I had this noise like many who have gone before me (hum so loud I could hear it while driving at almost any speed) and read about it, but couldn't find any pics with my feeble search skills. Thanks to U102768 for the help which got me my solution in the thread below!

https://www.jaguarforums.com/forum/x...-noise-111624/

Once I got this thing out and held it in my hand it was almost literally silent while running. Its not the motor/fan itself making the noise for me, its vibration against all the other plastic stuff around it. Mine only has one hold-down screw at bottom so I put new foam tape around the cutout in the dash and some on the outer face of the aspirator (assuring not to change opening) and tightened slowly til noise was lowest before putting panel back on. Cant' hear it when the car is idling anymore (with climate control off or on)! Can still hear it when I turn car off, which is fine. I know when it shuts off when I lock it in garage with windows open.

Pic #1 The label on the part. 6W838B506-AB
Pic #2- back of Rside of dashpiece. Just wanted to show you where the clips are. There are 2 more on the LH side-way to the edge. Its always scary as hell pulling these things for me but I used only my fingers and worked off one side at a time keeping "flex" of the whole piece to a minimum.
Pic #3 Looking down the inside of the aspirator. That white part is the sensor. The fan is down past it inside of there. You can see the tab with a hole in it at bottom for my one screw hold-down.
Pic #4 Thats the aspirator
Pic #5 Shows where aspirator is on RH side of steering wheel. You see the dash piece is displaced.