The fractional Rothermel model: Mittag-Leffler kinetics of wildfire front propagation from a stochastic hierarchical parcel cascade, with application to heading fire collapse

Abstract. In the Rothermel model the heading wind factor is an unbounded power law of midflame wind speed, unphysical at high wind and patched with an ad hoc cap, and its projected-wind extension collapses the heading front above a low fuel-dependent threshold. We recast the rate law as the fractional Rothermel model, a fractional ignition kinetics in which, with memory in the ignition waiting time, unburned-fuel survival solves a Riemann-Liouville relaxation equation with Mittag-Leffler solution, giving a wind factor that recovers the power law at low wind and saturates at high wind. We fix the fractional order rather than fitting it, grounding it in a stochastic parcel cascade whose random branching number yields distributed-order dynamics and whose effective order factorizes into a cascade exponent and a subordinator index. We prove a critical-plateau theorem governing collapse: it occurs only above a fuel-dependent threshold, is confined to a bounded wind window, vanishes for coarse low-exponent fuels, and becomes classical persistent collapse only in the unbounded-plateau limit, and we prove these structural results for every fractional order in the relevant range. Grounding the plateau in documented reaction intensities, short grass is the standard fuel whose window onsets and is cured within operational winds, and there the predicted spread rate matches in shape the field-based grassland relationship of Cheney and colleagues, which the classical power law overshoots. A fractional front framework adds ignition memory and long-range spotting.