Fingering instability in wildfire fronts

Abstract

A two-dimensional model for the evolution of the fire line – the interface between burned and unburned regions of a wildfire – is formulated. The fire line normal velocity has three contributions: (i) a constant rate of spread representing convection and radiation effects; (ii) a curvature term that smooths the fire line; and (iii) a Stefan-like term in the direction of the oxygen gradient. While the first two effects are geometrical, (iii) is dynamical and requires the solution of the steady advection–diffusion equation for oxygen, with advection owing to a self-induced ‘fire wind’, modelled by the gradient of a harmonic potential field. The conformal invariance of this coupled pair of partial differential equations, which has the Péclet number $\textit {Pe}$ as its only parameter, is exploited to compute numerically the evolution of both radial and infinitely long periodic fire lines. A linear stability analysis shows that fire line instability is possible, dependent on the ratio of curvature to oxygen effects. Unstable fire lines develop finger-like protrusions into the unburned region; the geometry of these fingers is varied and depends on the relative magnitudes of (i)–(iii). It is argued that for radial fires, the fire wind strength scales with the fire's effective radius, meaning that $\textit {Pe}$ increases in time, so all fire lines eventually become unstable. For periodic fire lines, $\textit {Pe}$ remains constant, so fire line stability is possible. The results of this study provide a possible explanation for the formation of fire fingers observed in wildfires.

Penguin Huddling: A Continuum Model

“…is the complex representation of the normal vector in the z−plane [34]. The function σ (θ) is found by solving the following integral equation [22,26,29]…”

“…Hence the choice M = N + 1 in (0, π) is made when numerically solving the integral equation ( 15) for σ (θ). The PG equation ( 23) becomes a system of n coupled ordinary differential equations (ODEs) determining the time evolution of the n Laurent coefficients (see [20,21,34]). In this work, the MATLAB routine ode15i was used to solve this ODE system.…”

“…Solidification and melting problems have also been extensively studied using analogous techniques, see [23][24][25][26][27][28] and [29]; the latter of these examines the freezing of porous media flow by freeze pipes, with the velocity of the free boundary dependent on both interior and exterior temperature gradients, similar to the full penguin problem. Free boundary problems arise in other natural processes, including, but not limited to, dendritic crystal growth [30,31] and flame front propagation [32][33][34]. Continua may also be used to model other collective behaviours [4,35,36] such as fish schooling [37], bird flocking [38], bat swarming [39,40] and ant colony formation [41,42].…”

“…As such, the wind flow and temperature transport in the region exterior to the penguin huddle can be modelled by the Laplace and the steady advectiondiffusion equations, respectively. This coupled pair of partial differential equations (PDEs) is conformally invariant [43] -a property which motivates the use of a conformal mapping approach in the numerical method [21][22][23][24]34]. The exterior region and the huddle boundary in the physical z−plane can be conformally mapped to the exterior and boundary of the unit disk in some "mathematical" (or "canonical") ζ −plane, and vice versa.…”