I saw the picture, read the title, too lazy to read through the rest.
This noise is normal in M cars, it's the clutch plates in the differential. The same noise comes out of my M3, and every other M3 really, if you try to listen for it.
If it really bothers you, then you can get an additive put inside your differential oil at the dealership. It will eliminate the noise by 2-3 days once added.
But really, it's not harmful to your differential, and it's all just a part of the M experience. If you wanted something without any quirks, you probably got the wrong car. M car's have a history of being quirky.