Pyrogenic vorticity from windward and lee slope fires

Research into dynamic bushfire behaviour conducted over the last few years has shown that bush-fires burning on lee-facing slopes can exhibit atypical forms of propagation under extreme weather conditions. For instance, recent numerical simulations have indicated that pyrogenic vorticity (the curl of the velocity field) is a key driver of rapid lateral fire spread - as a consequence, this type of fire propagation is being referred to as Vorticity-driven Lateral Spread (VLS). The VLS phenomenon has also been reproduced in laboratory-scale experiments, some of which show clear vortex structures. The presence of vorticity in the experiments and simulations suggests that additional insights into the physical processes may be drawn from direct examination of the fluid dynamical equation governing vorticity. We neglect Coriolis effects and divergence but do not make the Boussinesq approximation, giving the governing vorticity equation: ^D_Dt = (? · ?)u + ¹₂ × ?p + ? × F, where u is the velocity field, ? = ?×u is the vorticity, ? is the density, p is the pressure and ?×F represents eddy viscosity and body forces. In this paper we provide a rudimentary theoretical analysis of the vorticity equation for the separate situations of fires burning on leeward and windward sides of a triangular ridge under the influence of a strong cross-wind and assumed separation of the flow in the lee of the ridge. We consider the behaviour of the right and left flanks of the fire with some simplifying assumptions about the effects of the fire on the local wind flows. The analyses indicate that on the leeward slope the fire should produce pyrogenic vorticity in the vertical direction due to tilting of ambient lateral vorticity (due to the separated boundary layer) by the plume of the fire. As vertical vorticity is produced it is enhanced through stretching by the accelerating buoyant updraft. Longitudinal vorticity is generated in the downwind direction by tilting of the vertical vorticity in addition to the presence of baroclinic torques caused by pyrogenic differences in density of the flow. The generation of pyrogenic vorticity over a fire on a lee slope arises due to the fact that separation of the flow in the lee of the ridge creates a sheet of vorticity at some height above the lee slope. This region of ambient vorticity occurs at a height where the pyrogenic velocity gradients are large enough to drive interactions that result in tilting of the ambient vorticity. For the windward fire case, the ambient vorticity only occurs close to the surface where the pyrogenic velocity gradients required to tilt the ambient vorticity are negligible. So while some longitudinal vorticity can arise through baroclinic forcing, the case of a windward fire is far less likely to produce the strong vertical vorticity required to drive the VLS phenomena. The same conclusions will also hold for fires burning on flat ground. While these analyses should be considered as preliminary and approximate, they do provide some useful insights into the dynamics of the VLS phenomenon. For instance they provide a physical explanation of why the VLS phenomenon occurs exclusively in connection with steep, lee-facing slopes, or with lee slopes that possess features such as sharp bluffs, which act to promote flow separation. Moreover they demonstrate the importance of the various driving factors, namely tilting of ambient lateral vorticity, stretching of vertical vorticity by Lagrangian acceleration of the buoyant plume and baroclinic forcing arising due to the effect of the heat of the fire on the density of the flow.