Lightning: Growing a Fractal Discharge with the Dielectric Breakdown Model

Growing a spark instead of drawing one

The Dielectric Breakdown Model (DBM), introduced by Niemeyer, Pietronero and Wiesmann in 1984, does not draw a lightning bolt as a curve — it grows one, cell by cell, from the physics of the electric field around it. The model solves Laplace's equation for the electric potential φ in the space surrounding the growing discharge, with the discharge channel itself held at one fixed potential and the far boundary at another:

∇² φ = 0                      (Laplace's equation between growth events)
p_i = φ_i^η / Σ_j φ_j^η      (probability the channel grows into cell i)

Why the result looks like a fractal

At every step, each empty cell adjacent to the existing channel is given a growth probability proportional to the local field raised to a power η. With η = 1 the model reduces exactly to diffusion-limited aggregation, and the resulting pattern has a fractal dimension of roughly 1.7 in two dimensions: a structure that is more than a line but far less than a filled area. Raising η concentrates growth probability even more sharply on the tips of existing branches, since field strength is highest where the channel is already sharpest, producing thinner, more branching structures — exactly the runaway feedback that makes real electrical trees look the way they do.

What a real stepped leader does

Natural lightning grows in a broadly similar way. A stepped leader descends from the cloud in discrete jumps of roughly 50 metres, pausing for tens of microseconds between steps while it re-ionises the air ahead along whichever direction the local electric field is strongest — the same "grow where the field is highest" rule the DBM encodes explicitly. Side branches form when the field briefly favours more than one direction at once, which is why real lightning bolts fork unevenly rather than splitting into perfectly even trees. Once a leader tip gets close enough to the ground (or to an upward streamer rising to meet it), the circuit closes and a return stroke races back up the ionised channel at up to a third of the speed of light, producing the bright flash and the thunder.

Simulating it: re-solve the field after every growth step

Because adding one new conductive cell changes the boundary of the Laplace problem, the field has to be recomputed — usually by relaxation or successive over-relaxation on a grid — before the next growth probability can be assigned. That coupling between geometry and field is what makes DBM expensive compared with a purely random fractal generator, and it is also exactly why the result is not just decorative noise: every branch grows where the physics genuinely says the field is strongest, biased by parameters such as a directional wind drift that tilts the background field and a jaggedness term that adds small-scale randomness on top of the field-driven bias.

Frequently asked questions

Try it live

Everything above runs in your browser — open Lightning Bolt Generator and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.