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How to use
Start with Short-lived contrail: the aircraft cruises at 10.7 km in air of -54.6 C. Its exhaust is hot and full of water vapour; as it mixes with the cold air the mixture moves along the mixing line on the first graph (vapour pressure against temperature). If that line crosses the blue water-saturation curve, droplets form and freeze into ice: a contrail. Whether it then lasts depends on the humidity of the surrounding air over ice.
Try the presets in order, then change one slider at a time: raise the humidity until the trail stops fading and starts to grow, lower the altitude until the contrail disappears, or switch the fuel to hydrogen. Time runs faster than reality (Time compression); the horizontal axis of the 2D view is the age of the air, stretched so that seconds and minutes are both visible. Space pauses, R restarts the flight.
The science
1. Why a contrail forms: the Schmidt-Appleman criterion. Burning a kilogram of fuel releases EI kilograms of water vapour and heat Q(1 - η) into the exhaust (the rest is propulsive work). As the plume mixes with ambient air its temperature and vapour pressure move along a straight line in the (T, e) plane, with slope
G = EIH2O cp p / ( ε Q (1 - η) ) [Pa/K], ε = 0.622, cp = 1004 J/(kg K)
starting at the ambient state (Ta, ea), ea = RHw ew(Ta). If the line touches the saturation curve over liquid water ew(T) at any point, droplets condense and freeze within a fraction of a second. The limiting case is a line tangent to ew: the tangent point has the temperature TLM with dew/dT = G, and the contrail forms when
Ta ≤ TLC, ea + G (TLM - Ta) = ew(TLM)
Typical values are G about 1.5 to 2 Pa/K and TLC about -40 to -50 C for kerosene at cruise level. Because 1 - η is in the denominator, more efficient engines raise G and make contrails possible at warmer temperatures; hydrogen (EI = 8.94 kg/kg, Q = 120 MJ/kg) gives a steeper line still. The page solves the tangency numerically and checks it against the published fit of Schumann (1996), TLM = -46.46 + 9.43 ln(G - 0.053) + 0.72 ln2(G - 0.053) in C.
2. Saturation vapour pressures are the Murphy and Koop (2005) fits, valid to about -100 C: for ice ln ei = 9.550426 - 5723.265/T + 3.53068 ln T - 0.00728332 T. The ratio ew/ei is about 1.5 near -55 C, so air with only 70 % relative humidity over water is already supersaturated over ice.
3. Persistence. An ice crystal grows or shrinks by vapour diffusion and heat conduction (Pruppacher and Klett):
r dr/dt = (Si - 1) / (Fk + Fd), Fk = (Ls/(RvT) - 1) Ls ρi / (K T), Fd = ρi Rv T / (D ei)
so r2 changes linearly with time. For Si < 1 a crystal of initial radius r0 vanishes after t = r02(Fk + Fd)/(2(1 - Si)): the trail is short-lived and its visible length is V t. For Si ≥ 1 the crystals grow and the contrail becomes contrail cirrus. The brightness shown is proportional to the cross-section of the ice divided by the plume width.
4. Wake vortices and spreading. The wing sheds a vortex pair of circulation Γ0 = m g / (ρ b0 V), with b0 = (π/4) b the vortex spacing for elliptic loading. The pair descends at w0 = Γ0/(2πb0) and entrains the exhaust; the trail sinks about 4 b0 (typically 100 to 150 m) before the vortices break up. In ice-supersaturated air, vertical wind shear s then stretches the sheet: width ≈ W0 + s · depth · age.
Checks (run in the page). Vapour pressure against tabulated values (triple point 611.65 Pa, -40 C 12.84 Pa, 20 C 2339 Pa); numerical tangent TLM against Schumann's fit; a brute-force scan of the mixing line against ew versus the analytic TLC (9 cases); the integrated crystal sublimation time against the closed form; the vortex time-scale; the standard-atmosphere pressure at 10 km.
Simplifications. Isobaric adiabatic mixing in a single plume, no soot or ice-number budget (the crystal radius after the jet phase is an input), no radiative effects or optical depth, a steady wind with constant shear, no sedimentation of the crystals, and a uniform humidity. Real contrail lifetimes also depend on plume dilution, which is not modelled; the lifetime shown is that of an isolated crystal in the ambient air.