What did this end up being? My sister just got a '98 2.0L (not even a month ago), and she is having what sounds to be the same problem.
It sounds like the a fan (except they are not moving) or the secondary air pump is running after the car is shut off. It will kick on randomly while she is driving which I expect is normal for the secondary air pump, but whatever it is will also kick on randomly after the car is shut off (sometimes it will shut off when she shuts off the car and it will start up later) untill the battery is completely drained. Not really familiar with VWs, and I have only had the chance to look at the car once, but the noise (it vibrates too) is coming from the black canister on the drivers side of the engine (it has a hose running from it to the air intake and a couple other places). I am going to have plenty of time to look at it tomorrow and I wanted to get some more ideas about what to look at. Thanks.
David
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