The plume

DilutionCopy Anchor Link

How fast an exhaust plume mixes is measured. In more than seventy encounters behind airliners at cruise, Schumann et al. (1998) found the mass of air a kilogram of fuel's exhaust has mixed with growing as

N = 7000 · (t / 1 s)^0.8

from 6 ms to hours. Everything about the plume at an age follows from N.

Its widthCopy Anchor Link

A metre of trail holds the exhaust of the fuel burnt over that metre, mixed with N times its mass of air. At the plume's density that is an area, and the plume's Gaussian cross-section has the standard deviation that holds it:

σ = √(N · fuel per metre / (2π ρ))

For the fighter at 11 km: 1.8 m at a second, 4.4 m at ten, 9 m at a minute.

Its iceCopy Anchor Link

Once frozen, the crystals take up or give back vapour far faster than the plume changes, so the plume is held at saturation over ice. Its ice per kilogram of air is the water it holds, the air's own and the exhaust's, less what saturation over ice leaves as vapour.

• In air under ice saturation, mixing brings in dry air until the ice is gone: the trail's end.
• In air over it, mixing brings in more vapour than it dilutes: the ice grows, slowly, for as long as the air stays supersaturated.

Its crystalsCopy Anchor Link

Every soot particle that activates holds one crystal, about 10¹⁵ per kilogram of kerosene, and they share the ice. A young contrail's crystals are under a micron: 0.6 µm at a second. Hydrogen, with no soot, makes a hundred times fewer, which grow to 6 µm from the same air.

How much light it stopsCopy Anchor Link

Each crystal's extinction cross-section is its geometric one times an efficiency from van de Hulst's anomalous diffraction, about two for crystals much larger than the light, less for smaller. Summed across the plume, that gives an extinction per metre of trail; through the plume's axis, side on, its optical depth is

τ = extinction per metre / (√(2π) σ)

about 0.9 for the fighter at a second, 0.4 at ten, 0.2 at a minute: opaque when young, and a thin veil as it spreads.

Fuel per metreCopy Anchor Link

The trail's ice and its width scale with the fuel each metre was laid with, which the history keeps for every stretch: a stretch flown at 2 g, with twice the drag, is thicker than one flown level.