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

Abstract

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 $p_{i\to j}=\mathrm{clip}[e^{-\beta+\gamma(h_j-h_i)},0,1]$, plus an additive wind bias. Simulations on lattices up to $L=8192$ reveal a sharp active-to-inactive transition whose critical suppression threshold $\beta_c$ is even in $\gamma$, decreases with $|\gamma|$, 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 on a fine $\beta$ grid at $L=2048$--$8192$ gives a correlation-length exponent $\nu=1.8\pm0.17$ from both the susceptibility peak and the width of the transition, and a front-velocity exponent $\delta=0.34\pm0.03$, identical for smooth and rough terrain; neither matches directed percolation ($\nu_\perp=0.73$, $\nu_\parallel-\nu_\perp\approx0.56$) or isotropic percolation ($\nu=4/3$, $\approx0.18$). The single-seed survival probability at $\beta_c$ is independent of $L$ and decays extremely slowly, with a running exponent falling from $\approx0.09$ to $\approx0.04$, excluding directed percolation and pointing to a survival probability that remains finite at criticality, consistent with the $L$-independent value $P^*\approx0.5$ at which $P_{\rm surv}$ drops to zero. 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.