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I just watched the Mayday S1E4 episode concerning Aeroperu Flight 603. Gut wretching to see the pilots in that situation. On listening to the recording tape, alarms were blaring constantly, some for maybe as long as 15 minutes straight. I can only imagine that that constant blaring adds stress and confusion to the cockpit. So I wanted to ask if pilots have the ability to silence alarms after a certain period of time?

Sometimes they can sometimes they cant, how to do it varies by aircraft but generally alarms can often be silenced by acknowledging they are going off and pressing some kind of button. On my little Piper Archer the alternator under volt it just a light and can't be shut off but if my G430 throws an alarm or warning it can usually be silenced.

On the 737 some alarms cant be silenced and some can

Cockpit aural warnings include the fire bell, take-off configuration
warning, cabin altitude, landing gear configuration warning,
mach/airspeed overspeed, stall warning, GPWS and TCAS. External aural
warnings are: The fire bell in the wheel well and the ground call horn
in the nose wheel-well for an E & E bay overheat or IRS’s on DC. Only
certain warnings can be silenced whilst the condition exists.

A lot of aircraft display failures on some kind of annunciator panel or glass cockpit equivalent. These lights typically can not be turned off until the error is addressed (fire suppressed, breaker pulled etc.). Master Caution lights (and audio warnings) can typically be shut off through the use of a Master Caution Reset switch.

Once notified, the pilot may cancel the master caution, but a
dedicated system or component annunciator light stays illuminated
until the situation that caused the warning is rectified. Cancelling
resets the master caution lights to warn of a subsequent fault event
even before the initial fault is corrected.

The FAA's full advice on warning and caution light design can be found here.

They shed more light on the topic in this AC

9. Clearing and Recalling Visual Alert Messages. Clearing visual alert messages from the current warning, caution, and advisory display
   allows the flightcrew to remove a potential source of distraction and
   makes it easier for the flightcrew to detect subsequent alerts. a. The
   following guidance should be applied for clearing and recalling or
   storing the visual alert messages:

(1) If a message can be cleared and
the condition still exists, the system should provide the ability to
recall any cleared visual alert message that has been acknowledged. 11
12/13/2010 AC 25.1322-1

(2) Either through a positive indication on the display or through
normal flightcrew procedures, a means should be provided to identify
if alert messages are stored (or otherwise not in view). b. The visual
alert message must be removed from the display when the condition no
longer exists (§25.1322(a)(3)).

In short, if a message can be cleared you should be able to pull it back up easily and if the problem goes away the alert should clear as well.

Some alarms, like those announced by the Master Warning System like fire bells, can be silenced by pushing the Mater Warning light and you do it as part of the memory procedure to deal with the warning. Other alarms, like ones for configurations that are wrong for the conditions, can only be silenced by correcting the configuration. Think gear warning horns or flap overspeed clackers.

I used to complain about the gear warning horn in the CRJs which has a dual tone that sounds almost exactly like a TV station off-the-air test pattern. It is extremely easy to blend into the background when the stress level is high and you are becoming mentally saturated. Fortunately, the GPWS announcement "Too Low, Gear" comes on as you get close to the ground to hopefully wake you up. Hopefully.... some years back a Britair CRJ200 crew landed wheels up in spite of the warnings because they were landing flapless and were mentally saturated by the stress of landing at 160kt (and I believe the GPWS callout may have been inhibited; not sure).