Losing your ability to keep the airplane precisely on altitude is becoming a bigger deal every day, as the skies are becoming more tightly packed. Because each situation is likely to be unique, there are no cut and dried rules that always apply.
Appendix 5 of AC 91-85 contains information on procedures that are unique to oceanic RVSM airspace where direct voice communications between pilots and air traffic control (ATC) is not available and provides several scenarios as examples.
You should also refer to: Special Procedures for In-flight Contingencies in Oceanic Airspace.
Everything here is from the references shown below, with a few comments in an alternate color.
[ICAO Doc 9574, ¶5.1.1 h)] the following contingency procedures should be adhered to after entering RVSM airspace:
[ICAO NAT Doc 007, ¶13.3.1] The general concept of these Oceanic in-flight contingency procedures is, whenever operationally feasible, to offset from the assigned route by 15 NM and climb or descend to a level which differs from those normally used by 500 ft if below FL410 or by 1000 ft if above FL410.
[ICAO NAT Doc 007, ¶13.3.2] The aircraft should leave its assigned route or track by initially turning at least 45° to the right or left whenever this is feasible. The direction of the turn should, where appropriate, be determined by the position of the aircraft relative to any organised route or track system (e.g. whether the aircraft is outside, at the edge of, or within the system). Other factors which may affect the direction of turn are: direction to an alternate airport, terrain clearance, levels allocated on adjacent routes or tracks and any known SLOP off sets adopted by other nearby traffic.
[ICAO NAT Doc 007, ¶13.3.3] An aircraft that is able to maintain its assigned flight level, after deviating 10 NM from its original cleared track centreline and therefore laterally clear of any potentially conflicting traffic above or below following the same track, should:
[ICAO NAT Doc 007, ¶13.3.4] An aircraft that is unable to maintain its assigned flight level (e.g due to power loss, pressurization problems, freezing fuel, etc.) should, whenever possible, initially minimise its rate of descent when leaving its original track centreline and then when expected to be clear of any possible traffic following the same track at lower levels and while subsequently maintaining a same direction 15 NM offset track, descend to an operationally feasible flight level, which differs from those normally used by 500 ft if below FL410 (or by 1000 ft if above FL410).
[ICAO NAT Doc 007, ¶13.3.5] Before commencing any diversion across the flow of adjacent traffic or before initiating any turn-back (180°), aircraft should, while subsequently maintaining a same direction 15 NM offset track, expedite climb above or descent below the vast majority of NAT traffic (i.e. to a level above FL410 or below FL280), and then maintain a flight level which differs from those normally used: by 1000 ft if above FL410, or by 500 ft if below FL410. However, if the pilot is unable or unwilling to carry out a major climb or descent, then any diversion or turn-back manoeuvre should be carried out at a level 500 ft different from those in use within the NAT HLA, until a new ATC clearance is obtained.
[ICAO NAT Doc 007, ¶13.3.6] If these contingency procedures are employed by a twin engine aircraft as a result of the shutdown of a power unit or the failure of a primary aircraft system the pilot should advise ATC as soon as practicable of the situation, reminding ATC of the type of aircraft involved and requesting expeditious handling.
Advisory Circular 91-85B, Authorization of Aircraft and Operators for Flight in Reduced Vertical Separation Minimum Airspace, 1/21/19, U.S. Department of Transportation
ICAO Doc 4444 - Air Traffic Management, 16th Edition, Procedures for Air Navigation Services, International Civil Aviation Organization, October 2016
ICAO Doc 7030 - Regional Supplementary Procedures, International Civil Aviation Organization, 2008
ICAO Doc 9574 - Manual on Implementation of a 300 m (1,000 ft) Vertical Separation Minimum Between FL 290 and FL 410 Inclusive, Second Edition, International Civil Aviation Organization, 2001
ICAO NAT Doc 001, Guidance and Information Material Concerning Air Navigation in the North Atlantic Region, Seventh Edition, January 2002
ICAO NAT Doc 007, North Atlantic Operations and Airspace Manual, v 2018-1
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