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 flash5 starts with a negative leader propagating downward in a stepping process with each step tens of metres. This stepped leader has a corona 1–10 m in width. Moore et al.15 first detected >1 MeV radiation from a stepped leader. It was then observed that each step emits an x-ray burst16, which intensifies 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.

Friction force of electron motion in air is maximum at an energy of 100 eV, which defines a critical electric field E c ≈ 30 MV/m14. Fields above E c 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 field between the leader tip and the ground and serve as seed electrons to undergo avalanche in air20. The electron flux quickly rises as exp(z/L), where L is the avalanche length. The electron energy follows a Boltzmann distribution exp(−k e /7.3 MeV), such that the average energy is 7.3 MeV. Latest data analysis21 shows that collimating relativistic electrons are required to explain observed x-rays from the stepped leader and should be either Boltzmann-distributed at 7 MeV or monoenergetic from 1 to 10 MeV.

The isolated x-ray bursts from the stepped leader are much shorter than 1 μs16. 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-10 ns23 and can be as short as 1 ns24. 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-defined relativistic electron bunch in a ball lightning event (see Fig. 1b). For simplicity, we assume that this bunch has a density profile n b = n b0 exp(−r2/2σ2), where n b0 is the peak density and σ is the characteristic radius. We take a bunch size (≃4σ) of tens of cm, i.e. ~1 ns in duration. As discussed later, a bunch with total electron number N b = (2π)3/2n b0 σ3 ≈ 1014 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 7–30 cm near the ground14 and should support the rapid amplification of these nanosecond bunches. Another mechanism25 predicts that the leader can directly generate ~1016 energetic electrons on a timescale of 1 ns 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-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 surface29. Therefore, we can write the radiation energy as

where 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 one30.

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(−z2/2σ2) 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 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 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.

Figure 2 PIC results of microwave generation. Distribution of the initial bunch field and microwave fields at times 0.8 ns, 1.5 ns and 2 ns. The field is normalized to the bunch peak field E b0 . In the leftmost panel, the bunch is left-going to the plasma surface at z = 0. The white circle marks the bunch region with a density of 0.5n b0 . The radiation is a reflection of the bunch field and propagates along z. Arrows point to the field propagation direction. Parameters are given in the text and Methods. Full size image

Microwave bubble formation

Laser solitons have been observed in both PIC simulations33,34 and experiments35,36,37 on relativistic laser-plasma interaction. The laser needs to exceed the relativistic field threshold E r = mcω/e38 and is typically multi-cycle. The plasma is underdense with an initial density n 0 < n c , where n c = ε 0 mω2/e2 is the critical density39. During the laser propagation in the plasma, the self-phase modulation effect40 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 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 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 × 1010cm−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 friction14,18 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

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 × 1011cm−3 and N b ≈ 3.7 × 1014.

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.

Figure 3 PIC results of microwave self-trapping and bubble formation. Snapshots of the microwave electric field E = |E x |, magnetic field , electron density n e and ion density n i from t = 1 ns to 11 ns. Vertical dashed line marks the plasma surface. Parameters are given in the text and Methods. Full size image

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 E2/4 ≈ 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.

Figure 4 PIC results of stable microwave bubble. (a) Snapshots of the microwave electric field E, magnetic field B, electron density n e and ion density n i at t = 19 ns. White arrows mark the magnetic field direction. (b) Field energy density and plasma density n e,i verses y across the bubble centre. (c) Evolution of the electric field, electric field energy W e and magnetic field energy W m in the bubble. Parameters are the same as Fig. 3. Full size image

Three-dimensional field 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.

Explanation of the diverse properties

The properties of ball lightning2,3,4,5 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 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 and , 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 Stenhoff4, 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 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 point4.

Appearance in aircraft

First, the avalanche electron energy 7.3 MeV is independent of the air density13, 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 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 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 cavity46, 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 theory47, 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 analysis12, the microwave bubble of Fig. 4 in reality should be ball-shaped as its micrometre-scale counterpart in laser-plasma experiments35,36,37. The full trapping of the field in Fig. 2 can account for the 62 ring-shaped ball lightning reports2.

Size

Ball lightning has a common diameter of 20–50 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 1 ns.

Sound

Hissing, buzzing or fluttering sounds from ball lightning have been reported, which can be perfectly explained by the microwave hearing effect48,49. At 0.1 mJ/cm2, 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 sighting50, 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 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 100 Hz, twice the frequency of the adjacent power lines (35 kV, 50 Hz). The latter is only 20 m from the ball and can produce a 50 Hz electric field of ~1 V/cm52 at the ball. This field can induce electron drift on the ball surface by tens cm (see Methods). This drift motion can perturb the spectral emission in the plasma shell. The spectral intensity should be independent of drift direction and varies at 100 Hz. The ball is attached to the soil on a hillside, where electrons cannot feel the oscillating field due to the screening effect. Thus, Si, Fe and Ca in soil glow steadily51.

Odour

Ionized air can produce O 3 and NO 2 5,53, 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, Stenhoff4 noticed that some superficial 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 microwave hearing effect. 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 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. 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 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 microwaves56.

Motion

Near the ground, ball lightning moves mostly horizontally at about 2 m/s2 and usually travels with the wind3. A light breeze typically at 1.5–3 m/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 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 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 1–5 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 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. 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 refilled by microwave heating. Non-neutral plasmas shown in Fig. 4b can further resists the recombination loss.