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When I picked up mine it had the same whine. I neglected it until I was 500 miles from home and it left me on the side of the road overheated with a seized pump and a snapped accessory belt. Save yourself a headache and just replace the PS pump.
Drain the fluid and refil it with DEX/MERC ATF. I use Valvoline MaxLife. I'm kinda limping mine along so I can do it when I do the front 60k this summer. I drain and refil about every 2 weeks, otherwise it has little to no power.
I'd give it a try, the fluid in there probably looks like mud.
As jthod suggested, flush and refill the power steering fluid. Do a search as there are several threads describing how to do this.
In some cases it will stop the whining, sometimes it lessens it, sometimes it has no affect. It's an interim fix until you can determine if the power steering pump needs to be replaced and then locate a replacement.
Yes, change the PS fluid, use a good synthetic ATF.
A PS fluid change every couple of years is a good idea anyway.
The pump in my '92 Saab started whining at about 70K miles; a fresh fill of Mobil 1 ATF quieted it down, and occasional fluid changes have kept it that way. For 110K more miles so far.
Just make sure it's the P/S pump. Mine had a whine when I bought it, the previos owner told me it was the p/s pump. Mine turned out to be an idler pulley.
Sometimes the pumps whine due to air in the system. My old 89 developed a whine that sounded like I had a turbo under the hood. I ended up having to do a vacuum-assisted purge to clear the air from the system. Never heard another peep from the pump for the remaining several years I had the car.