Is it the high pitched whining noise? It sounds to me like it could be a bearing noise but you said the car has to be run for awhile first. I’m still leaning toward pump noise. When the car is driven for a while it comes on then when you shut it off and immediately restart it the noise is gone for a little while again? It could be a pump and it takes a little while to build enough pressure to make the noise.
I had an odd squeak that came and went if I touched tensioner. seems rotating tensioner was just enough to make it go wonky. the spring tension was just off enough to make serp belt chirp. replace tensioner? it looked ok to me but actually was making noise. it was spring, not the tensioner bearing.
I never did a model-year or engine description. Am I missing something? Oh well, I won’t look up TSBs to see if there’s any clues. That’s where I look first with this type issue. I’m out unless more info is submitted. CSA
I couldn’t hear much either except what may be a bearing noise. Should be fairly easy for a shop to determine if its water pump, alternator, or the idler pulley. Just a side comment but I had my idler pulley give out 50 miles from home and jump the belt. I did manage to hobble to a parts place and get another one but I had tools with me. Same problem with a worn water pump pulley at 20 below but not so lucky on that one. Always best to take care of a problem as soon as you notice it.