The 3D companion sim runs a 1-D chain of fault patches. This one is the genuine 2-D fault plane underneath that simplification: a grid of patches spanning both along-strike position x and down-dip depth y, each obeying the same linear slip-weakening friction and steady tectonic loading (Rice 1993; Ida 1972 / Palmer & Rice 1973), now coupled to four elastic neighbours instead of two:
τ_strength(δ,y) = τ_d + (τ_s − τ_d)·B(y)·max(0, 1 − δ/Dc)
τ_local(x,y,t) = τ_bg(t) − k_load·s_{x,y} + k·(s_left+s_right+s_up+s_down − 4·s_{x,y})
slip rate v = max(0, τ_local − τ_strength) / η, η = μ / (2·Vs)
B(y) = 1 + 14·(y/W)⁶ + 7·(1 − y/W)¹⁰ (brittle-ductile + near-surface strengthening)
B(y) is what makes this a real 2-D result and not just a flattened 3-D scene: real faults only rupture seismically between a shallow velocity-strengthening layer and a deeper brittle-ductile transition, so strength is boosted near both the top and bottom rows of the grid. A 1-D chain has no depth axis and cannot represent that at all — here it emerges as a genuine down-dip arrest while the rupture keeps growing along strike, producing the elongated, strike-parallel rupture patches real earthquakes actually have. The −k_load·s term (a loader-plate spring, standard in Burridge-Knopoff-style fault models) is also new here: without it a patch that fully weakens never re-locks, because background stress keeps climbing but nothing relieves it locally — confirmed by direct simulation of the 1-D model's own equations, which never lets a rupture arrest. Here each patch's own slip relieves the very stress that drove it, so the fault actually re-locks and the loading → rupture → arrest cycle repeats.
- Stress Drop Δτ — more stored energy per patch drives a faster, larger rupture in both directions.
- Crustal Rigidity μ — sets the elastic coupling and the shear-wave speed reference used for the supershear threshold.
- Slip-Weakening Dc — the breakdown-zone size; larger Dc spreads fracture energy over more slip and slows nucleation.
- Loading Rate — how fast background tectonic stress rebuilds between events (compressed to seconds here).
Watch the two rupture-front speeds separately: along-strike frequently crosses the shear-wave speed Vs into supershear (as in the 1999 İzmit and 2001 Kunlun strike-slip earthquakes), while down-dip almost always stays sub-Rayleigh and gets capped well before it reaches the grid edge — the seismogenic width is not assumed here, it is measured from where B(y) actually stopped the rupture. Magnitude uses the directly-measured ruptured area and mean slip: M₀ = μ·Area·⟨slip⟩, Mw = ⅔log₁₀M₀ − 6.07 (Hanks & Kanamori, 1979).