Stepped Leader Growth: Lattice Dielectric Breakdown Model (2D)

Discharge Parameters

Branching exponent η 2.5

Cloud field strength 1.0

Step interval (μs) 35

Grounded structure position 0.0

Controls

Live Readouts

Leader initiating…

Elapsed time

0 μs

Avg. descent speed

0 ×10⁵ m/s

Active branch tips

Steps taken

How it works (2D lattice model)

The 3D version of this simulator approximates the driving electric field with a hand-written formula. This 2D companion instead does what the real dielectric breakdown model (Niemeyer–Pietronero–Wiesmann, 1984) actually specifies: it numerically solves Laplace's equation for the electric potential φ on a lattice, every single frame, with successive-over-relaxation (SOR):

∇²φ = 0   subject to   φ = 1 on the leader channel (and cloud plane)
                        φ = 0 on the ground row and the grounded rod

SOR update:  φ(x,y) ← φ(x,y) + ω·[ ¼(φ_left+φ_right+φ_up+φ_down) − φ(x,y) ]

Every empty lattice cell touching the growing channel is a growth candidate. Its local field is the actual potential drop across that one lattice bond, E = 1 − φ(site) — literally the discrete electric field the model just solved for, not a hand-tuned proxy. The next cell to ionize is drawn from the true field-weighted distribution over every candidate at once (roulette-wheel selection), exactly like the textbook DBM growth rule:

P(site i) = E(site i)^η / Σⱼ E(site j)^η ,   E(site) = 1 − φ(site)

η (branching exponent) — low η spreads growth probability almost evenly across the whole perimeter, producing a bushy fractal; high η concentrates it on the single strongest-field cell, funnelling the channel almost straight down.
Cloud field strength — adds an extra downward-growing bias term (representing space-charge build-up near the tip) on top of the solved Laplacian field, without which the two rows would compete for the exact same physics regardless of the slider.
Step interval — the assumed real microseconds between steps, used only to convert lattice steps into the elapsed-time and descent-speed readouts (real stepped leaders average ≈1–2.6×10⁵ m/s).
Grounded structure position — a second φ=0 sink on the lattice. Because it is a genuine second boundary condition, the Laplace solver itself shows a steeper local potential drop (a stronger discrete field) toward it as the leader approaches — real electrostatic attraction falling directly out of the PDE, not an added attraction rule.

When the growth process reaches the ground row or a cell next to the rod, the channel attaches and the return stroke fires. Distances: 1 lattice cell ≈ 18 m.

field solved live on a 96×92 lattice

Leader channel Branch (deeper generation) Grounded structure