Topographic Disorder, Wind Coupling, and Directional Fire Spread: Critical Behavior in a Terrain-Weighted Forest Fire Model

We introduce the Terrain-Weighted Forest Fire Model (TFFM), a lattice model in which fire spreads on a spatially correlated Gaussian height field with the asymmetric bond probability $p_{i\to j}=\mathrm{c}\mathrm{l}\mathrm{i}\mathrm{p}[e^{-\beta +\gamma (h_j-h_i)},0,1]$, plus an additive wind bias. Simulations up to $L=8192$ reveal a sharp active-to-inactive transition whose critical suppression threshold ${\beta }_c$ is even in $\gamma$, decreases with $\mathrm{\mid }\gamma \mathrm{\mid }$, and decreases as the terrain correlation length ${\sigma }_h$ is reduced: slope asymmetry acts as a suppressant because downhill bonds are penalized and fire stalls at local elevation maxima. For rough terrain and low tree density, the fire fails to percolate even at zero suppression. Finite-size scaling gives a front-velocity exponent $\delta =0.34\pm 0.02$, identical for smooth and rough terrain and matching neither directed percolation (${\nu }_{\parallel }-{\nu }_{\perp }\approx 0.56$) nor isotropic percolation ($\approx 0.18$). The single-seed survival probability at ${\beta }_c$ is independent of $L$ and decays extremely slowly, with a running exponent falling from $\approx 0.09$ to $\approx 0.04$, excluding directed percolation and suggesting a survival probability that remains finite at criticality, consistent with the $L$-independent value $P^{\ast }\approx 0.5$ at which $P_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{v}}$ drops to zero. The correlation-length exponent cannot be resolved at these sizes, consistent with slow crossover corrections from quenched correlated disorder. Wind raises ${\beta }_c$ by a factor of 2–4, produces a sharp onset of downwind fire-scar drift at weak coupling, and, at high terrain coupling, decreases the burned fraction at boundary crossing—a terrain-wind competition effect absent from isotropic bond-disorder models. The model yields fire-risk thresholds and fire-scar signatures comparable to satellite burn-scar data.