I have the same car 2001S - tiptronic with 60k miles. I had a similar sound on start up. I concure with harrycb, take off the serpentine belt and start the motor. That's what I did, and the whining stopped. I turned off the engine and turned the alternater, power steering pump, water pump, A/C, and pulleys by hand. Everything felt fine, no funny noises. I put the belt back on and started the engine. Then I used the old mechnic's trick of using a large screwdriver as a stethescope. Put the tip of the screwdriver on the side of the alternator, power steering pump, A/C, and water pump, and put your ear on the handle of the screwdriver. BE CAREFUL!! Don't wear anything loose that could get caught in the serpentine belt. The water pump was making a random ticking sound that I could not hear without using the screwdriver/stethescope. I replaced the waterpump and put in the L/N low temp thermostat. (I recommend taking two days to replace the water pump. The first day to remove it. By the time you get it out, you will be cranky. Trust me. I followed both the Bently manual and 101 projects. I used radiator hose pinch off clamps so I didn't have to drain the cooling system. Once the bolts that hold the water pump are loosened up, coolant starts running out of the block, if you are under the car, there will be coolant running down your arms all the way to your arm pits. Your hands will be sticky and oddly the handles on wrenches will be slick. There is not a lot of room to work. By the time I got it out, my language was not fit for civilized people. The next day, I put it all together, added coolant, and purged the cooling system. ) There were no visible signs the water pump was going. After I removed the water pump, I held it up and looked behind water pump pulley and I could see where collant had been weeping out at the pump shaft. Impossible to see this with the water pump in situ. The whining noise, from the motor, is gone.
