Pointy-Headed Fires: On the Convex Duality Between Fire Shapes and Spread Rates in Fire Growth Models

Background: some widely used wildland fire behavior models like FARSITE propagate fire fronts by computing the front-normal velocity (spread rate) as a function of local inputs and the front-normal direction. Such models are sometimes observed to cause the collapse of crown fires into sharp wedge shapes that eliminate heading fire behavior. Aims: we set out to document this phenomenon, and more generally understand the relationships between fire shapes and spread rate functions. Methods: the phenomenon is studied both mathematically and through simulation experiments. Non-smooth fire fronts are theorized mathematically by an Eikonal partial differential equation ($H(x, \tau, D\tau) = 1$), where the unknown $\tau(x)$ is the time-of-arrival function and the Hamiltonian $H(x, t, p)$ is positively homogeneous and possibly non-convex in $p$; convex analysis is used to study viscosity solutions in constant conditions. Results: we show that a fire spread model preserves the smoothness of fire fronts if and only if it is equivalent to using the Huygens principle. Non-trivially, this is equivalent to a convexity criterion on the inverse spread rate profile, which is then the polar dual of the Huygens wavelet; this corresponds to Hamiltonian-Lagrangian duality. The relevance of smoothness-destroying models to crown fire is debated. Exact analytical formulas are derived for fire growth in spatially constant conditions. Conclusions: our understanding of fire spread models is improved by solving the spread equations in more general ways than previously known. In particular, the collapse of heading crown fires into sharp shapes is now explained. Smoothness-destroying spread models cannot be simulated by algorithms based on travel time like cellular automata; their general well-definedness remains an open question. Fire modelers can use these findings to guide their search for improved crown fire models, and more generally to verify the accuracy of numerical implementations.