1. A warm body in cold air
Each bird keeps its core near Tb = 38 °C. Heat leaves through plumage, fat and skin (conduction in series) and then from the outer surface by convection and radiation:
q = (Tb - Ts) / R = h (Ts - Tair) + ε σ (Ts4 - Tenv4).
The model solves this for the surface temperature Ts with Newton's method; the loss is Q = Aexposed q.
2. Insulation in series
The layers add like resistors: R = Lf/kf + Lfat/kfat + Lskin/kskin (plumage about 2.7 cm with k about 0.037 W/m/K, fat about 1.5 cm at 0.2, skin 3 mm at 0.4; approximate). The feathers carry roughly 90 % of the resistance. Wind pumps air into the plumage, modelled as kf → kf (1 + 0.06 U) (an assumed coefficient).
3. Wind and the convection coefficient
The trunk is a cylinder of diameter d in cross flow. The Churchill-Bernstein correlation gives Nu = 0.3 + 0.62 Re1/2 Pr1/3 [1 + (0.4/Pr)2/3]-1/4 [1 + (Re/282000)5/8]4/5, h = Nu kair/d, mixed with 3 W/m²/K of free convection. Air properties change with temperature.
4. Metabolism and the fat budget
Resting heat production follows Kleiber's law P = 3.4 m3/4 W (about 44 W for 30 kg). A bird makes at least this much, and more when its loss exceeds it (capped at 4 times). Fat supplies about 37 MJ per kg, so dmfat/dt = -M / 37 MJ. A 9 kg reserve at 45 W lasts about 85 days, the right order for a fasting male (approximate).
5. Why huddling works
A neighbour at distance d hides the arc 2 asin(Rb/d) of the bird's circumference (Rb = 0.20 m). The union of these arcs gives the sheltered fraction of the side area; the back and head (20 % of the surface) stay exposed. The wind a bird feels is weighted by how much of the windward half-circle is open. Inner birds are almost completely covered, so they sit at the resting rate. Packing density sets the spacing s = sqrt(2 / (√3 ρ)) of a hexagonal lattice, so tighter packing hides more arc.
6. Warm air inside: a 2-D heat field
A depth-averaged field solves ρc H ∂T/∂t = keff ∇2T - ρc H u·∇T - hv (T - Tair) + q on a 0.25 m grid. Each bird is a source of its heat output; the wind advects the warm air, and the huddle itself slows the wind (shelter factor grows with bird density). Explicit upwind/central scheme with automatic sub-stepping. The effective diffusivity and the vertical exchange hv are tuned toy values chosen so that a dense huddle warms the air between birds by several to some tens of degrees (still air gives more; wind carries the heat away), the order reported in the literature; birds feel this local air temperature, which closes the feedback loop.
7. Rotation and fairness
In real huddles a wave of small steps lets the whole group slowly turn over (Gerum et al. 2013) and huddles form and dissolve like a phase transition with temperature and wind (Richter et al. 2018; Zitterbart et al. 2011). Here every bird compares its heat output with the colony mean; the stressed ones walk along the edge to the lee side, pushing neighbours aside through soft non-overlap constraints. Fairness is measured with Jain's index J = (Σx)2 / (n Σx2) of the energy each bird has spent (1 means everybody paid the same) and with the coefficient of variation.
8. When does a huddle form?
The map in the Phase tab shows the huddle drive φ: a smooth function of the loss of a lone bird divided by its resting rate (0 below 0.95, 1 above 1.5). The target packing density is interpolated between a loose 1.2 birds/m² and your packing setting by φ. This is a modelling choice, not a measured threshold.
9. Side note: human wind chill
The familiar index 13.12 + 0.6215 T - 11.37 v0.16 + 0.3965 T v0.16 (T in °C, v in km/h) is an empirical fit for a human face, not for a feathered bird; the readout is shown only for comparison.
10. Honest simplifications
Birds are discs; the 3-D shape is only used for display. Covered surfaces exchange no heat; the sky is a fixed offset below air temperature; birds hold their core temperature (a bird that would need more than 4 times its resting rate is only counted as overloaded); there is no ground conduction or chick-on-feet brooding. The self-checks verify the numerics, not the biology.
References
Gerum et al. 2013, New J. Phys. 15, 125022 (huddle waves). Ancel et al. 1997, Nature 385, 304 (energy saving in huddling penguins). Richter et al. 2018, J. Phys. D: Appl. Phys. (phase transitions in huddling emperor penguins). Zitterbart et al. 2011, PLoS ONE 6, e20260 (coordinated movements in a huddle). Churchill and Bernstein 1977, J. Heat Transfer 99, 300. Kleiber 1932, Hilgardia 6. Jain, Chiu and Hawe 1984, DEC-TR-301. Osczevski and Bluestein 2005, Bull. AMS 86, 1453.