The wake

The vortex pairCopy Anchor Link

A wing's lift leaves behind it a pair of counter-rotating vortices, rolled up within a few spans from its sheet. For an elliptic loading they are π/4 of the span apart, and by Kutta–Joukowski each carries the circulation

Γ = L / (ρ V b₀),    b₀ = π/4 · span

Each sinks in the other's downwash at

w = Γ / (2π b₀)

For the fighter at 11 km: Γ ≈ 196 m²/s and w ≈ 2.8 m/s. For a 70 t airliner, whose span spreads the same lift wider: Γ ≈ 307 m²/s, w ≈ 1.8 m/s.

Into the vorticesCopy Anchor Link

The exhaust leaves the engines between and below the vortices, and is drawn round the one on its side. How long that takes is the time the vortex turns the flow through a radian at the engine's distance from its core, 2π d² / Γ. A fighter's engines, close to the centreline, are well inboard of its cores, and their trails are drawn out to them within a second or two. An airliner's engines sit under the wing, nearer the cores.

Down, and stoppedCopy Anchor Link

The pair sinks, and carries its trails down with it. Sinking through stably stratified air, it warms by compression and becomes buoyant against the air round it, which slows it. In air of buoyancy frequency N,

descent(t) = w / N · sin(N t),    until N t = π/2

so it stops at w / N after a quarter period. N is 0.012 rad/s under the tropopause and 0.021 above it, from the standard atmosphere's lapse rate.

Linked, and goneCopy Anchor Link

The air's own turbulence bends the two vortices towards each other until they touch and reconnect into rings, the Crow instability, and the wake stops sinking. Sarpkaya fit the time it takes in the pair's own time, t₀ = b₀ / w, against the turbulence measured by the pair's descent:

ε* = (ε b₀)^⅓ / w

T = 9                       ε* < 0.001
T = 9.18 − 180 ε*           ε* < 0.0121
T^¼ · exp(−0.7 T) = ε*      ε* < 0.2535
T = 0.8039 ε*^−¾            beyond

with ε the eddy dissipation rate, 10⁻⁷ m²/s³ in the weak turbulence of the cruise levels. The pair sinks until T t₀, then stays where it is. An airliner's wake links in about two minutes, after the stratification has already stopped it; a fighter's, smaller and faster, in about half a minute.

A harder pull makes a stronger pair that sinks faster, and links sooner in proportion: the fighter's trails drop about the same 90 m at any g, only sooner.

PushedCopy Anchor Link

Pushed under 0 g the wing lifts downwards: the vortices turn the other way, and the pair rises above the path at the same speed it would have sunk, carrying its trails up.

That is the primary wake. In the real atmosphere some of the exhaust is left behind at flight level as the pair descends, and the rings the linked vortices make spread and decay; neither is drawn.