It sounds like a "ground loop" or ciculating ground type of problem, where ground locations are at different potentials and you have current flowing through them. Check the voltage between the alternator terminal and somewhere on the engine with the engine running. It would be nice to see how much current flows through the alternator at idle, but most people don't have an ammeter for such. Alternatively, it could be a bad connection somewhere. Check the power wiring for frayed, oxidized wire, or loose connections.

So, have you performed any electrical work to this car that may be causing this? Any aftermarket accessories, even a security system? How quickly will the battery die after last using the car?