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Wake Vortex Initial Circulation

Also known as wake vortex strength · wake turbulence circulation · vortex circulation of an aircraft · initial circulation of the wake · Gamma 0 wake · wake vortex strength heavy aircraft · trailing vortex circulation · wake vortex from weight span and speed · aircraft wake strength · wake turbulence separation strength · lift generated circulation

Γ0=4 M gπ ρ s V\Gamma_0 = \frac{4\,M\,g}{\pi\,\rho\,s\,V}
sig figs

Worked example: Schumann A380: 508 t, 79.8 m, 250 m/s, 0.39 kg/m3 → 815.5 m2/s (from b0 and w0). Press Try an example to run it live, then adjust anything.

Enter your known values, leave one input blank, and solves for the missing one. Tap a variable’s symbol to see what it means, with a typical value. Change any unit and the answer follows.

Constant used: built into this formula, no need to enter
gn=9.80665 m/s2g_{n} = 9.80665\ \text{m/s}^{2}Standard acceleration of gravity · exact

Where Wake Vortex Initial Circulation fits

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Wake Vortex Initial Descent Speed

See also

Wake Vortex Initial Descent Speed: How fast the vortex pair sinks from this circulation and the vortex spacing, and so how soon it is out of the flight level.Standard Atmosphere Density from Altitude (11 to 20 km): The air density at a cruise altitude in the standard atmosphere, which this page needs.Stall Speed: The slow end of the same trade: a heavy aircraft flying slow makes the strongest wake.Load Factor in a Coordinated Level Turn: In a turn the lift is greater than the weight, and the wake it leaves is stronger by the same factor.
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Wake Vortex Initial Circulation explained

sVΓ0Γ0M gair ρ

An aircraft trails two vortices behind it, and this page gives the strength of each as a circulation Γ0\Gamma_0 from the aircraft's mass MM, wing span ss, true airspeed VV and the air density ρ\rho. It is the value as the vortices leave the aircraft, in square metres per second.

The formula comes from the lift that holds the aircraft up. The weight MgM g equals the lift. For a wing with elliptic loading the lift is the air density times the speed times the circulation times the spacing of the vortices, ρ V Γ0 b0\rho\,V\,\Gamma_0\,b_0, and that spacing is b0=πs/4b_0 = \pi s/4 for a span ss. Setting Mg=ρ V Γ0 πs/4M g = \rho\,V\,\Gamma_0\,\pi s/4 and solving for Γ0\Gamma_0 gives the page's relation, which Ulrich Schumann gives in his contrail cirrus model paper (2012) as the first step of its wake vortex model.

Read the answer by its denominator. The strength is greater for a heavy, slow, high-flying aircraft with a short span: a heavier aircraft has more weight to carry, and thinner air, a lower speed or a shorter span leave less to carry it with, so the vortex pair must be stronger. That is why wake turbulence separation is set by the weight class of the leader. For the large airliner Schumann uses as an example, an A380 at 250 m/s with a mass of 508 tonnes, a span of 79.8 m and an air density of 0.39 kg/m³, the equation gives 815.24 m²/s, with the vortex pair 62.7 m apart. Schumann's printed digits give Γ0=2π b0 w0\Gamma_0 = 2\pi\,b_0\,w_0 with b0=62.7b_0 = 62.7 m and w0=2.07w_0 = 2.07 m/s, which is 815.5 m²/s, and the equation's 815.24 is 0.03 per cent less, within the three digits the paper prints.

The relation assumes an elliptically loaded wing whose lift equals the weight; in a turn the lift is greater than the weight, and the wake it leaves is stronger by the same factor. The page gives the strength as it leaves the aircraft; the vortices then decay with the turbulence of the air, over some minutes, and sink at the speed the wake vortex descent speed page gives. The sources are Schumann's 2012 paper in Geoscientific Model Development, section 2.5, and the review by Gerz, Holzapfel and Darracq (2002).

Wake Vortex Initial Circulation formula

Γ0=4 M gπ ρ s V\Gamma_0 = \frac{4\,M\,g}{\pi\,\rho\,s\,V}
Where
  • Γ0\Gamma_0= Initial circulation of each vortex (m²/s)
  • MM= Aircraft mass (t)
  • ρ\rho= Air density (kg/m³)
  • ss= Wing span (m)
  • VV= True airspeed (m/s)
Where this comes from
  • Schumann, U. (2012). A contrail cirrus prediction model. Geoscientific Model Development, 5(3), 543-580 (open access, CC BY 3.0; read on 23 Oct 2026, section 2.5, the wake vortex sub-model: 'wake vortex separation b_0 = pi s_a/4, initial circulation Gamma_0 = 4 M_a g/(pi rho s_a V_a), effective time scale t_0 = 2 pi b_0^2/Gamma_0, initial velocity scale w_0 = Gamma_0/(2 pi b_0)', with s_a the wing span, M_a the aircraft mass and V_a the true airspeed; and its example, an A380 observed during the CONCERT campaign with V_a = 250 m/s, M_a = 508 Mg, s_a = 79.8 m and rho = 0.39 kg/m3, for which 'b_0 = 62.7 m, t_0 = 30.3 s, w_0 = 2.07 m/s'). Read it
  • Gerz, T., Holzapfel, F., & Darracq, D. (2002). Commercial aircraft wake vortices. Progress in Aerospace Sciences, 38(3), 181-208 (the wanted list's source; the DOI record was checked in Crossref, the review is behind a paywall and was not opened here; Schumann (2012) cites Holzapfel (2003) for the same sub-model). Read it
Revised 30 October 2026
Cite this page

formula.expert. (2026, October 30). Wake Vortex Initial Circulation Formula & Calculator. https://formula.expert/solvers/wake-vortex-initial-circulation

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