Wildfire Spread Map (2D): Wind Ellipse, Fuel Moisture and Ember Spotting

What is simulated

The same fire world as the 3D page (a 64 m x 64 m forest floor in 1 m cells, a campfire at the centre, embers, water and firebreaks) drawn as a map. The map adds what 3D hides: the arrival-time colouring of burnt ground (red = early, violet = late), the cleared ring, the typical ember range and the dashed single-ellipse prediction.

Spread rate and shape

Head rate R = R0 eta_M (1 + c_w U^1.4) with the Rothermel moisture damping eta_M = 1 - 2.59 r + 5.11 r^2 - 3.52 r^3, r = M/Mx, which is zero at the extinction moisture. Wind stretches the fire into an ellipse with length-to-breadth ratio L/B = 0.936 e^(0.2566 U) + 0.461 e^(-0.1548 U) - 0.397 (Anderson, U in mi/h); from the ignition point, R(psi) = R_head (1 - e)/(1 - e cos psi), e = sqrt(1 - (B/L)^2). Slope doubles the rate per 10 degrees uphill (rule of thumb, multiplicative). The constants R0 and c_w are illustrative.

Why a map and not a formula

In uniform fuel the dashed ellipses are the answer. With wet patches, firebreaks, bare ground and spot fires the fire arrival time is the solution of a minimum-travel-time problem on the 16-neighbour lattice, and the coloured map is that solution. The lattice slightly distorts very narrow fires (check 2 reports the error).

Embers and the cleared ring

Embers rise with the plume speed w ~ 2 (g I/(rho c_p T))^(1/3) and fall with terminal speed v_t (linear drag, tau = v_t/g): z(t) = z0 - v_t t + (w0 + v_t) tau (1 - e^(-t/tau)), x(t) = U (t - tau (1 - e^(-t/tau))). The yellow dashed circle is the downwind range of a typical ember (1.6 m/s); the green circle is your cleared radius. Landing on fuel starts a spot fire with probability 0.6 (1 - M/Mx), an assumption.

Intensity

Byram I = H w R and flame length L_f = 0.0775 I^0.46; campfire flame height from Heskestad L = 0.235 Q^(2/5) - 1.02 D; radiant ignition distance sqrt(chi Q/(4 pi q_crit)) with chi = 0.3 and 12.6 kW/m^2.

Campfire safety (real life)

Check for fire bans, build on bare ground, clear fuel well beyond the stones, keep water and a shovel at hand, never leave the fire alone and put it out until the ashes are cold.

Simplifications

• One fuel type per cell, no crown fire, no wind change inside the forest, no fire-induced wind.
• Spread constants, ember emission and ignition probabilities are illustrative: use the page for trends, not for prediction.
• Slope and wind are combined multiplicatively.

Checks

Six boxes under the controls compare the page's functions with independent calculations (ellipse shape, lattice spread, damping polynomial, RK4 ember flight, radiation integral, drying law).