Whine after start-up could be the aux. AIR pump, previous posts discuss. To find out, open the hood, start the engine, then get out and look in the left front corner of the engine compartment, ahead of the left front wheel, that's where the pump is and you should hear it running.
If the whine became noticable because the AIR pump got louder, it could be the pump is going out or the filter is clogged.
It is also possible what you hear is power steering. Again, before a cold start, open the hood, start the engine then go listen to see if it's the steering pump. If you hear the pump whining with the engine at idle and no load on the power steering, check the fluid level. It might be low. If adding fluid or changing the fluid does not eliminate the noise, then you are probably looking at power steering pump replacement. Don't put it off, because if the pump is failing, metal debris may be getting into the fluid and that may, also, damage the steering gear. A steerting rack will cost a lot more than a pump.