Lightning · grown from the equations of electricity
Electric Fire
Every flash on this page is computed: the leader feels its way down through the electric field one step at a time, the return stroke lights it from the ground up, and the thunder you hear is put together from that bolt’s own shape.
Scroll ↓ to walk through one flash
01 · charging
A cloud charges itself
Inside a thunderstorm, soft hail (graupel) falls through rising ice crystals. Where they collide they swap electric charge, and the updraft carries the light, positively charged crystals to the top. The main negative charge settles near the −15 °C level, about six kilometres up.
On the ground a field mill feels it. In fair weather the air near the ground carries about +100 volts per metre; under the storm that reverses and grows to several thousand.
Air needs about three million volts per metre to break down. Balloons sent into storms rarely find more than a few hundred thousand, so how a flash gets started is still argued about.
02 · the stepped leader · slowed ×700
Feeling its way down
A flash to ground begins with a faint channel of ionised air reaching down from the cloud in jumps of about 30 metres, pausing a few tens of microseconds between them, at about 150 km/s on average. It forks as it goes. In 1934 Basil Schonland and H. Collens caught it on film with a camera whose lenses spun, a design Charles Vernon Boys had described in 1926.
Here it is grown by the dielectric breakdown model (Niemeyer, Pietronero and Wiesmann, 1984). The page solves for the electric potential around the channel. Each step goes to a neighbouring cell at random, with odds that rise steeply with the local field, as fieldη.
03 · attachment
Who gets struck
Nothing on the ground is chosen until the leader’s tip is within the striking distance, about 10 × I0.65 metres for a peak current of I kiloamperes (91 m at 30 kA). Then the nearest grounded tip sends up a leader of its own, and the flash goes wherever the two meet.
That is why tall, pointed things get hit, and why Franklin’s rod works. Engineers roll an imaginary 45 m sphere over a building to see which parts it could touch.
04 · the return stroke
The flash runs upward
Once connected, the channel discharges from the ground up. A wave of current climbs it at around a third of the speed of light, carrying typically 30,000 amperes and heating a core a few centimetres across to some 30,000 °C. That is the blinding flash; the faint leader before it is invisible to the eye.
Then, a few hundredths of a second apart, faster “dart” leaders run back down the same hot channel and set off more strokes, typically three or four in all. That is why lightning flickers.
05 · thunder
Thunder, read from the bolt
Every few metres of channel heats explosively and sends out a pulse of sound, loudest at right angles to that stretch of channel. You hear each piece at its own distance divided by the speed of sound, about 3 seconds per kilometre. Here the bolt is coloured by when its sound reaches you, and lights up as you hear it.
Pieces that face you and lie at the same distance arrive together as a clap; the rest roll in as rumble. Air soaks up the high notes with distance. And because air cools with height, sound from low down bends upward and passes over a distant listener, so thunder is rarely heard beyond about 25 km.
06 · 1752
The kite and the key
Benjamin Franklin proposed testing whether lightning was electricity by catching it on a tall pointed rod. In May 1752 Thomas-François Dalibard did so at Marly-la-Ville, near Paris. Franklin’s own version, flown in Philadelphia that summer (June, according to Joseph Priestley’s 1767 account), used a silk kite with a wire at its tip and a key at the end of the wet string.
…the pointed wire will draw the electric fire from them…Franklin to Peter Collinson, 19 October 1752
The kite was never struck. The storm’s field charged the wet string: its loose fibres stood apart, sparks jumped from the key to his knuckle, and he charged a Leyden jar from it. In August 1753 Georg Wilhelm Richmann was killed in St Petersburg trying a similar experiment.
07 · the instrument deck
Your storm
Tap the sky to start a leader there.
Part two · 1777 and now
Lightning in a block
In 1777 Georg Christoph Lichtenberg sprinkled dust on a charged plate of resin and found it gathered into branching stars along the paths of tiny sparks. Today the same figures are made inside clear acrylic. An electron accelerator fires a beam into the block; the electrons come to rest a short way inside and stay trapped there as a sheet of charge. Strike the block with a sharp point and the charge rushes out through branching channels that crack the plastic as they go, frozen for good.
The same model as the leader, with one change: the trapped electrons are the source of the field. Lower η fills the block with fine fronds, higher η makes fewer, longer channels.
“Classic model” is the 1984 set-up: a point held at one voltage inside a circle held at another, with no trapped charge. At η = 1 the pattern’s fractal dimension comes out near 1.7, the value found in 1984 and the same as for diffusion-limited aggregation. At η = 2 it drops to about 1.4.
Part three · working
How the storm is computed
The leader. A vertical slice of air 2.4 km wide and 1.8 km tall is cut into 20 m cells. The cloud’s base, at 1.26 km, is held at the cloud’s potential and the ground at zero, together with the tower, house and tree. The potential is relaxed with over-relaxation (SOR) each time the channel grows. The channel counts as a conductor that loses a little potential along its length. Each step goes to one of the eight neighbouring cells with odds proportional to (potential drop / distance)η.
Attachment. When the lowest tip comes within the striking distance of the ground or a grounded object, that point grows an upward leader by the same rule, in reverse. The flash attaches where the two touch.
Drawing. Each 20 m step is split into eight pieces with small random kinks. The channel also wanders in depth, which matters for the thunder.
Timing and light. The leader’s front comes down at 1.5 × 105 m/s. Return strokes climb at 108 m/s. Branches light when the wave passes their junction, and each branch carries current in proportion to the root of its length. A stroke’s light is a sum of decays (40 µs, 0.6 ms, 6 ms, 40 ms), integrated over each video frame’s exposure. In real time a stroke is therefore one blinding frame and an afterglow, and slowed down it is a wave. Later strokes follow only the main channel, each led by a dart leader at 2 × 106 m/s.
Inside the cloud. The channel climbs to the negative charge near 6 km and spreads sideways for several kilometres. This part is a random walk, not a field solution. It is seen only as glow through the cloud, but it supplies most of the thunder’s length.
Thunder. Following A. A. Few’s description and A. Glassner’s synthesis, every 2.5 m piece sends out an N-shaped pressure pulse 4 to 8 ms long, weighted toward directions within about 30° of its normal and spread as 1/r. It arrives after r / c, with c = 331.3 √(1 + T/273.15) m/s. Pieces are grouped by distance and low-passed where air absorption reaches 6 dB (absorption taken as about 5 dB/km at 1 kHz, rising as f1.7). Sound from a piece at height z is cut beyond about √(2Rz), where R ≈ 88 km is the curvature of sound rays in still air cooling by 6.5 K per km. A few dozen softened echoes stand in for the land around you. Wind is ignored.
Field mill. The ground field comes from three charges and their images in the ground: +40 C at 10 km, −40 C at 6 km and +4 C at 2 km, on top of the fair-weather +100 V/m. Each flash removes part of the negative charge, which then rebuilds over tens of seconds.
Lichtenberg block. Poisson’s equation with a uniform trapped charge, grounded rim and channels, with growth ∝ (potential)η. In classic mode there is no charge, a circular boundary and four-neighbour growth, as in 1984.
Checks
Computed in the page unless marked.
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Part four · people
Catching lightning
Sources
- B. Franklin, letter to Peter Collinson, 19 October 1752 (the kite), printed in the Pennsylvania Gazette and read to the Royal Society that December.
- J. Priestley, The History and Present State of Electricity, London (1767).
- C. V. Boys, “Progressive lightning”, Nature 118, 749 (1926).
- B. F. J. Schonland and H. Collens, “Progressive lightning”, Proc. R. Soc. A 143, 654 (1934).
- A. A. Few, “Power spectrum of thunder”, J. Geophys. Res. 74, 6926 (1969).
- L. Niemeyer, L. Pietronero and H. J. Wiesmann, “Fractal dimension of dielectric breakdown”, Phys. Rev. Lett. 52, 1033 (1984).
- A. Glassner, “The digital ceraunoscope: synthetic thunder and lightning”, IEEE Computer Graphics and Applications 20 (2000).
- M. B. Hastings, “Fractal to nonfractal phase transition in the dielectric breakdown model”, Phys. Rev. Lett. 87, 175502 (2001).
- H. J. Christian et al., “Global frequency and distribution of lightning as observed from space by the Optical Transient Detector”, J. Geophys. Res. 108(D1), 4005 (2003).
- V. A. Rakov and M. A. Uman, Lightning: Physics and Effects, Cambridge University Press (2003).
- J. R. Dwyer and M. A. Uman, “The physics of lightning”, Physics Reports 534, 147 (2014).
- IEEE Std 998 and NFPA 780: the electrogeometric model and the rolling-sphere method for lightning protection.