I'm not certain if its the same sound I was having on my E90 320d. I've posted the details here:
http://www.bmwfanatics.co.za/showthread.php?tid=30009
For me, the sound is more like a chirping sound, more than a whine and it did go away after a few km of driving. In short, it was a pulley that needed replacing and I had it replaced twice. Dealers said it was due to manufacture fault or something like that.
But like the other gentlemen have said on this thread, take your car to your nearest BMW dealership and ask them to check it out. They should be able to tell you just by listening to the sound. But be aware that if they can't hear it when you bring it in, they might request to keep the car overnight to hear the sound it makes on a proper cold start.
Cheers and keep us posted, especially if the problem isn't the pulley/belt that needs replacing.
:=):