to be particularly sensitive to the microwave 70 . Occurring at 0.1 J/cm 2 56 , nerve damage can lead to a numbness in the limbs 71 . In our theory, the microwave reaches ~1 J/cm 2 for the ball formation, which is enough to induce both microwave hearing and nerve damage on witnesses.
Electron drift in air. In an electric field E f sin(2πf t) with frequency f, electrons in air gain a drift velocity V e = μ e E f sin(2πft) 13 , where μ e is the electron mobility. The amplitude of electron drift is then . Taking E f = 1 V/cm, f = 50 Hz and μ e = 0.6 m 2 /V/s 18 , we have δ ≈ 38 cm. Ions have δ ≈ 0.1 mm due to its small mobility.
Air convection. 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 Δ T is small, the convection speed is given by = ∆ u gD T T / 0 72 , where g = 9.8 m/s 2 is the gravitational acceleration, D is the bubble size and T 0 is the initial air temperature. If thermal energy is transferred primarily by the air convection, one has H = C p ρ 0 SuΔ T, where H is the total heat power of bubble, C p is the specific heat capacity of air, ρ 0 is the air density, and S = π(D/2) 2 is the cross sectional area of bubble. From these relations, we obtain At the room temperature T 0 = 293 K, we have ρ 0 = 1.2 kg/m 3 and C p = 1 × 10 3 J/K/kg 73 . For a bubble with D = 30 cm and H = 100 W, the convection speed is u ≈ 23 cm/s and temperature increase is Δ T ≈ 5 K. We also get the Reynolds, Peclet, and Rayleigh numbers of this system as 4.5 × 10 3 , 3.2 × 10 3 , and 1.4 × 10 7 respectively. These dimensionless numbers confirm that convection is the dominant mechanism of heat transport 72 .