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In the answers to this question about Denver's widely-scattered pinwheel of runways and this question about Schiphol's runway out in the middle of nowhere, it's mentioned that parallel runways are required, by both European and U.S. regulations to be over a kilometre apart in order to be able to have simultaneous departures or simultaneous arrivals on both runways. In contrast, if one runway is departing an aircraft while a parallel runway is landing an aircraft, the runways need only be a few hundred metres apart.

Why is there such a large runway-separation requirement for simultaneous arrivals or simultaneous departures from parallel runways? It can't be for aircraft separation, since an aircraft is allowed to land on a runway quite close to a parallel runway that is simultaneously departing an aircraft, despite the possibility of this resulting in a head-on collision if one or both aircraft drift off track, so what, in fact, is the reason for this?

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    $\begingroup$ Aircraft on parallel runways both go in the same direction, so that a departing plane and arriving plane would both be flying in the same direction. There would be no head-on collision. $\endgroup$ – abelenky May 1 '18 at 16:37
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enter image description here
(YouTube) Much closer than 1 km at SFO.

1 km (~3,300') is not the minimum (see this illustration from the FAA AIM). The centerline spacing for parallel approaches can go as low as 750' (~230 m), a prime example is San Francisco:

enter image description here
(Google Earth) Centerline spacing at SFO.

But getting very close like at SFO requires special training, procedures, charts, controllers, and ground equipment (namely the precision runway monitor system - PRM). In lack of PRM and the associated items, the centerline separation allows sufficient reaction time against any transgression (e.g., pilot deviation).

Wake turbulence is a factor for the trailing plane whether on the same runway or a parallel one, but RECAT has helped reduce the over conservative separation (ICAO).

For your last point on departures and arrivals on parallel runways, a minimum centerline spacing is required:

enter image description here
(FAA)


Further reading:

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The simple answer seems to be for wake turbulence. This FAA document, ORDER JO 7110.308C, seems to say it is for wake turbulence reasons in para. 9:

The geometry of the approach, as well as the lateral separation between the two approaches and prevailing local meteorological conditions, provide the wake turbulence avoidance necessary for reduced separation dependent approach operations.

Figure 3 shows the aircraft arriving in a staggered fashion, 1 nautical mile diagonal separation with runways 2500 ft apart (parallel runways separated by less than 2,500 feet, also referred to as, Closely Spaced Parallel Runways (CSPR)). The text also describes how the lead aircraft follow a lower glide path than the following aircraft such that the lead's wake turbulence isn't a factor for the following aircraft.

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It is usually to guarantee sufficient aircraft separation when visual separation is not available (bad weather). See also Why is one of two parallel runways sometimes closed in foggy weather?

It can't be for aircraft separation, since an aircraft is allowed to land on a runway quite close to a parallel runway that is simultaneously departing an aircraft, despite the possibility of this resulting in a head-on collision if one or both aircraft drift off track

There can't be a head-on. Unstated in that requirement is that both runways are used in the same direction. If the arriving aircraft has perform a missed approach, it has procedures to direct it away from the other runway. Also, the departing aircraft will be advised. Under normal circumstances where the landings are executed, there's no overlap of tracks in the air, only on the runways.

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