Could be belt slippage until it warms up enough to burn the glazing off? Maybe a bearing is running dry on lube and makes noise until the grease is warmed up enough to turn liquid and run into the appropriate areas. Are you sure its coming from the alternator itself? Try removing the belt and running the engine for a short amount of time and see if the noise is still there. If it is then you need to look harder to locate the noise. If the noise goes away try spinning all of the pulleys by hand and see if there is any play or noise out of them. They should all spin smoothly with very slight resistance. If you get one that you can flip by hand and it keeps on trucking for a long time then the bearing is worn and should be replaced. The only exception is the alternator, it should spin freely with almost no resistance. The crank pulley obviously wont spin by hand very easily, but there isnt much to go wrong with them aside from losing the rubber in harmonic balancer, but those dont make noise, they come apart when that happens.