The ball bearings in the fan have dust contamination. It sounds fine at low RPM, but when the fan ramps up to full speed, the contaminated bearings dont spin right, and cause acoustic resonation of the stator assembly, which makes the fan spin out of balance, and make all kinds of terrible noise.
Two solutions.
1) Order a replacement cooling fan from someplace like amazon, and install it. (Recommended. Seriously, they are NOT that expensive.)
2) Disconnect the fan, and carefully disassemble it. You will need a pair of hemostats and a dental pick. Usually, there are 2 to 3 clear/white plastic slip rings that hold the fan blade assembly on the stator shaft. Carefully remove these, then gently push on the shaft with the hemostats, until it pops loose. Be very careful not to lose any of the small pieces, or break any of the fan blades!! Once disassembled, you will find a very tiny (1/2cm, usually) ball bearing race assembly. Take this, and submerge it in something like WD40, and work it. Some brown gunk will come out. That's the dust contamination that is making it shriek. Once you have removed all the dirt this way, repeat the above, but use electrical contact cleaner to remove all the WD40. (Liquid oil encourages the contamination to return with vengeance. Remove it all with the contact cleaner bath.) Once the bearing race is all nice and clean, reassemble the fan, and replace the little plastic retainer rings very carefully with the hemostats. (Not recommended-- high risk of losing tiny parts-- only postpones the inevitable-- improper cleaning encourages more contamination. Contamination is a sign the seals have failed.)