Wake Vortex Initial Descent Speed

Behind an aircraft there is a pair of vortices of opposite sign, and each one pushes the other down. The sinking speed is the speed that one vortex induces at the other's centre: $w_0 = \frac{\Gamma_0}{2\pi\,b_0}$, where $\Gamma_0$ is the initial circulation of each vortex (its strength) and $b_0$ is the distance between them. A stronger pair sinks faster, and a pair set wider apart sinks more slowly.

The page uses the spacing of an elliptically loaded wing, $b_0 = \frac{\pi}{4}\,s$ for a span $s$, which turns the relation into $w_0 = \frac{2\,\Gamma_0}{\pi^{2}\,s}$. The inputs are then just the circulation and the span. Schumann prints both relations as the velocity scale of his wake vortex model (2012, section 2.5). Where the circulation itself comes from, the weight, span, speed and air density of the aircraft, is the subject of the initial circulation page, and a banked aircraft makes a stronger wake, in proportion to its load factor.

Schumann works an example of an A380 at 250 m/s and 508 tonnes, with a span of 79.8 m. The vortex spacing is 62.7 m, the descent speed is 2.07 m/s and the time scale is 30.3 s. From a circulation of 815.24285 m²/s the page gives 2.0702 m/s. Run backwards, 2.07 m/s over a 79.8 m span is a circulation of 815.16 m²/s, and 815.24285 m²/s sinking at 2.0702098 m/s means a span of 79.8 m.

The sinking speed is the reason an aircraft that follows another at the same height should be kept above and behind it, and why the wake of a landing aircraft settles toward the runway. The page gives only the first sinking speed. In stratified air, and the stratosphere above the tropopause is stratified, the descent stops after a few hundred metres, because the sinking air warms as it is compressed, ends up warmer and lighter than its surroundings, and is pushed back up; the page on the Brunt-Vaisala frequency gives the frequency at which the air pushes the vortices back. The vortices also decay with the turbulence in the air.