Alternator...

Ok I just added a cpl of pics, I took earlier today to send to another helpful member that I was trying to understand from...
That green wire that I am concerned about that is just laying by the intake, is that supposed to be grounded somewhere ?
The alternator is a Powermaster (I believe at least 100 amps or so) it's seems to be one of those Serpentine pulleys in the front of it, but the car had no other fancy pulley bracket, all in one etc...
Now the fact that at times I've had to have the car jump started ...and it works go good for a few weeks has me in a spin. The garage looked at it and it showed 13.7-13.9 V. and they checked the alternator and the shop said it's working fine. I can't figure this out.
Most importantly, as well remember back in the day, we checked the alternator by removing the battery cable once the car started to run and if it kept running it was ok, if not, you needed an alternator.
Well the alternator was checked and the shop said it's fine....but when the cable is removed, its "nights out in Moparville"
Any help, ideas will truly be appreciated, any correction will sincerely be rejuvenating...to my well being, lol
Regardless, if someone can tell me where that wire (green) goes, I'd love to know and if it will help with my "suspicious" electrical problem.
Sincerely and still...
MAPS