Lift, drag and wing loading
Every bird here is a point mass with L = ½ρV²S·CL and D = ½ρV²S·(CD0 + CL²/πeAR). The best glide ratio is about ½√(πe·AR/CD0), so long narrow wings (albatross, aspect ratio about 14) glide at 21:1 and short broad wings (eagle, about 6) at 11:1. Speed scales with √(weight / wing area): the albatross carries 14.5 kg per m² of wing, the hummingbird 1.8.
Thermals
Sun-heated dry fields and rock warm the air above them; it rises as a column 70-200 m wide at 1-5 m/s, strongest over bright dry ground and weaker over forest and water, and capped by a cumulus cloud near the top of the convective layer. A bird gains height only where the air rises faster than it sinks (about 0.4-1.4 m/s), and circling tightly raises that sink. Eagles centre by comparing lift at the two wing tips.
Ridge lift and dynamic soaring
Wind forced up a slope gives w ≈ U·∇h, which is why the cliffs and the ridge work on windward sides. Over the sea the wind slows towards the waves; an albatross climbs into the faster wind aloft, turns downwind and dives, and each loop takes energy from the shear. The autopilot's loop parameters were found by random search on this very flight model, and it only sustains itself above roughly 14 m/s of wind at 40 m (real albatrosses manage with somewhat less).
V formation
A wing sheds a pair of vortices; outboard of them the air rises. A follower in that upwash needs less thrust by about (w/V)·(L/D). With the geese's own numbers the model gives roughly 15-20 % at the sweet spot, in line with the up-to-20 % estimates for real flocks; being directly behind a leader instead means downwash and a penalty.
Murmurations
Starlings do not watch a metric radius: each bird reacts to its 7 nearest neighbours however far away they are (Ballerini et al., 2008). That keeps the flock cohesive at any density. A predator pass makes the evasion spread as a wave because every bird copies its neighbours.
Migration fuel
Fat holds about 37 kJ/g. Flight costs a species-specific multiple of its basal metabolic rate (about 8-22x when flapping, about 1.5x when soaring), so a thermal or a formation saves real fat. The world is a few km wide, so each metre you fly stands for hundreds of metres of the real route, and the real leg lengths are approximate.
What is simplified
- Point-mass flight with an automatic pitch trim; no wing-beat aerodynamics (flapping is a power-limited thrust, calibrated to published flight/basal ratios).
- Thermals are idealised shapes; wind is a smooth shear profile with light turbulence.
- Formation followers get a small steering assist for station keeping; the upwash and savings are computed, not scripted.
- Small birds are drawn up to several times larger than life and the hummingbird's wing beat is slowed so they can be seen; physics uses true sizes.
- Fat, power and range numbers are order-of-magnitude, not measurements.