Could the whining have something to do with my power steering pump? It starts when the ignition is set to "on" (without the engine running), which leads me to believe that it must be coming from some electrical part.

Hey thanks, I'd appreciate that!
... and also for the spell check

My S60 2.5T 2004 is whining/whirring. The sound seems to be coming from the driver side up near the front of the engine compartment. The noise begins when the ignition switch is set to "on", continues while the engine is running, and stops about 30 seconds after the engine has been turned ...