My issue is very similar to predundant's, except my Taurus is a 2001 instead of a 2003, so I don't think my alternator is controlled by the computer... Also, the previous owner installed a new battery and alternator in May of 2013, presumably to fix the same or similar problem. Therefore, I think the damage to the battery and/or alternator may be a symptom rather than the cause...
I bought my Taurus in January 2014 and it ran fine until this week. It was raining, so I had my headlights, windshield wipers, air conditioning, and CD player going as I headed down the expressway. The first thing I noticed was my CDs started skipping and the volume cut way down. I turned it to the radio and got nothing but static. Next, I noticed the car seemed to surge and fade, as if it was getting inconsistent power. Then the windshield wipers started slowing down and the air conditioning all but shut off. Realizing that I had a problem, I exited the expressway onto surface streets and turned off the A/C
and radio. Whenever I pressed on the brakes, the ABS
light came on. A few seconds later, the air bag light came on, then the battery light. The tach needle went to zero and my gas gauge fluctuated from empty to half a tank. I managed to make it home and pulled into my parking space and shut down the engine. I immediately tried to restart the engine and got nothing but the clicks of a dead battery. The next day i walked up to my local AutoZone and bought a 750 amp jump starter. After letting it charge my battery for about 10 minutes, the engine started right up. It ran fine for about 2 days and the same thing happened. Again, I left the jump starter connected to my battery for about 15 minutes and it started right up and has been fine for the last 2-3 days (but I don't drive very much, i work from a home office)
I could replace the battery and/or the alternator -- just as the car's previous owner had done in May of 2013 -- but I think that's just dealing with the symptom rather than the cause. The Check Engine light hasn't come on or indicted any problems, so my mechanic can't get any code to help.